LIBRARY .UNIVERSITY OF CALIFORNIA DAVIS \pqrs (2), since Or is equal to unity by construction. The contribution of the mesh PQRS to the total normal induction is equal to area FQR8 x -y^^ x cos Q area Ruvw , • / . x = e X jy^^ — by equation (1) = e X area pqrs by equation (2). Thus the contribution of the mesh to the total normal induction is equal to e times the area cut off a sphere of unit radius with its centre at by a cone having the mesh for a base and its vertex at 0. By dividing up any finite portion of the surface into meshes and taking the sum of the contributions of each mesh, we see that the total normal induction over the surface is equal to e times the area cut off a sphere of unit radius with its centre at by a cone having the boundary of the surface as base and its vertex at 0. Let us now apply the results we have obtained to the case of a closed surface. First take the case when is inside the surface. The total normal induction over the surface is equal to e times the sum of the areas cut off the unit sphere by cones with their bases on the meshes and their vertices at 0, and since the meshes completely fill up the closed surface the sum of the areas cut off the unit sphere by the cones will be the area of the sphere, which is equal to T. E. 2 18 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I 47r, since its radius is unity. Thus the total normal in- duction over the closed surface is 47re. Next consider the case when is outside the closed surface. Draw a cone with its vertex at cutting the closed surface in the areas PQRS, P'Q'RS'. Then the magni- FiG. 5. tude of the whole normal induction over the area FQRS is equal to that over the area P'Q'R'S', since they are each equal to e times the area cut off by this cone from a sphere whose radius is unity and centre at 0. But over the surface PQRS the electric intensity points along the outward drawn normal so that the sign of the component resolved along the outward drawn normal is positive ; while over the surface P'Q'R'S' the electric intensity is in the direction of the inward drawn normal so that the sign of its component along the outward drawn normal is negative. Thus the total normal attraction over PQRS is of opposite sign to that over P'Q'R'S', and since they are equal in magnitude they will annul each other as far as the total normal induction is concerned. Since the whole of the closed surface can be divided up in this way by cones with their vertices at 0, and since the two sections of each of these cones neutralize each other, the total normal induction over the closed surface will be zero in this case. lOJ GENERAL PRINCIPLES OF ELECTROSTATICS. 19 We thus see that when the electric field is due to a small body with a charge e the total normal induction over any closed surface enclosing the charge is 47re, while it is equal to zero over any surface not enclosing the charge. We have therefore proved Gauss's theorem when the field is due to a single small electrified body. We can easily extend it to the general case when the field is due to any distribution of electrification. For we may regard this as arising from a number of small bodies having charges ^j, e.., ^3 ... &c. Let N be the component along the outward drawn normal to the surface of the resultant electric intensity, N^ the component in the direction due to ei, N^ that due to 63 and so on ; then If ft) is the area of the mesh at which the normal electric intensity is iV, the total normal induction over the surface is SiV^o) that is, the total normal electric induction over the surface due to the field is equal to the sum of the normal in- ductions due to the small charged bodies of which the field is supposed to be built up. But we have just seen that the normal induction over a closed surface due to any one of these is equal to 47r times its charge if the body is inside the surface, and is zero if the body is out- side the surface. Hence the sum of the normal induc- tions due to the several charged bodies, i.e. that due to the actual field, is 47r times the charge of electricity inside the closed surface over which the normal intensity is taken. 2—2 20 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I 11. Electric intensity at a point outside a uniformly charged sphere.— Let us now apply the theorem to find the electric intensity at any point in the region outside a sphere uniformly charged with elec- tricity. Let be the centre of the sphere, P sl point outside the sphere at which the electric intensity is required. Through P draw a spherical surface with its centre at 0. Let R be the electric intensity at P. Since the charged sphere is uniformly electrified the direction of the intensity will be OP, and it will have the same value R at any point on the spherical surface through P with its centre at 0. Hence since at each point on this surface the normal electric intensity is equal to R ; the total normal induction over the sphere through P is equal to 72 X (surface of the sphere), i.e. Rx^tir. OP^. By Gauss's theorem this is equal to 47r times the charge enclosed by the spherical surface, that is to 47r times the charge on the inner sphere. If e is this charge we have therefore i2x47rOP'- = 47re, Hence the intensity at a point outside a uniformly electrified sphere is the same as if the charge on the sphere were concentrated at the centre. 12. Electric intensity at a point inside a uni- formly electrified spherical shell. — Let Q be a point inside the shell, R the electric intensity at that point. Through Q draw a spherical surface, centre ; then as before, the normal electric intensity will be constant all 13] GENERAL PRINCIPLES OF ELECTROSTATICS. 21 over this surface. The total normal induction over this sphere is therefore R x area of sphere, i.e. E X 47r . 0Q\ By Gauss's theorem this is equal to 47r times the charge of electricity inside the spherical surface passing through Q, hence as there is no charge inside this surface, 4>7rR X OQ' = 0, or R = 0. Hence the electric intensity at any point inside a uni- formly electrified spherical shell vanishes. 13. Infinite Cylinder uniformly electrified. — We shall next consider the case of an infinitely long circular cylinder uniformly electrified. Let P be a point outside the cylinder at which we wish to find the electric intensity. Through P describe a circular cylinder coaxial with the electrified one, draw two planes at right angles to the axis of the cylinder at unit distance apart, and consider the total normal induction over the closed surface formed by the curved surface of the cylinder through P and the two plane ends. Since the electrified cylinder is infinitely long aud is symmetrical about its axis, the electric intensity at all points at the same distance from the axis of the cylinder will be the same, and will by symmetry be along a radius drawn through P at right angles to the axis of the cylinder. Thus the electric intensity at any point on the plane ends of the cylinder will be in the plane of these ends, and will therefore have no component at right angles to them, the plane ends will therefore contribute nothing 22 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I to the total normal induction over the surface ; at each point of the cylindrical surface the electric intensity is at right angles to the curved surface and equal to R. The total normal induction over the surface is therefore R X (area of the curved surface of the cylinder). But since the length of the curved surface is unity its area is equal to 27rr where ?' is the distance of P from the axis of the cylinder. If E is the charge per unit length on the electrified cylinder then by Gauss's theorem the total normal induction over the surface is equal to 4<7rE. The total normal induction is however equal to Rx 27rr, hence R X 27rr = ^ttE, r Thus in the case of the cylinder the electric intensity varies inversely as the distance from the axis of the cylinder. We can prove in the same way as for the uniformly electrified spherical shell that the electric intensity at any point inside a uniforaily electrified cylindrical shell vanishes. 14. Uniformly electrified infinite plane. — In this case we see by symmetry (1) that the electric intensity will be normal to the plane, (2) that the electric intensity will be constant at all points in a plane parallel to the electrified one. Draw a cylinder PQR8, Fig. 6, the axis of the cylinder being at right angles to the plane, the ends of the cylinder being planes at right angles to 14] GENERAL PRINCIPLES OF ELECTROSTATICS. 23 the axis. Since this cylinder encloses no electrification the total normal induction over its surface is zero by Gauss's 'D' Fig. 6. theorem. But since the electric intensity is parallel to the axis of the cylinder the normal intensity vanishes over the curved surface of the cylinder. Let F be the electric intensity at a point on the face PQ, this is along the outward-drawn normal if the electrification on the plane is positive, F' the electric intensity at a point on the face R8, co the area of either of the faces PQ or BS, then the total normal induction over the surface PQRS is equal to Fco-F'(o; and since this vanishes by Gauss's theorem F = F\ or the electric intensity at any point, due to the infinite uniformly charged plane, is independent of the distance of the point from the plane. It is, therefore, constant in magnitude at all points in the field, acting upwards in the region above the plane, downwards in the region below it. To find the magnitude of the intensity at P. Draw through P (Fig. 7) a line at right angles to the plane and prolong it to Q, so that Q is as much below the plane as P is above it. With PQ as axis describe a right circular cylinder bounded by planes through P and Q parallel to the electrified plane. Consider now the total normal induction over the surface of this cylinder. The electric 24 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I intensity is everywhere parallel to the axis of the cylinder, and has, therefore, no normal component over the curved p R Q Fig. 7. surface of the cylinder, the total normal intensity over the surface thus arises entirely from the flat ends. Let R be the magnitude of the electric intensity at any point in the field, « the area of either of the flat ends of the cylindrical surface. Then the part of the total normal induction over the surfaces PQRS due to the flat end through P is Rco. The part due to the flat end through Q will also be equal to this and will be of the same sign, since the intensity at Q is along the outward-drawn normal. Thus since the normal intensity vanishes over the curved surface of PQRS the total normal induction over the closed surface is 2Ra). If a is the quantity of electricity per unit area of the plane the charge of elec- tricity inside the closed surface is aco ; hence by Gauss's theorem 2Rco = iiiro-co, or R = 27ro-. By comparing this with the results given in Arts. 11 and 13 the student may easily prove that the intensity due to the charged plane surface is half that just outside a charged spherical or cylindrical surface having the same pharge of electricity per unit area, 16] GENERAL PRINCIPLES OF ELECTROSTATICS. 25 15. Lines of Force. A line of force is a curve drawn in the electric field, and such that its tangent at any point is parallel to the electric intensity at that point. 16. Electric Potential. This is defined as follows : The electric potential at a point P exceeds that at Q by the work done by the electric field on a body charged with unit of electricity when the latter passes from P to Q. The path by which the unit of electricity travels from P to Q is immaterial, as the work done will be the same whatever the nature of the path. To prove this suppose that the work done on the unit when it travels along the path PAQ is greater than when it travels along the path PBQ. Since Fig. 8. the work done on the unit of electricity when it goes from P to Q along the path PBQ is equal to the work which must be done to bring the unit from Q to P along QBP, we see that if we make the unit travel round the closed curve PAQBP the work done on the unit when it travels along PAQ is greater than the work spent in bringing it back from Q to P along the path QBP. Thus though the unit of electricity is back at the point from which it started, and if the field is entirely due to charges of electricity, everything is the same as when we started, we 26 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I have, if our hypothesis is correct, gained work. This is not in accordance with the principle of the Conservation of Energy, and we therefore conclude that the hypothesis on which it is founded, i.e. that the work done on unit electric charge when it travels from P to Q depends on the path by which it travels is incorrect. Since electric phenomena only depend upon differences of potential it is immaterial what point we take as the one at which we call the potential zero. In mathematical investigations it simplifies the expression for the potential to assume as the point of zero potential one at an infinite distance from all the electrified bodies. If P and Q are two points so near together that the electric intensity may be regarded as constant over the distance PQ then the work done on unit charge when it travels from P to Q is P x PQ, if F is the electric intensity resolved in the direction PQ. If Vp, Vq denote the potentials at P and Q respectively, then since by definition Vp — Vq is the work done on unit charge when it goes from P to Q we have Vp-Vq^FxPQ, hence ^=^T^ ^^^' thus the electric intensity in any direction is equal to the rate of diminution of the potential in that direction. Hence if we draw a surface such that the potential is constant over the surface (a surface of this kind is called an equipotential surface) the electric intensity at any point on the surface must be along the normal. For since the potential does not vary as we move along the surface, 17] GENERAL PRINCIPLES OF ELECTROSTATICS. 27 we see by equation 1 that the component of the electric intensity tangential to the surface vanishes. Conversely a surface over which the tangential com- ponent of the intensity is everywhere zero will be an equipotential surface, for since there is no tangential in- tensity no work is done when the unit charge moves along the surface from one point to another ; that is, there is no difference of potential between points on the surface. The surface of a conductor placed in an electric field must be an equipotential surface when the field is in equilibrium, for there can be no tangential electric in- tensity, otherwise the electricity on the surface would move along the surface and there could not be equili- brium. It is this fact that makes the conception of the potential so important in electrostatics, for the surfaces of all bodies made of metal are equipotential surfaces. 17. Potential due to a uniformly charged sphere. The potential at P is the work done by the electric field when unit charge is taken from P to an infinite distance. Let us suppose that the unit charge travels from P to an infinite distance along a straight line passing through the centre of the sphere. Let QRSThe a series of points t-tt Fig. 9. very near together along this line. If e is the charge on the sphere, its centre, the electric intensity at Q is e/OQ", while that at R is e/OR^; as Q and R are very near together these quantities are very nearly equal, and we may take 28 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I the average electric intensity between Q and R as equal to e/OQ . OR, the geometric mean of the intensities at Q and R. Hence the work done by unit charge as it goes from Q to jK is equal to QR OQ.OR OQ OR ' Similarly the work done by the charge as it goes from i^ to >Sf is _e e OR OS' as it goes from S to T e e OS~dT' and so on. The work done by the charge as it goes from Q to T is the sum of these expressions, and this sum is equal to e e OQ~OT' and we see by dividing up the distance between the points into a number of small intervals and repeating the above process that this expression will be true when Q and T are a finite distance apart : and that it always represents the work done by unit charge as long as Q and T are two points on a radius of the sphere. The potential at P is 18] GENERAL PRINCIPLES OF ELECTROSTATICS. 29 the work done when the unit charge goes from P to an infinite distance, and is therefore by the preceding result equal to ^ . This is also evidently the potential at P of a charge e placed at if the dimensions of the body over which the charge is spread are infinitesimal in comparison with OP. 18. The electric intensity at any point inside a closed equipotential surface which does not en- close any electric charge vanishes. We shall prove that the potential is constant throughout the volume enclosed by the surface, then it will follow by equation (1), Art. 16, that the electric intensity vanishes through- out this volume. For if the potential is not constant it will be possible to draw a series of equipotential surfaces inside the given one ; let us consider the equipotential surface for which the potential is very nearly, but not quite, the same as for the given surface ; as the difference of potential between this and the outer surface is very small the two surfaces will be close together, and they cannot cut each other, for if they did any point in their intersection would have two different potentials. Suppose for a moment that the potential at the inner surface is greater than that at the outer. Let P be a point on the inner surface, Q the point where the outward drawn normal at P to the inner surftxce cuts the outer surface. Then since the electric intensity from P to Q is equal to {Vp — Vq)JPQ we see that since by hypothesis Vp — Vq is positive, the normal 30 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I electric intensity over the second surface is everywhere in the direction of the outward -drawn normal to the surface, and therefore the total normal electric induction over the surface will be positive, hence there must be a positive charge inside the surface, as the total normal induction over the surface is by Gauss's theorem proportional to the charge enclosed by the surface. Hence, as by hypothesis there is no charge inside the surface, we see that the potential over the inner surface cannot be greater than that at the outer surface. If the potential at the inner surface were less than that at the outer then the normal electric intensity would be everywhere in the opposite direction, and we can show by Gauss's theorem as before that this would require a negative charge inside the surface. Hence as there is no charge either positive or negative the potential at the inner surface can neither be greater nor less than at the outer surface, and must therefore be equal to it. In this way we see that the potential inside the surface must have the same value as at the surface, and since the potential is constant the electric intensity will vanish inside the surface. 19. It follows from this that if we have a hollow, conducting surface there will be no electrification on its inner closed surface unless there are electrified bodies inside the hollow. Let Fig. 10 represent the conductor with a cavity inside it. To prove that there is no elec- trification at P a point on the surface, take any closed surface enclosing a small portion a of the inner surface near P; by Gauss's theorem the charge on a is pro- portional to the total normal electric induction over the surface surrounding a. The electric intensity is however 20] GENERAL PRINCIPLES OF ELECTROSTATICS. 31 zero everywhere over this surface. It is zero over the part of the surface outside the cavity because this part of the Fig. 10. surface is in a conductor, and when there is equilibrium the electric intensity is zero at any point in a conductor : the electric intensity is zero inside the cavity because the inner surface being the surface of a conductor is an equi- potential surface, and as we have just seen the electric intensity inside such a surface is zero unless the surface encloses electric charges. Thus since the electric in- tensity vanishes at each point on the surface surrounding a, the charge at a must vanish ; in this way we can see that there is no electrification at any point on the inner cavity. The electrification is all on the outer surface of the conductor. 20. The result proved in Art. 18 that when the force between two charged bodies varies inversely as the square of the distance between them the electric intensity vanishes throughout the interior of an electrified con- ductor leads to the most rigorous proof of the truth of this law by experiment. 32 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I Let us for simplicity confine our attention to the case when the electrified conductor is a sphere positively electrified. Fig. 11. Consider the state of things at a point P inside a sphere whose centre is 0, Fig. 11 : through P draw a plane at right angles to OP. The electrification on the portion of the sphere above this plane produces an electric intensity in the direction PO, while the electrification on the portion of the sphere below the plane produces an electric intensity in the direction OP. When the law of force is the inverse square these two intensities balance each other, the greater distance from P of the electri- fication below the plane being compensated by the larger electrified area. Now suppose that the law of force varies as r~P, then if p is greater than 2 the force diminishes more quickly as the distance increases than when the law of force is the inverse square, so that if the larger area below the plane was just sufficient to -compensate for the greater distance when the law of force was the inverse square it will not be sufficient to do so when p is greater than 2, thus the electrification on the portion of the sphere above the plane will gain the upper hand and the resultant electric intensity will be in the direction PO. Again, if p is less 20] GENERAL PRINCIPLES OF ELECTROSTATICS. 33 than 2 the force will not diminish so rapidly when the distance increases as when the law is the inverse square, so that if the greater area below the plane is sufficient to compensate for the increased distance when the law of force is the inverse square it will be more than sufficient to do so when p is less than 2 ; in this case the electri- fication below the plane will gain the upper hand, and the electric intensity at P will be in the direction OP, Now suppose we have two concentric metal spheres connected by a wire, and that we electrify the outer sphere positively, then if ^ = 2 there will be no electric intensity inside the outer sphere, and therefore no movement of electricity on the inner sphere which will remain un- electrified. If p is greater than 2 we have seen that the electric intensity due to the positive charge on the outer sphere will be towards the centre of the sphere, i.e. the force on a negative charge will be from the inner sphere towards the outer. Negative electricity will therefore flow from the inner sphere, which will be left with a positive charge. Next suppose that p is less than 2, the electric in- tensity due to the charge on the outer sphere will be from the centre of the sphere, the direction of the force acting on a positive charge will therefore be from the inner sphere to the outer, positive electricity will therefore flow from the inner sphere to the outer, so that the inner sphere will be left with a negative charge. Thus according as p is greater than, equal to or less than 2 the charge on the inner sphere will be positive, zero or negative. By testing the state of electrification on the inner sphere we can therefore test the law of force. This is what was done by Cavendish in an experiment T. E. 3 34 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I made by him, and which goes by his name. The following is a description of a slight modification due to Maxwell of Cavendish's original experiment. The apparatus for the experiment is represented in Fig. 12. Fig. 12. The outer sphere A, made up of two tightly fitting hemispheres, is fixed on an insulating stand, and the inner sphere fixed concentrically with the outer one by means of an ebonite ring. Connection between the inner and outer spheres is made by a wire fastened to a small metal disc B which acts as a lid to a small hole in the outer sphere. When the wire and the disc are lifted up by a silk string the electrical condition of the inner sphere can be tested by pushing an insulated wire con- nected to an electroscope (or preferably to a quadrant electrometer, see Art. 60) through the hole until it is in contact with the inner sphere. The experiment is made 20] GENERAL PRINCIPLES OF ELECTROSTATICS. 35 as follows : when the two spheres are in connection a charge of electricity is communicated to the outer sphere, the connection between the spheres is then broken by lifting the disc by means of the silk thread; the outer sphere is then discharged and kept connected to earth ; the testing wire is then introduced through the hole and put into contact with the inner sphere. Not the slightest effect on the electroscope can be detected, showing that if there is any charge on the inner sphere it is too small to affect the electroscope, v To determine the sensitiveness of the electroscope or electrometer, a small brass ball sus- pended by a silk thread is placed at a considerable distance from the two spheres. After the outer sphere is charged (suppose positively) the brass ball is touched and then left insulated ; in this way the ball gets by induction a negative charge amounting to a calculable fraction, say a, of the original charge communicated to the outer sphere. Now when the outer sphere is connected to earth this negative charge on the ball will induce a positive charge on the outer sphere which is a calculable fraction, say y8, of the charge on the ball. If we disconnect the outer sphere from the earth and discharge the ball this positive charge on the outer sphere will be free to go to the electroscope if this is connected to the sphere. When the ball is not too far away from the sphere this charge is sufficient to deflect the electroscope, i.e. a fraction a/3 of the original charge on the sphere is sufficient to deflect the electro- scope, showing that the charge on the inner sphere in the Cavendish experiment could not have amounted to a/8 of the charge communicated to the outer sphere. If the force between two charges is assumed to vary as r~P, we can calculate the charge on the inner sphere and express 3—2 36 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I it in terms of p, hence knowing from the Cavendish experiment that this charge is less than ayS of the original charge we can calculate that p must differ from 2 by less than a certain quantity. In this way it has been shown that^ differs from 2 by less than 1/20,000. 21. Definition of surface density. When the electrification is on the surface of a body, the charge per unit area is, when the electrification is uniform over the surface, called the surface density of the electricity ; when the electrification is variable the surface density at any point is the limiting value of the ratio of the charge on a small area o) of the surface surrounding the point to &>, when ft) is made indefinitely small. 22. Coulomb's Law. The electric intensity R at a point P close to the surface of a conductor surrounded by air is at right angles to the surface and is equal to 47rcr where a is the surface density of the electrification. The first part of this law follows from Art. 16, since the surface of a conductor is an equipotential surface. Fig. 13. To prove the second part take a small area at P (Fig. 13) and through the boundary of this area draw the cylinder 23] GENERAL PRINCIPLES OF ELECTROSTATICS. 37 whose generating lines are parallel to the normal at P. Let this cylinder be truncated at T and 8 by planes parallel to the tangent plane at P. The total normal electric induction over this cylinder is R(o, where R is the normal electric intensity and w the area of the cross section. For Rio is the part of the total intensity due to the end T of the cylinder, and this is the only part of the surface of the cylinder which contributes anything to the total normal induction. For the intensity along that part of the curved surface of the cylinder which is in air is tangential to the surface and therefore has no component along the normal, while since the electric intensity vanishes inside the conductor the part of the surface which is inside the conductor will not con- tribute anything to the total induction. If a is the surface density of the electrification at P the charge inside the cylinder is two- ; hence by Gauss's theorem R(o = ^ircoa or R — ^TTo: The result expressed by this equation is known as Coulomb's Law. It requires modification when the con- ductor is not surrounded by air, but by some other in- sulator. See Art. 71. 23. Energy in the electric field. Let us take the case of a number of conductors placed in an electric field ; let E^ be the charge on the first conductor, Vi its poten- tial, E.2 the charge on the second conductor, V^ its poten- tial, and so on, we shall show that the potential energy of this system of conductors is equal to To prove this we notice that the potentials of the 88 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I conductors will depend upon the charges of electricity on the conductors, in such a way that if the charge on every part of the system is increased m times, the potential at every point in the system will also be increased m times. To find the energy in the system of conductors we shall suppose that they are originally uncharged, and at potential zero, and that a charge J^i/n is brought from an infinite distance to the first conductor, a charge E^jn is brought from an infinite distance to the second con- ductor, a charge -£^3/71 to the third conductor, and so on. After this has been done, the potential of the first con- ductor will be Vxjn, that of the second V^jn, and so ou. Let us call this the first stage of the operation. Then bring from an infinite distance charges E^jn to the first conductor, E^jn to the second, and so on. When this has been done the potentials of the conductors will be 2 Fi/?i, 2 V^jn, Call this the second stage of the operation. Keep repeating this process until the first con- ductor has the charge E^ and the potential F,, the second conductor the charge E^ and the potential Fg Then in the first stage the potential of the first con- ductor is zero at the beginning, and Vyjn at the end ; the work done in bringing up to it the charge E^jn is therefore E V . . gi'eater than but less than — . ^ ; similarly the work spent in bringing up the charge E.iln to the second con- E V doctor is efreater than zero but less than — . — . ° n n Therefore Qi the work spent in the first stage of the operations is >o<^,{E,r,+E,r,+E,v^+...]. Ill 23] GENERAL PRINCIPLES OF ELECTROSTATICS. 39 In the second stage of the operations the potential of the first conductor is VJn at the beginning, 2Fi/n at the end, so that the work spent in bringing up the charge EJn E V to the first conductor is greater than — . — , leas than 71 n W 2T^ — . — ^ ; similarly the work spent in bringing up the charge E^/n to the second conductor is greater than — . — , E '2V less than — . — ^. Thus Qa the work spent in this n n ^ stage is >\{E,V,-VE,V^ + E,V,+ .,.) Similarly Q^ the work spent in the third stage is >-,{E,V, + E,V,+ E,V,+ ...) 11/ <-^(E,V, + E,V,+ E,V,+ ...) and Qn the work spent in the last stage is <-,{E,V, + E,V, + E,V^+...). Thus Q the total amount of work spent in charging the conductors is equal to Qi + Q2 + . . . Q«, and is therefore <^-±^±^±^^iE,V, + E,V, + E,V, + ...). 40 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I (n-l)n n.(n + l) '•^' ^ 2/1^ (A,K,+ ...)< 2^^, (^.F,4-...) or >|(l-i)(^.7. + ...)32^3, Pa&E^. Hence by Art. 25 we see that when the first con- ductor has the charge ^1, the second the charge E^, the 27] GENERAL PRINCIPLES OF ELECTROSTATICS. 43 third the charge E^, and so on, Ti the potential of the first conductor will be given by the equation F2 the potential of the second conductor by the equation if F, is the potential of the third conductor F3 = pi3^i +p^E^ +Ps3^3 + •.. , If we solve these equations we get ^1=^11 ^^1 + ^21^2+^31^3+..., ^2 = ^12^14-^22^2+^32^3+..., where the qs are functions of the ps and only depend upon the configuration of the system of conductors. The g's are called coefficients of capacity when the two suffixes are the same, coefficients of induction when they are different. 27. We shall now show that the coefficients which occur in these equations are not all independent, but that i?2i=i?i2. To prove this let us suppose that only the first and second conductors have any charges, the others being with- out charge and insulated. Then we may imagine the system charged, by first bringing up the charge E^ from an infinite distance to the first conductor and leaving all the other conductors uncharged, and then when this has been done bringing up the charge E^ from an infinite distance to the second conductor. The work done in bringing the charge E^ up to the first conductor will be 44 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I the energy of the system when the first conductor has the charge E^ and the other conductors are without charge, the potential of the first conductor is in this case pnE^ , so that by Art. 23 the work done is ^E^.p^E^ or ^^n^i^. To find the work done in bringing up the charge E^ to the second conductor let us suppose that this charge is brought up in equal instalments each equal to E^ln. Then the potential of the second conductor before the first instalment is brought up is by Art. 26 equal to p^^Ej, after the first instalment has arrived it is p^E. -\-p<>>— . ^ n Hence the work done in bringing up the first instalment will be between ;>„J5:,f and (p,^_E,+p^f]^\ n v^ ^ n J n Similarly the work done in bringing up the second in- stalment E2/H will be between „ Eo\ Eq 1 / r, ^Eo\ Eo p,,E, +P.-)- and [p,,E, + p^ -j - , and the work done in bringing up the last instalment of the charge will be between (i>12^1+JP22 ^ ' ^— and (pi.E. + p,^— '^ thus the total amount of work done in bringing up the charge E^ will be between „^ 1 + 2 + 3 + ?i-l p,,E,E, + -^ p^E,' _, ^ 1 + 2 + 3+n ^2 and Pi2^i^2 + -^ i>22^2 , 28] GENERAL PRINCIPLES OF ELECTROSTATICS. 45 that is, between but if 11 is very great these two expressions become equal to p^J^^E^ + \p,^^E^, which is the work done in bringing the charge E^ to the second conductor when the first conductor has already received the charge E^ \ hence the work done in bringing up first the charge E^ and then ^,is hPnE:+P.,E,E,-v\p,^E:, It follows in the same way that the work done when the charge E.^, is first brought to the second conductor and then the charge E^ to the first is hP,.E,' + P.,EA+ip,A'> but since the final result is the same in the two cases, the work required to charge them must be the same ; hence iP^^^:' +P.AE, + hPnE.' = \pJ^'.^P.JE,E, + l^„^;^ i.e. P2i=Pvi- It follows from the way in which the g's can be ex- pressed in terms of the ^'s, that q.zi = qv2- 28. Now py2 is the potential of the second conductor when unit charge is given to the first, the other con- ductors being insulated and without charge, and p^^i is the potential of the first conductor when unit charge is given to the second. But we have just seen that ^21 =i?i2, hence the potential of the second conductor when insulated and without charge due to unit charge on the first is equal to the potential of the first when insulated and without charge due to unit charge on the second. 46 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I 29. Let us consider some examples of this theorem. Let us suppose that the first conductor is a sphere with its centre at 0, and that the second conductor is very small and placed at P, then if P is outside the sphere we know by Art. 17 that if unit charge is given to the sphere the potential at P is increased by 1/OP. It follows from the preceding article that if unit charge be placed at P the potential of the sphere when insulated is increased by 1/OP. If P is inside the sphere then when unit charge is given to the sphere the potential at P is increased by Ija where a is the radius of the sphere. Hence if the sphere is insulated and a unit charge placed at P the potential of the sphere is increased by 1/a. Thus the increase in the potential of the sphere is independent of the position of P as long as it is inside the sphere. Since the potential inside any closed conductor which does not include any charged bodies is constant, by Art. 18, we see by taking as our first conductor a closed surface, and as our second conductor a small body placed at a point P anywhere inside this surface, that since the potential at P due to unit charge on the conductor is independent of the position of P, the potential of the conductor when insulated due to a charge at P is inde- pendent of the position of P. Thus however a charged body is moved about inside a closed insulated conductor the potential of the conductor will remain constant. An example of this is afforded by the experiment described in Art. 5 ; the deflection of the electroscope is independent of the position of the charged bodies inside the insulated closed conductor. 30] GENERAL PRINCIPLES OF ELECTROSTATICS. 47 30. Again, take the case when the first conductor is charged, the others insulated and uncharged ; then so that ^"=^- Now suppose that the first conductor is connected to earth while a charge B,^ is given to the second conductor, all the other conductors being uncharged ; then since Fj = we have 0=pi,E,+p,^E^, El pi2 F2 by the preceding equation. Hence if a charge be given to the first conductor, all the others being insulated, the ratio of the potential of the second conductor to that of the first will be equal in magnitude but opposite in sign to the charge induced on the first conductor, when connected to earth, by unit charge on the second conductor. As an example of this, suppose that the first conductor is a sphere with centre at 0, and that the second conductor is a small body at a point P outside the sphere ; then if unit charge be given to the sphere, the potential of the body at P is ajOP times the potential of the sphere, where a is the radius of the sphere; hence by the theorem of this article when unit charge is placed at P, and the sphere connected to the earth, there will be a negative charge on the sphere equal to a/ OP. Another example of this result is when the first 48 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I conductor completely surrounds the second ; then since the potential inside the first conductor is constant when all the conductors inside are free from charge, the potential of the second conductor when a charge is given to the first conductor will be the same as that of the first. Hence from the above result it follows that when the first con- ductor is connected to earth, and a charge given to the second, the charge induced on the first conductor will be equal and opposite to that given to the second. Another consequence of this result is that if >S^ be an equipotential surface when the first conductor is charged, all the others being insulated, then if the first couductor be connected to earth the charge induced on it by unit charge on a small body P remains the same however P may be moved about, provided that P always keeps on the surface S. 31. As an example in the calculation of coefficients of capacity and induction, we shall take the case when the conductors are two concentric spherical shells. Let a be the radius of the inner shell, which we shall call the first conductor, h the radius of the outer shell, which we shall call the second conductor. Let E^, E.^ be the charges of electricity on the inner and outer shells re- spectively, Fi, Fa the corresponding potentials of these shells. Then if there were no charge on the outer shell the charge E^ on the inner would produce a potential E^ja on its own surface, and a potential E^jh on the surface of the outer shell ; hence, Art. 26, 1 1 32] GENERAL PRINCIPLES OF ELECTROSTATICS. 49 The charge Er, on the outer shell would, if there were no charge on the inner shell, make the potential inside the outer shell constant and equal to the potential at the surface of the outer shell. This potential is equal to E^jh, so that the potential of the first conductor due to the charge E^ on the second is E^fh, which is also equal to the potential of the second conductor due to the charge E^ ; hence. Art. 26, 1 1 We have therefore Cb Solving these equations, we get ab -^ ah b — a ^ b — a E.= r^F.-r^n, b-a b — a Hence _ ab _ _ ^^ __^ We notice that qi2 is negative ; this, as we shall prove later, is always true whatever the shape and position of the two conductors. 132. Another case we shall consider is that of two spheres the distance between whose centres is very large compared with the radius of either. Let a be the radius of the first sphere, b that of the second, R the distance T. E. 4 50 GENERAL PRINCIPLES OF ELECTROSTATICS. [CH. I between their centres, E^, E.2 the charges, F,, V2 the potentials of the two spheres. Then if there were no charge on the second sphere, the potential at the surface of the first sphere would if the distance between the spheres were very great be approximately EJa, while the potential of the second sphere would be approximately EjIR'j hence 1 1 approximately. Similarly if there were no charge on the first sphere, and a charge E^ on the second, the potential at the first sphere would be E^/R, that at the second E.Jb, approxi- mately ; hence we have approximately _1_ _ 1 Pn-j^, P^^-l' So that approximately ^'~R^J' Solving these equations we get „ aR^ ^^ ahR ^^ ^1 = ^55 — n *^i- R^-ab ' R?-ab ^ obR ,^ bR' R'^-ab ' ' R-'-ab hence aR^ _ _ _ ahR _ bR^ 33] GENERAL PRINCIPLES OF ELECTROSTATICS. 51 We see that as before q^^ is negative. We also notice that ^11 and q^^ get larger the nearer the spheres get to- gether. 33. Electric Screens. As an example of the use of coefficients of capacity we shall consider the case of three conductors, A, B, C, and shall suppose that the first of these conductors A is as in the Fig. 14 inside the Fio. 14. third conductor C, which is supposed to be a closed surface, while the second conductor B is outside G. Then if El, Vi\ E' the area at B. Let F be the electric intensity and co the area enclosed by the tubes at A. Then applying Gauss's theorem (Art. 10) to the tubular surface formed by the prolongations backwards of the tubes through A we get Fay - Ray' = 0. 5—2 68 LINES OF FORCE. [CH. II But if a is the surface density of the electrification at jB, we have when the medium surrounding B is air, by Coulomb's law (Art. 22) R = 47ro-, hence Foa = ^iraw , but o-ft)' is the charge of electricity on B, it is therefore equal to N the number of Faraday tubes which start from B, and which pass through A, hence J?'a) = 47ri\r, or if (o is unity Thus the electric intensity in air is 47r times the number of Faraday tubes passing through unit area of a plane drawn at right angles to the electric intensity. 41. The properties of the Faraday tubes enable us to prove with ease many important theorems relating to the electric field. Thus, for example, we see that on the conductor at the highest potential in the field the electrification must be entirely positive ; for any negative electrification would imply that Faraday tubes arrived at the conductor, these tubes must however arrive at a place which is at a lower potential than the place from which they start. Thus, if the potential of the conductor we are considering is the highest in the field it is impossible for a Faraday tube to arrive at it, for this would imply that there was some other conductor at a still higher potential from which the tube could start. Similar reasoning shows that the electrification on the conductor or conductors at the lowest potential in the 42] LINES OF FORCE. 69 field must be entirely negative. Now take the case when one conductor has a positive charge while all the other conductors are connected to earth; we see from the last result that the charges on the uninsulated conductors must be all negative, and since the potentials of these conductors are all equal and the same as that of the earth, no Faraday tubes can pass from one of these conductors to another, or from one of these to the earth. Hence all the tubes which fall on these conductors must have started from the conductor at highest potential. Thus the sum of the number of tubes which fall on the uninsulated con- ductor cannot exceed the number which leave the posi- tively charged conductor, that is, the sum of the negative charges induced on the conductors connected to earth cannot exceed the positive charge on the insulated con- ductor. 42. These results give us important information as to the coefficients of capacity defined in Art. 26. For let us take the first conductor as the insulated one with the positive charge; then since V^, V^... are all zero we have, using the notation of that Article, E, = qnV„ E^=qi2V,y ^3 = ^13^1 Since E^ and Fj are positive, while E^, E^, &c. are all negative, we see that q^ is positive, while q^iy qis are all negative. Again, since the positive charge on the first conductor is numerically not less than the sum of the negative charges on the other conductors, El is numerically not less than ^2 + -^a + • • •> i.e. qu is numerically not less than q^^ + ^13 + 5'i4 + • • •• 70 LINES OF FORCE. [CH. II If one of the conductors, say the second, completely surrounds the first, and if there is no conductor other than the first inside the second, and if all the conductors except the first are at zero potential, then all the tubes which start from the first must fall on the second. Thus the negative charge on the second must be numerically equal to the positive charge on the first (see Art. 30). There can be no charges on any of the other conductors, for all the tubes which fall on these conductors must come from the first conductor, the tubes from this conductor are however completely intercepted by the second surface. Thus if the second conductor incloses the first conductor, and if there are not any other closed conductors between the first and the second, then g„ = — ^j.,, and ^13, ^14, ^15... are all zero. 43. Expression for the Energy in the Field. When we regard the Faraday tubes as the agents by which the phenomena in the electric field are produced we are naturally led to suppose that the energy in the electric field is in that part of the field through which the tubes pass, i.e. in the dielectric between the conductors. We shall now proceed to find how much energy there must be in each unit of volume if we regard the energy as dis- tributed throughout the electric field. We have seen Art. 23 that the electric energy is one half the sum of the products got by multiplying the charge on each conductor by the potential of that conductor. We may regard each unit charge as having associated with it a Faraday tube, which commences at the charge if that is positive and ends there if the charge is negative. Let us now see how the energy in the field can be expressed in terms of these 43] LINES OF FORCE. 71 tubes. Each tube will occur twice in the expression for the electric energy ^S^F, the first time corresponding to the positive charge at its origin, the second time cor- responding to the negative charge at its end. Thus since there is unit charge at each end of the tube the con- tribution of each tube to the expression for the energy- will be J (the difference of potential between its begin- ning and end). The difference of potential between the beginning and end of the tube is equal to XR . PQ ; where PQ is a small portion of the length of the tube, so small that along it R, the electric intensity, may be re- garded as constant: the sign 2) denotes that the tube between A and B, A being a unit of positive and B a unit of negative charge, is to be divided up into small pieces similar to PQ, and that the sum of the products of the length of each piece into the electric intensity along it is to be taken. Thus the whole tube AB contributes to the electric energy JSi2 . PQ, which is equivalent to sup- posing that each unit length of the tube contributes an amount of energy equal to one half the electric intensity. Any finite portion CD of the tube will therefore contribute an amount of energy equal numerically to one half the difference of potential between G and D. We may there- fore regard the energy of the field as due to each of the Faraday tubes having associated with it an amount of energy per unit length numerically equal to one half the electric intensity. Let us now consider the amount of energy per unit volume. Take a small cylinder surrounding any point P in the field with its axis parallel to the electric intensity at P, its ends being at right angles to the axis. Then if R is the electric intensity at P, I the length of the 72 LINES OF FORCE. [CH. II cylinder, the amount of energy due to each tube in the cylinder is ^Rl. If co is the area of the cross section, N the number of tubes passing through unit area, the number of tubes in the cylinder is Nay. Thus the energy in the cylinder is iRlNo), but in air, see Art. 40, 47riV^=E; thus the energy in the cylinder is but lo) is the volume of the cylinder, hence the energy per unit volume is equal to Stt* Thus we may regard the energy as distributed through- out the field in such a way that in each unit of volume there is an amount of energy equal to R^/Sir. 44. If we divide the field up into a series of equi- potential surfaces, the potentials of successive surfaces decreasing in arithmetical progression, and then draw Fig. 23. 45] LINES OF FORCE. 73 another series of cylindrical surfaces cutting these equi- potential surfaces at right angles, such that the number of Faraday tubes passing through the cross section of each of these cylindrical surfaces is the same for all the cylinders, the electric field will be divided up into a number of cells which will all contain the same amount of energy. For the potential difference between the places where a Faraday tube enters and leaves a cell is the same for all the cells ; thus the energy of the portion of each Faraday tube inside a cell will be constant for all the cells, and since there are the same number of Faraday tubes inside each cell, the energy in each cell will be constant. 45. Force on a conductor regarded as arising from the Faraday Tubes being in a state of tension. We have seen, Art. 37, that on each unit of area of a charged conductor there is a pull equal to ^Ba-, where a is the surface density of the electricity, and R the electric intensity. Now o- is equal to the number of Faraday tubes which fall on unit area of the surface, hence the force on the surface will be the same as if each of the tubes exerted a pull equal to ^R. Thus the mechanical forces in the electric field are the same as would be exerted if we supposed the Faraday tubes to be in a state of tension, the tension at any point being equal to one half the electric intensity at that point. Thus the tension at any point of a Faraday tube is numerically equal to the energy per unit length of the tube at that point. If we have a small area co at right angles to the electric intensity, the tension over this area is equal to iNRo), 74 LINES OF FOKCE. [CH. II where N is the number of Faraday tubes passing through unit area, and R is the electric intensity. By Art. 40 47r Hence the tension parallel to the electric intensity is The tension across unit area is therefore equal to Stt ' 46. This state of tension will not however leave the dielectric in equilibrium unless the electric field is uniform, that is unless the tubes are straight. If however there is in addition to this tension along the lines of force a pres- sure acting at right angles to them and equal to R'^/Sir per unit area the dielectric will be in equilibrium, and since this pressure is at right angles to the electric intensity it will not affect the normal force acting on a conductor. To show that this pressure is in equilibrium with the tensions along the Faraday tubes, consider a small volume whose ends are portions of equipotential surfaces and whose sides are lines of force. D Fig. 24. Let US now consider the forces acting on this small volume parallel to the electric intensity at A. The forces are the tensions in the Faraday tubes and the pressures at 46] LINES OF FORCE. 75 right angles to the sides. Resolve these parallel to the outward-drawn normal at A. The number N' of Faraday tubes which pass through A is the same as the number which pass through B. If R, R are the electric in- tensities at A and B respectively, then the tension exerted in the direction of the outward-drawn normal by the Faraday tubes at A will be N'Rj^, while the tension in the opposite direction exerted by the Faraday tubes at B is N'R' cos e/2, where e is the small angle between the direction of the Faraday tubes at A and B. Since e is a very small angle we may replace cos e by unity ; thus the resultant in direction of the outward-drawn normal at A of the tension in the Faraday tubes is N'{R-R)I2. Let N be the number of tubes passing through unit area, w, co' the areas of the ends A and B respectively ; then. Art. 40, -., ^^ R R' , JS = Jy(0= -r— 0) = -. ft), 47r 47r so that the resultant in the direction of the outward- drawn normal at A is 4<7r since R'(o' = Ro) , we may write this as RR / , \ or approximately, since R is very nearly equal to R, 76 LINES OF FORCE. [CH. II Let us now consider the effect of the pressure p at right angles to the lines of force ; this has a component in the direction of the outward-drawn normal at A as in consequence of the curvature of the lines of force the normals to them at all points of the surface are not at right angles to the outward-drawn normal at A ; the angle between the pressure and the normal at A will always however be nearly a right angle. If this angle is -^ — ^ at a point where the pressure is p', the component of the pressure along the normal at A will be proportional to p sin 6. But since p' only differs from p, the value of the pressure at J., by a small quantity, and 6 is small, the com- ponent of the pressure will be equal to p sin 0, if we neglect the squares of small quantities ; that is, the effect along the normal at A of the pressure over the surface will be approximately the same as if that pressure were uniform. To find the effect of the pressure over the sides we re- member that a uniform hydrostatic pressure over any closed surface is in equilibrium ; hence the pressures over the sides G, D will be equal and opposite to the pressures over the ends A and B, but the pressure over these ends is pa)' — po) ; hence the resultant effect in the direction of the outward-drawn normal at A of the pressure over the sides is p (ft) — ft)'). Combining this with the effect due to the tension in the tubes we see that the total force parallel to the outward-drawn normal on the element is g- (ft)'- ft)) -h;) (ft) -ft)'); • 1 .. B? NR this vanishes ii P = 'E~ — ~n~ • 46] LINES OF FORCE. 77 Thus the introduction of this pressure will maintain equi- librium as far as the component parallel to the electric intensity is concerned. Now consider the force at right angles to the electric intensity. Let PQRS, fig. 25, be the section of fig. 24 by the plane of the paper, PS, QR being sections of equi- potential surfaces, and PQ, SR lines of force. Let t be the depth of the figure at right angles to the plane of the paper. We shall assume that the section of the figure by the plane through PQ at right angles to the plane of the paper is a rectangle. Let R be the electric intensity along PQ, R' that along SR, s the length PQ, s' that of SR. Since the difference of potential between P and Q is the same as that between S and R, Rs = R's. Consider the forces parallel to PS. First take the tensions along the Faraday tubes ; those at PS will have no component along PS : in each tube at Q there is a tension i2/2, the component of which along PS is (i2 8in^)/2, where 6 is the angle between PS and QR. Let PS and QR meet in 0, RS PQ PQ-SR s-s' 6 = OR OQ OQ-OR RQ 78 LINES OF FORCE. [CH. II Thus the component of the tensions at Q along PS is 2 • EQ ' The number of tubes which pass through the end of the figure through RQ at right angles to the plane of the paper is N. QR.t, where N is the number of tubes which pass through unit area. The total component along PS due to the tensions in these tubes is thus Now the component along PS due to the pressures at right angles to the electric intensity is equal to pst — p'st, where p and p' are the pressures over PQ, RS respectively. Tf J^' r R'' fR^ R'^ ,\ this is equal to ( «~ * "" q ^ *' ) ^ = — (6'' — s) t, (since Rs = Rs), OTT or approximately, since R' is very nearly equal to Ry = £(.'-.). Thus the component in the direction of PS due to the tensions is equal and opposite to the components due to t he pressures ; thus the two are in equilibrium as far as the 48] LINES OF FORCE. 79 components at right angles to the electric intensity are concerned. But we have already proved that the tensions and pressures balance as far as the component along the direction of the electric intensity is concerned ; thus the system of pressures and tensions constitutes a system in equilibrium. 47. This system of tensions along the tubes of force and pressures at right angles to them is thus in equilibrium at any part of the dielectric where there is no charge, and gives rise to the forces which act on electrified bodies when placed in the electric field. Faraday introduced this method of regarding the forces in the electric field ; he expressed the system of. tensions and pressures which we have just found, by saying that the tubes tended to con- tract and that they repelled each other. This conception enabled him to follow the processes of the electric field without the aid of mathematical analysis. 48. The student will find much light thrown on the effects produced in the electric field by the careful study from this point of view of the diagrams of the tubes of force given in Art. 38. Thus take as an example the diagram given in Fig. 18, which represents the lines of force due to two charges A and B of opposite sign, the ratio of the charges being 4:1. We see from the diagram that though more tubes offeree start from the larger charge A, and the tension in each of these is greater than in a tube near the smaller charge B, the tubes are much more symmetrically distributed round A than round B. The symmetrical distribution of the tubes round A makes the pulls exerted on A by the taut Faraday tubes so nearly counterbalance each other that the resultant pull of these 80 LINES OF FORCE. [CH. II tubes on A is only the same as that exerted on B by the tubes starting from it ; as these, though few in number, are less symmetrically distributed, and so do not tend to counter- balance each other to nearly the same extent. The tubes of force in the neighbourhood of the point of equilibrium are especially interesting. Since the charge on A is four times that on B, only \ of the tubes which start from A can end on B, the remaining | must go off to other bodies, which in the case given in the diagram are supposed to be at any infinite distance. The point of equilibrium corre- sponds as it were to the ' parting of the ways ' between the tubes of force which go from A \^7rt _ surface of the sphere I 53] CONDENSERS. 87 thus the capacity in this case is equal per unit area of surface to l/47r times the distance between the con- ductors. The case of two spheres whose distance apart is very small compared with their radii is however approxi- mately the case of two parallel planes ; hence the capacity of such planes per unit area of surface is equal to l/47r times the distance between the planes. This is proved directly in Art. 56. If after the spheres are charged the inner one is insu- lated, and the outer one removed to an infinite distance (to enable this to be done we may suppose that the outer sphere consists of two hemispheres fitted together, and that these are separated and removed), the charge on the sphere will remain equal to e, i.e. to v— — F, but the potential of the sphere will rise ; when it is alone in the field the potential will be eja, i.e. — a Thus by removing the outer sphere the potential difference between the sphere and the earth has been increased in the proportion of 6 ioh — a. By making h — a very small compared with 6, we can in this way increase the potential difference enormously and make it capable of detection by means which would not have been suffi- ciently sensitive before the increase in the potential took place. It was by the use of this principle that Volta suc- ceeded in demonstrating by means of the gold-leaf electro- scope and two metal plates, the difference of potential between the terminals of a galvanic cell ; this difference is 88 CAPACITY OF CONDUCTORS. [CH. Ill SO small that the electroscope is not deflected when the cell is directly connected to it ; by connecting the ter- minals of the cell to two plates placed very close together, then severing the connection between the plates and the cell and removing one of the plates, Volta was able to increase the potential of the other plate to such an extent that it produced an appreciable deflection of an electro- scope with which it was connected. Work has to be done in separating the two con- ductors ; this work appears as increased electric energy. Thus, to take the case of the two spheres, when both spheres were in position the electric energy, which, by Art. 23 is equal to ^lEV, is 2b-a' ' When the outer sphere is removed the potential of the sphere is e/a, so that the electric energy is 2 a' 2 {h-df ' and has thus been increased in the proportion of h to h— a. 54. Let us now take the case when the inner sphere is connected to earth while the outer sphere is at the potential V. In this case we can prove exactly as before that the charge on the inner sphere is equal and opposite to the charge on the internal surface of the outer sphere, and that if e is the charge on the inner sphere ah ^j — a 55] CONDENSERS. 89 In this case in addition to the positive charge on the internal surface of the outer sphere there will, since its external surface is at a higher potential than the sur- rounding conductors, be a positive charge on this surface. If c is the radius of the external surface of the outer sphere, we must have the total charges on the two spheres = Vc. Since the charge on the inner surface of the outer sphere is equal and opposite to the charge on the inner sphere, the charge on the external surface of the outer sphere must be equal to Vc. Thus the total charge on the outer sphere is equal to ab h — a V+cV. 55. This charge on the outside of the outer sphere will be affected by the presence of other conductors ; let us suppose that outside the external sphere there is a small sphere connected to earth ; let r be the radius of this sphere, i^ the distance of its centre from the centre of the concentric spheres. Let e be the total charge on the two concentric spheres, e" the charge on the small sphere. The potential due to e' at a great distance R from is e'/M, similarly the potential due to e" is at a distance R equal to e'/R. Since the surface of the outer sphere is at the po- tential F, we have ^ c^R' and since the potential of the small sphere is zero, we have 90 CAPACITY OF CONDUCTORS. [CH. Ill hence F=--!l — -^^ ^ re that is, the presence of the small sphere increases the charge on the outer sphere in the proportion of 1 to 1 - rc\R\ It is only the charge on the external surface of the outer sphere which is affected. The charges on the inner sphere and on the internal surface of the outer sphere are not altered by the presence of conductors outside the system. 56. Parallel Plate Condensers. Condensers are frequently constructed of two parallel metallic plates ; the theory of the case when the plates are so large and close together that they may be regarded as infinite in area is very simple. In this case the Faraday tubes passing between the plates will be straight and at right angles to the plates; the electric intensity between the plates is constant since the Faraday tubes are straight ; let R be its value, then if d is the distance between the plates, the work done on unit charge of electricity as it passes from the plate when the potential is high to the one where the potential is low is Rdy this by definition is equal to F, the differ- ence of potential between the plates. Hence Y = Rd. I 57] CONDENSERS. 91 If (T is the surface-density of the charge on the plate at high potential, — a will be that on the plate of low potential, and by Coulomb's law, Art. 22, R = 47rcr, hence V= ^urad, "=4^ W- if V is equal to unity, a- is equal to 1 ^ird' The charge on an area A of one of the plates when the potential difference is unity is thus Aj^iTrd, this by definition is the capacity of the area A, We arrived at the same result in Art. 53 from the consideration of two concentric spheres. The electrical energy of the condenser is, by Art. 23, equal to \tEV, in this case this is equal to V'A Sird ' or if E is the charge on one of the planes 2'7rdE' 57. Guard Ring. In practice it is of course im- possible to have infinite plates, and when the plates are finite, then as the diagram, Fig. 21, Art. 38, shows the Faraday tubes near the edges of the plate are no longer straight, and the electrification ceases to be uniform, and given by the expression (1), Art. 56. Thus to express the 92 CAPACITY OF CONDUCTORS. [CH. Ill quantity of electricity on the finite plane, we should have to add to the expression a correction for the inequality of the distribution over the ends of the plates. This correction can be calculated, but the necessity for it may be avoided in practice by making use of a device due to Lord Kelvin, and called the guard ring. Fig. 27. Suppose one of the plates, say the upper one, is divided into three portions flush with each other and separated by the narrow gaps E, F. Then if, in charging the condenser the portions A, B, G are connected metallically with each other, the places where the electrification is not uniform will be on ^ and C, so that apart from the effects of the narrow gaps E, F, the electrification on B will, if we neglect the effect of the gap, be uniform and equal to S/4f7rd, where S is the area of the plate B, the capacity of ^ is thus equal to Sj^^ird. If, as ought to be the case, the widths of the gaps at E and F are very small compared with the distance between the plates, we can easily calculate the effect of the gaps. For if the gaps are very narrow the electrification of the lower plate will be approximately uniform. The Faraday tubes in the neighbourhood of the gaps will be distributed as in Fig. 28. We see 58] CONDENSERS. 93 from this, if we consider one of the gaps E, that all the Faraday tubes which would have fallen if there had been I Fig. 28. no gap on a plate whose breadth was E, will fall on one or other of the plates A and B, Fig. 28, and from the symmetry of the arrangement half of these tubes will fall on B, the other half on A ; thus the actual amount of electricity on B will be the same as if we supposed B to extend half way across the gap, and to be uniformly charged with electricity whose surface density is Vj^ird. We see then that, allowing for the effects of the gaps, the capacity of B will be equal to S'l^ird, where S>' = area of plate B + J (the sum of the areas of the gaps E and F). If the plate B is not at zero potential there will be some electrification on the back of the plate arising from Faraday tubes which go from the back of B to other conductors in its neighbourhood and to earth. The elec- trification of the back of B may be obviated by covering this side of ^, J5, G with a metal cover connected with the earth. It can also be obviated by making B the low potential plate (i.e. the one connected to earth), care being taken that the other conductors in the neighbour- hood are also connected to earth. 58. Capacity of two coaxial cylinders. Let us take the case of two coaxial cylinders, the inner one being 94 CAPACITY OF CONDUCTORS. [CH. Ill at potential F, the outer one being at potential zero. Then if E is the charge per unit length on the inner cylinder, — E will be the charge per unit length on the inner surface of the outer one, since all the Faraday tubes which start from the inner cylinder end on the outer one. The electric intensity at a distance r from the axis of the cylinders is, by Art. (13), equal to r Thus the work done on unit charge when it goes from the outer surface of the inner cylinder to the inner surface of the outer, is equal to '^2E dr. a r where a is the radius of the inner cylinder, h the radius of the inner surface of the outer. This work is however by definition equal to F, the difference of potential between the cylinders, hence '^2E J a ^ dr When V is unity E the charge per unit length is equal to ^ 2 log-' and this, by definition, is the capacity per unit length of the condenser. 58] CONDENSERS. 95 If the cylinders are very close together, and i{b — a = t, t will be small compared with a ; in this case the capacity per unit length 2 log-- = — ■ approximately 2- a = 1- ~2 t _ 27ra ~~ 4f7rt ' Now 27ra is the area of unit length of the cylinder, hence the capacity per unit area is l/4i7rt ; we might have deduced this result from the case of two parallel planes. When the two cylinders are concentric, there is no force tending to move the inner cylinder; thus since the system is in equilibrium, the potential energy if the charges are given must be either a maximum or a mini- mum. The equilibrium is however evidently unstable, for if the inner cylinder is displaced the forces in the electric field will tend to make the cylinders come into contact with each other and thus increase the displace- ment. Since the equilibrium is unstable the potential energy is a maximum when the cylinders are coaxial. [The potential energy however is, by (Art. 23), equal to 96 CAPACITY OF CONDUCTORS. [CH. Ill when C is the capacity of the condenser. Thus if the potential energy is a maximum the capacity must be a minimum. Thus any displacement of the inner cylinder will produce an increase in the capacity, but since the capacity is a minimum when the cylinders are coaxial, the increase in the capacity will be proportional to squares and higher powers of the distance between the axes of the cylinders. 59. Condensers whose capacities can be varied. For some experimental purposes it is convenient to use a condenser whose capacity can be altered continuously, and in such a way that the alteration in the capacity can be easily measured. For this purpose a condenser made of two parallel plates, one of which is fixed, while the other can be moved by means of a screw, through known dis- tances, always remaining parallel to the fixed plate, is useful. In this case the capacity is inversely proportional to the distance between the plates, provided that this dis- tance is never greater than a small fraction of the radius of the plates. Another arrangement which has been used for this purpose is shown in Fig. 20. It consists of three EC B O Fig. 29. coaxial cylinders, two of which, AB, CD, are of the same radius and are insulated from each other, while the third, EF, is of smaller radius and can slide parallel to its axis. The cylinder EF is connected metallically with CD, so 60] CONDENSERS. 97 that these two are always at the same potential, the cylinder AB is at a different potential, then when the cylinder EF is moved about so as to expose different amounts of surface to AB the capacity of the system will alter, and the increase in the capacity will be pro- portional to the increase in the area of the surface of EF brought within AB. 60. Electrometers. Consider the case of two parallel conducting planes • let V be the potential difference between the planes, d their distance apart. The force on a conductor per unit area is by Art. 37, equal to where R is the electric intensity at the conductor and a the surface density ; but while (T= — R by Coulomb's law ; we see therefore that the attraction of one plate on the other is per unit area equal to Stt d^ ' Hence the force on an area A of one of the plates is equal to A V Thus if we measure the mechanical force between the plates we can deduce the value of F, the potential differ- ence between them. This is the principle of Lord Kelvin's attracted disc electrometer. This instrument measures T. E. 7 98 CAPACITY OF CONDUCTORS. [CH. Ill the force necessary to keep a moveable disc surrounded by a guard ring in a fixed position ; when this force is known the value of the potential difference is given by the expression (1). Quadrant Electrometer. The effect measured by the instrument just described varies as the square of the potential difference; thus when the potential difference is diminished the attraction between the plates diminishes with great rapidity. For this reason the instrument is not suited for the measurement of very small potential differences. To measure these another electrometer, also due to Lord Kelvin, called the quadrant electrometer, is frequently employed. This instrument is represented in Fig. 30 : it consists of a cage, made by the four quadrants A, B, C, I); each quadrant is supported by an insulating stem, while the opposite quadrants A and G are connected by a metal wire, as are also B and -D ; thus A and C are always at the same potential and so also are B and D. Each pair of quadrants is in connection with an electrode, E or F, by means of which it can easily be put in metallic connection with any body outside the case of the instrument. Inside the quad- rants and insulated from them is a flat piece of aluminium shaped like a figure of eight. This is suspended by a silk fibre and can rotate with the flat side horizontal about a vertical axis. A fine metal wire from the lower surface of this aluminium needle hangs and dips into some sulphuric acid contained in a glass vessel, the outside of which is coated with tin-foil and connected with earth. This vessel, with the conductors inside and outside, forms a condenser of considerable capacity ; it requires therefore 60] CONDENSERS. 99 a large charge to alter appreciably the potential of this jar, and therefore of the needle. To use the instrument Fig. 30. charge up the jar to a high potential (7; the needle will also be at the potential G. Now if the two pairs of quadrants are at the same potential, the needle is inside a conductor symmetrical about the axis of rotation of the needle, and at one potential. There will evidently be no couple on the needle arising from the electric field, and the needle will take up a position in which the couple arising from the torsion of the thread supporting the needle vanishes. If, however, the two pairs of quadrants are not at the same potential the needle will swing round until, if there is 7—2 100 CAPACITY OF CONDUCTORS. [CH. Ill nothing to stop it, the whole of its area will be inside the pair of quadrants whose potential differs most widely from its own. As it swings round, however, the torsion of the thread produces a couple tending to bring the needle back to the position from which it started. The needle finally takes up a position in which the couple due to the torsion in the thread balances that due to the electric field. The angle through which the needle is deflected gives us the means of estimating the potential difference between the quadrants. The way in which the couple acting on the needle depends upon the potentials of the quadrants and the needle can be illustrated by considering a case in which the electric principles involved are the same as in the quadrant electrometer, but where the geometry is simpler. Let E, F (Fig. 31) be two large co-planar surfaces in- sulated from each other by a small air gap. Let G be Fig. 31. another plane surface, parallel to E and F, and free to move in its own plane. Let t be the distance between G and the planes E and F. Let A,B,C hQ the potentials of the planes F, E, G respectively. Let I be the width of the planes at right angles to the plane of the paper. If XI is the force tending to move the plane G in the direction of the arrow, then if this plane be moved through a short distance x in this direction the work done by the 60] CONDENSERS. 101 electric forces is Xlx. If the electric system is left to itself, i.e. if it is not connected to any batteries, &c., so that the charges remain constant, this work must have been gained at the expense of the electric energy, we have therefore, by the principle of the Conservation of Energy, Xh = decrease in electric energy, the charges remaining constant when the plane G is displaced through the distance x ; or by Art. 36, Xlx = increase in electric energy, the potentials remain- ing constant when the system suffers the same displacement. (1) Consider the change in the electric energy when the plane G is moved through a distance x. The area of G opposite to F will be increased by Ix, and in consequence the energy will be increased by the energy in a parallel plate condenser, whose area is Ix, the potentials of whose plates are A and G respectively, and the distance be- tween the plates is t ; this by Art. 56 is equal to At the same time as the area of G opposite to F is in- creased by Ix, that opposite to E is decreased by the same amount, so that the electric energy will be decreased by the energy in a parallel plate condenser whose area is Ix, the potentials of the plates B and G and their distance apart t, this by Art. 56 is equal to 102 CAPACITY OF CONDUCTORS. [CH. Ill Thus the total increase in the electric energy when G is displaced through x, the potentials being constant is equal to Thus by equation (1) or ^ = 4^(^-^>|^-l(^+^> \i G — A is greater than G — B, X is positive, that is, the plate G tends to bring as much of its surface as it can over the plate from which it differs most in potential. In the quadrant electrometer the electrical arrange- ments are similar to the simple case just discussed, hence the force will vary with the potential differences in a similar way. Hence we conclude that the couple tending to twist the needle in the quadrant electrometer from the quadrant whose potential is B to that whose potential is A, will be proportional to (B-A)\o-l{A + B) we may put it equal to n{B-A)\c'-l(A+B)Y where n is some constant. 60] CONDENSERS. 103 This, when the needle is in equilibrium, will be balanced by the couple due to the torsion in the sus- pension of the needle. This couple is proportional to the angle 9 through which the needle is deflected. Let the couple equal md. Hence we have when the needle is in equilibrium me = n(B-A)[c-'^(A + B)\, e = l(B-A)\c-l(A^B)^ (2). If, as is generally the case when small differences of potentials are measured, the jar containing the sulphuric acid is charged up so that its potential is very high com- pared with that of either pair of quadrants, G will be very large compared with A or B, and therefore with li^ + B), SO that the expression (2) is very approximately e=-(B-A)a Hence in this case the difference of potential is pro- portional to the deflection of the needle. This furnishes a very convenient method of comparing differences of potentials, and though it does not give at once the ab- solute measure of the potential, this may be deduced by measuring the deflection produced by a standard po- tential difference of known absolute value such as that between the electrodes of a Clark's cell. The quadrant electrometer may also be used to measure large differences of potential ; to do this, instead 104* CAPACITY OF CONDUCTORS. [CH. Ill of charging the jar independently, connect the jar and therefore the needle to one pair of quadrants, say the pair whose potential is A, then since C = A the expression (2) becomes thus the needle is deflected towards the pair of quadrants whose potential is B, and the deflection of the needle is in this case proportional to the square of the potential differ- ence between the quadrants. Thus if the quadrants are connected respectively to the inside and outside coatings of a condenser, the deflection of the electrometer will be proportional to the energy in the condenser. 61. Test for the equality of the capacity of two condensers. This can easily be done in the following Fig. 32. way. Suppose A and B, Fig. 32, are the plates of one condenser, G and D those of another. First connect A to 0, and B to D, and charge the condensers by connecting A and B with the terminals of a battery or some other suitable means. Then disconnect A and B from the battery. Disconnect A from C and B from D, then if the capacities of the two condensers are equal, their charges will be equal since they have been charged to equal potentials. The charge in A will be equal and opposite to that in D, while that in B will be equal and 62] CONDENSERS. 105 opposite to that in G. Thus if A be connected with D and C with B the positive charge on the one plate will counterbalance the negative on the other, so that if after this connection has been made A and B are connected with the electrodes of the electrometer, no deflection will occur. 62. Comparison of two condensers. If a con- denser whose capacity can be varied is available, the capacity of a condenser can be compared with known capacities by the following method. Let A and B (Fig. 33) be the plates of the condenser whose capacity is required, G and D, E and F, G and E, the plates of three condensers whose capacities are known ; connect the plates B and G together and to one electrode of an electrometer, also connect F and G together and to the other electrode of the electrometer. D and E are to be connected together and to one electrode of a battery, induction coil or other means of producing a difference of potential, while A and H are to be connected together and to the other pole of this battery. In general there will be a deflection of the electrometer ; if there is, then 106 CAPACITY OF CONDUCTORS. [CH. Ill we must alter the capacity of the condenser whose capacity is variable until this deflection vanishes, show- ing that the plates BG, FG are at the same potential. When this is the case a simple relation exists between the capacities. Let Ci, Ca, C3, C4, be the capacities of the condensers AB, CD, EF, GH respectively, let V, be the potential of A and H\ x the potential of BG and FG, V the potential of DE. To fix our ideas, let us suppose that V is greater than Vq, then there will be a negative charge on -4, a positive one on B, a negative charge on G, and a posi- tive one on D ; then since B and G form an insulated system which was initially without charge, the positive charge on B must be numerically equal to the negative charge on G. The positive charge on B while the negative one on G is numerically equal to G,{V-w), which is a positive quantity ; hence since these are equal we have G,{x-r,) = c,{V-x) (1). Again, since there is no deflection of the electrometer, the potential of F and G is the same as that of B and G, and is therefore equal to x, while since F and G are insulated the positive charge on G must be numerically equal to the negative charge on F. The positive charge on G is equal to c,(x-r,), 03] CONDENSERS. 107 while the negative charge on F is numerically equal to since these are equal G,{x-V,) = C,{V-x) (2); comparing equations (1) and (2), we see that or G,= G, ' thus if we know the capacities of the other condensers we know Cj. Thus if we have standard condensers whose capacities are known we can measure the capacity of other con- densers. Other methods of determining capacity which require for their explanation a knowledge of the principles of electro-magnetism, will be described in the part of the book dealing with that subject. 63. Leyden jar. A convenient form of condenser called a Leyden jar is represented in Fig. 34. The O v^ Fig. 34. 108 CAPACITY OF CONDUCTORS. [CH. Ill condenser consists of a vessel made of thin glass; the inside and outside surfaces of this vessel are coated with tin-foil. An electrode is connected to the inside of the jar in order that electrical connection can easily be made with it. If A is the area of the tin-foil, t the thickness of the glass, i.e. the distance between the surfaces of tin-foil, then if the interval between the tin-foil was filled with air the capacity would be approximately A^ since this case is approximately that of two parallel planes provided the thickness of the glass is very small compared with the radius of the vessel. The effect of having glass within the tin-foil plates will, as we shall see in the next chapter, have the effect of increasing the capacity so that the capacity of the Leyden jar will be where K is a, quantity which depends on the kind of glass of which the vessel is made. K varies in value from 4 to 10 for different specimens of glass. 64. If we have a number of condensers we can con- nect them up so as to make a condenser whose capacity is either greater or less than that of the individual condensers. Thus suppose we have a number of condensers which in the figures are represented as Leyden jars, and suppose we connect them up as in Fig. 35, that is, connect all the insides of the jars together and likewise all the outsides, this is called connecting the condensers in parallel. We 64] CONDENSERS. 109 thus get a new condenser, one plate of which consists of all the insides, and the other plate of all the outsides of Fig. 35. the jars. If G is the capacity of the compound condenser, Q the total charge in this condenser, V the difference of potential between the plates, then by definition Q = GV. Tf Qi, Qi, Q-iy ••• are the charges in the first, second, third, etc. condensers, Oj, Cg, Cg, ... the capacities of these condensers but Q = Qi + Q, + Qa + ... = (Ci + 0, + (^3 + ...) K hence C = Ci + Cg + O3 + . . . , or the capacity of a system of condensers connected in this way, is the sum of the capacities of its components. Thus the capacity of the compound system is greater than that of any of its components. Next let the condensers be connected up as in Fig. 36, where the condensers are insulated, and where the outside of the first is connected to the inside of the second, the outside of the second to the inside of the third, and so on. This is called connecting the condensers up in cascade or in series. One plate of the compound system thus 110 CAPACITY OF CONDUCTORS. [CH. Ill formed is the inside of the first condenser, the other plate is the outside of the last. Fig. 36. Let G be the capacity of the system, G^, G.^, G.^, the capacities of the individual condensers; then since the condensers are insulated the charge on the outside of the first is equal in magnitude and opposite in sign to the charge on the inside of the second, the charge on the outside of the second is equal in magnitude and opposite in sign to the charge on the inside of the third, and so on. Since the charge on the inside of any jar is equal and opposite to the charge on the outside, we see that the charges on the jars are all equal. Let Q be the charge on any jar, F,, V^ ... the differences of potential between the inside and outside of the first and second jars. Then Ul O2 O3 if V is the difference of potential between the outside of the last jar and the inside of the first, then 65] CONDENSERS. Ill V SO that Q = -^ ? = , 7T ' ri • 77" + • • • Vi O2 O3 but since G is the capacity of the compound condenser of which Q is the charge, and V the potential difference, Q = GV, , 1111 hence 77 = 7r + 7T+p+---, thus the reciprocal of the capacity of the condenser made by connecting up in cascade the series of condensers, is equal to the sum of the reciprocals of the capacities of the condensers so connected up. We see that the capacity of the compound condenser is less than that of any of its constituents. 65. If we connect a condenser of small capacity in cascade with a condenser of large capacity, the capacity of the compound condenser will be slightly less than that of the small condenser ; while if we connect them in parallel, the capacity of the compound condenser is slightly greater than that of the large condenser. 66. As another example on the theory of condensers, let us take the case when two condensers are connected in parallel, the first having before connection a charge Q^, the second the charge Q2. Let Gi and G2 be the capacities of these condensers respectively. When they are put in connection they form a condenser whose capacity is Ci + G^, id whose charge is Qi+Qi- Now the electric energy in a charged condenser is me half the product of the charge into the potential ifference, and since the potential difference is equal to 112 CAPACITY OF CONDUCTORS. [CH. Ill the charge divided by the capacity; if Q is the charge, G the capacity, the energy is 2 C Thus the total electric energy in the two jars before they are connected is 1 Q^\lQ.' 2 G,2 G^ after they are connected it is Now 1 {Q^ + Q.\ 2 G + G, ■ 2V0, +Cj" 1 (Qi + Q.y 2 (0. + 0/) ^ {G,'Q^'-hG,%^-2GAQ^Q.) 2GA{GrA-G,) 1 (G,Q^-C,Q,y, 2G,G,(G, + G,) an essential positive quantity which only vanishes if Q,IG, = Q.IC,. that is, when the potentials of the jars before connection are equal. In this case the energy after connection is the same as before the connections are made. If the potentials are equal before connection, connecting the jars will evidently make no difference, as all that con- nection does is to make the potentials equal. In every other case electric energy is lost when the connection is made ; this energy is accounted for by the work done by the spark which passes when the jars are put in connection. CHAPTER IV. Specific Inductive Capacity. 67. Specific Inductive Capacity. Faraday found that the charge in a condenser between whose plates a constant difference of potential was maintained depended upon the nature of the dielectric between the plates, the charge being greater when the interval between the plates was filled with glass or sulphur than when it was filled with air. Thus the 'capacity' of a condenser (see Art. 51) de- pends upon the dielectric between the plates. Faraday's original experiment by which this result was established was as follows : he took two equal and similar condensers, A and B, of the kind shown in Fig. (37), made of concentric spheres ; in one of these, B, there was an opening by which melted wax or sulphur could be run into the interval be- tween the spheres. The insides of these condensers were connected together, as were also the outsides, so that the potential difference between the plates of the condenser was the same for A as for B. When air was the dielectric between the spheres Faraday found, as might have been expected from the equality of the condensers, that any charge given to the condensers was equally distributed between A and J5; when however the interval in B vva-s T. E. 8 114 SPECIFIC INDUCTIVE CAPACITY. [CH. IV filled with sulphur and the condensers again charged he found that the charge in B was three or four times that Fig. 37. in A : proving that the capacity of B had been in- creased three or four times by the substitution of sulphur for air. This property of the dielectric is called its specific inductive capacity. The measure of the specific induc- tive capacity of a dielectric is defined as the ratio of the capacity of a condenser the region between whose plates is entirely filled by this dielectric to the capacity of the same condenser when the region between its plates is entirely filled with air. As far as we know at present the specific inductive capacity of a dielectric in a con- denser does not depend upon the difference of potential established between the plates of that condenser, that is, upon the electric intensity acting on the dielectric. We may therefore conclude that at any rate for a wide range 67] SPECIFIC INDUCTIVE CAPACITY. 115 of electric intensities the specific inductive capacity is independent of the electric intensity. The following table contains the values of the specific inductive capacities of some substances which are of frequent occurrence in a physical laboratory: Solid paraffin 2-29. Paraffin oil 1-92. Ebonite 315. Sulphur 3-97. Mica 6-64. Dense flint-glass 7-37. Light flint-glass 6-72. Turpentine 2-23. Distilled water 76. Alcohol 26. The specific inductive capacity of gases depends upon the pressure, the difference between K, the specific in- ductive capacity, and unity being directly proportional to the pressure : this law however does not seem to hold at very low pressures. The specific inductive capacity of some gases at atmospheric pressure is given in the following table ; the specific inductive capacity of air at atmospheric pressure is taken as unity: Hydrogen -999674. Carbonic acid 1 000356. Carbonic oxide 10001. defiant gas 1000722. 68. It was the discovery of this property of the di- electric which led Faraday to the view we have explained, 8—2 116 SPECIFIC INDUCTIVE CAPACITY. [CH. IV in Art. 38, that the effects observed in the electric field are not due to the action at a distance of one electrified body on another, but are due to effects in the dielectric filling the space between the electrified bodies. The results obtained in Chapters II. and ill. were deduced on the supposition that there was only one dielectric, air, in the field ; these require modification in the general case when we have any number of dielectrics in the field. We shall now go on to consider the theory of this general case. We assume that each unit of positive electricity by whatever medium it is surrounded is the origin of a Faraday tube, each unit of negative electricity the ter- mination of one : let us consider from this point of view the case of two parallel plate condensers A and B, the plates of A and B being at the same distance apart, but while those of A are separated by air, those of B are separated by a medium whose specific inductive capacity is K. Let us suppose that the charge per unit area on the plates of the condensers A and B is the same. Then since the capacity of the condenser B is K times that of A and since the charges are equal, the potential difference between the plates of B is only 1/K of that between the plates of A. Now if Vp is the potential at P, Vq that at Q, R the electric intensity along PQ ; then, whatever be the nature of the dielectric, when PQ is small enough to allow of the intensity along it being regarded as constant, R.PQ=Vp-Vq (1), for by definition R is the force on unit charge, hence the 68] SPECIFIC INDUCTIVE CAPACITY. 117 left-hand side of this expression is the work done on unit charge as it moves from P to Q, and is thus by definition, Art. 16, equal to the right-hand side of (1). The electric intensity between the plates both of A and B is uniform, and at any point equal to the difference of potential between the plates divided by the distance between the plates, this distance is the same for the plates A and B, so that the electric intensity between the plates of A is to that between the plates of B as the potential difference between the plates of A is to that between the plates of B. That is, the electric intensity in ^ is iiT times that in B. Consider now these two condensers. Since the charges on unit area of the plates are the same the number of Faraday tubes passing through the dielectric between the plates is the same, while the electric intensity in B is only l/K that in air. Hence we conclude that when the same number of Faraday tubes pass through unit area of a dielectric whose specific inductive capacity is K as through unit area in air, the electric intensity in the dielectric is ^ of the electric intensity in air. By Art. 40 we see that if N is the number of Faraday tubes passing through unit area in air, R the electric intensity in air, hence when N tubes pass through unit area in a medium whose specific inductive capacity is K, R, the electric in- tensity in this dielectric is given by the equation 118 SPECIFIC INDUCTIVE CAPACITY. [CH. IV 69. Polarization in a dielectric. We define the polarization in the direction PQ where P and Q are two points close together as the excess of number of Faraday tubes which pass from the side P to the side Q over the number which pass from the side Q to the side P of a plane of unit area drawn between P and Q at right angles to PQ. We may express the result in Art. 68 in the form (electric intensity in any direction at P) = Y=r (polarization in the dielectric in that direction at P). K. The polarization in a dielectric is mathematically identical with the quantity called by Maxwell the electric displacement in the dielectric. 70. Thus the polarization along the out ward -drawn normal at P to a surface is the excess of the number of Faraday tubes which leave the surface through unit area at P over the number entering it. If we divide any closed surface up as in Art. 9 into a number of small meshes, each of these meshes being so small that the polarization over the area of any mesh may be regarded as constant, then if we multiply the area of each of the meshes by the normal polarization at this mesh, the sum of the products taken for all the meshes which cover the surface is defined to be the total normal polarization over the surface. We see that it is equal to the excess of the number of Faraday tubes which leave the surface over the number which enter it. Now consider any tube which does not begin or end inside the closed surface, then if it meets the surface at all it will do so at two places, P and Q ; at one of these 70] SPECIFIC INDUCTIVE CAPACITY. 119 it will be going from the inside to the outside of the surface, at the other from the outside to the inside. Such a tube will not contribute anything to the total normal polarization over the surface, for at the place where it leaves the surface it contributes + 1 to this quantity, which is neutralized by the — 1 which it contributes at the place where it enters the surface. Now consider a tube starting inside the surface, this will leave the surface but not enter it, or if the surface is bent so that the tube cuts the surface more than once, it will leave the surface once oftener than it enters it. This tube will therefore contribute + 1 to the total normal polarization : similarly we may show that each tube which ends inside the surface contributes — 1 to the total normal polarization. Thus if there are N tubes which begin inside the surface, and M tubes which end inside, the total normal polarization is equal to N — M. But each tube which begins inside the surface corresponds to a unit positive charge, each tube which ends in the surface to a unit negative one, so that iV— if is the difference between the positive and negative charge inside the surface, that is, it is the total charge inside the surface. Thus we see that the total normal polarization over a closed surface is equal to the charge inside the surface. Since the normal polarization is equal to {Kl4i'n) times the normal intensity where K is the specific inductive capacity, which for air is equal to unity, we see that when the dielectric is air the preceding theorem is identical with Gauss's theorem. Art. 10. In the form stated above it is applicable whatever dielectrics may be in the field, when in general Gauss's theorem as stated in Art. 10 ceases to be true. 120 SPECIFIC INDUCTIVE CAPACITY. [CH. IV 71. Modification of Coulomb^s equation. If o- is the surface density of the electricity over a conductor then cr Faraday tubes pass through unit area of a plane drawn in the dielectric above the conductor at right angles to the normal. Hence a is the polarization in the dielectric in the direction of the normal to the con- ductor. Hence by Art. 69 if jK is the normal electric intensity This is Coulomb's equation generalized, so as to apply to the case when the conductor is in contact with any dielectric, 72. Expression for the Energy. The student will see that the process by which the expression ^%(EV) was in Art. 23 proved to represent the electric energy of the system will apply whatever the nature of the dielectric may be, as will also the immediate deduction from it that the energy is the same as if each Faraday tube possessed an amount of energy equal per unit length to one-half the electric intensity. The expression for the energy per unit volume how- ever requires modification. Consider as in Art. 43 a cylinder whose axis is parallel to the electric intensity and whose flat ends are at right angles to it, let I be the length, Q) the area of one of the ends, P the polarization, R the electric intensity. Then the portion of each Faraday tube inside the cylinder has an amount of energy equal to 73] SPECIFIC INDUCTIVE CAPACITY. 121 the number of such tubes inside the cylinder is equal to Pro, hence the energy inside the cylinder is equal to ileoPB, but l(o is the volume of the cylinder, hence the energy per unit volume is equal to iPB; but by Art. 69 P=^R, so that the energy per unit volume is equal to Thus for the same electric intensity the energy is K times as great as it is in air. Another expression for the energy per unit volume is 2^ p. K ' so that for same polarization the energy in the dielectric is only 1/J^th part of what it is in air. We see, as in Art. 45, that the tension in each Faraday tube will still equal one-half the electric intensity (jR) ; the tension across unit area in the dielectric will therefore be — — , the lateral pressure will also be equal to KR-JStt. OTT 73. Conditions to be satisfied at the boundary of two media of different specific inductive capa- city. Suppose that the line AB represents the section 122 SPECIFIC INDUCTIVE CAPACITY. [CH. IV by the plane of the paper of the plane of separation of two different dielectrics ; let the specific inductive ca- pacities of the upper and lower media respectively be 8 R Fig. 38. Let US consider the conditions which must hold at the surface. In the first place we see that the electric intensity parallel to the surface must be the same in both media; for if they were different, that in the medium K^ being the greater, we could get an infinite amount of work by making unit charge travel round the closed circuit PQRS, PQ being just above, and RS just below the surface of separation. For if PQ is the direction of the tangential component T^ of the electric intensity in the upper medium, the work done on unit charge as it goes from P to Q is Ti . PQ ; as QR is ex- cessively small the work done on or by the charge as it goes from Q to i^ may be neglected if the normal in- tensity is not infinite, the work required to take the unit charge back from JK to >Sf is T^. RS, if T^ is the tangential component of the electric intensity in the lower dielectric, and we may neglect the work done or spent in going from S to P. Thus since the system is brought back to the state from which it started, the work done must vanish, hence T^PQ — T^RS must be zero. But since PQ = RS this requires that T^ = T^ or the tangential components of the electric intensity must be the same in the two media. 73] SPECIFIC INDUCTIVE CAPACITY. 123 Next suppose that a is the density of the free electricity on the surface separating the two media. Draw a very flat circular cylinder shown in section at FQRS, the axis of this cylinder being parallel to the normal to the surface of separation, the top face of this cylinder being just above, the lower face just below this surface. The length of this cylinder is very small com- pared with its breadth ; the area of the curved surface of the cylinder will be very small compared with the area of its ends, and by making the cylinder sufficiently short we can make the ratio of the area of the curved surface to that of the ends as small as we please ; hence in considering the total normal polarization over a very short cylinder, we may leave out the effect of the curved surface and consider only the flat ends of the cylinder. But since the cylinder encloses the charge aco, if co is the area of one end of the cylinder, the total normal polarization over its surface must be equal to aco. If N^ is the normal polarization in the first medium measured upward the total normal polarization over the top of the cylinder is iV^o); if iV^2 is the normal polarization measured upward in the second medium, the total normal polariza- tion over the lower face of the cylinder is — iV^gW ; hence the total normal polarization over the cylinder is iVjO) - N^o). Since this, by Art. 70, is equal to aw, we have When there is no charge on the surface separating the two dielectrics, these conditions become (1) that the tangential electric intensities, and (2) the normal polariza- tions must be the same in the two media. 124 SPECIFIC INDUCTIVE CAPACITY. [CH. IV 74. Refraction of the lines of force. Suppose that Ri is the resultant electric intensity in the upper medium, R2 that in the lower ; 61,62 the angles these make with the normal to the surface of separation. The tan- gential intensity in the first medium is Ri sin 61, that in the second is ii.2 sin 6^, and since these are equal jRisin^i= 2^2 sin 6^ (1). The normal intensity in the upper medium is Ri cos 6^ , hence the normal polarization in the upper medium is K,R, cos (9i/47r, that in the second is iTa^a cos ^a/^Tr, and since, if there is no charge on the surface, these are equal we have ^R,cos6, = f' R,cos6., (2); 477 47r dividing (1) by (2), we get -j^ tan 61 = j^ tan 61 ill ^2 hence, if K^> K^, 6^ is > 6^, thus when the tube enters a medium of greater specific inductive capacity from one of less, it is bent away from the normal. This is shown in the diagram Fig. 39 (from Lord Kelvin's Reprint of Papers on Electrostatics and Mag- netism), which represents the lines of force when a sphere, made of paraffin or some material whose specific inductive capacity is greater than unity, is placed in a field of uni- form force such as that between two infinite parallel plates. An inspection of the diagram shows the tendency of the tubes to run as much as possible through the sphere ; this is an example of the principle that a system always tends to get the potential energy as small as possible. 74] SPECIFIC INDUCTIVE CAPACITY. 125 We saw, Art. 72, that when the polarization is P the energy per unit volume is lirP'^IK, thus this for the Fig. 39. same value of P is less in paraffin than it is in air; that is when the same number of tubes pass through the paraffin they have less energy in unit volume than when they pass through air, there is therefore a tendency for the tubes to flock into the paraffin. The reason that all the tubes do not run into the sphere is that those which are some distance away from it would have to bend down in order to reach the paraffin, they would therefore have to greatly lengthen their path in the air, and the increase in the energy consequent upon this would not be compensated for in the case of the tubes some distance originally from the sphere by the diminu- tion in the energy when they got in the sphere. In Fig. 40 (from Lord Kelvin's Reprint of Papers on Electrostatics and Magnetism) the effect produced by a conducting sphere on a field of uniform force is given for comparison with the effects produced by the paraffin 126 SPECIFIC INDUCTIVE CAPACITY. [CH. IV sphere. It will be noticed that the paraffin sphere pro- duces effects similar in kind though not so great in intensity as the conducting sphere. This observation is true for all electrostatic phenomena, for we find that Fig. 40. bodies having a greater specific inductive capacity than the surrounding dielectric behave in a similar way to conductors. Thus they deflect the Faraday tubes in the same way though not to the same extent ; again, a conduc- tor tends to move from the weak to the strong parts of the field, so likewise does a dielectric surrounded by one of smaller specific inductive capacity. Again, the electric intensity inside a conductor vanishes, the electric intensity just inside a dielectric of greater specific inductive capacity than the surrounding medium is less than that just outside. As far as electrostatic phenomena are concerned an in- sulated conductor behaves like a dielectric of infinitely great specific inductive capacity. 75] SPECIFIC INDUCTIVE CAPACITY. 127 75. Force between two small charged bodies immersed in any dielectric. If we have a small body with a charge e immersed in a medium whose specific inductive capacity is K, then the polarization at a dis- tance r from the body is e/4!7rr\ To prove this describe a sphere radius r with its centre at the small body, then the polarization P will be uniform over the surface of the sphere and radial ; hence the total normal polarization over the surface of the sphere will equal P x (surface of the sphere), i.e. P x 47r?'^; but this, by Art. 70, is equal to e, hence P X 4f7rr^ — e, P-^r^ «■ But if R is the electric intensity, then by Art. 69 Hence by (1) ^ = :^2' the repulsion on a charge e' is Re\ i.e. the repulsion between the charges when separated by a distance r in a dielectric whose specific inductive capacity is K is only l/ii"th part of the repulsion between the charges if they were separated by the same distance in air. Thus when the charges are given the mechanical forces in the field are diminished by the interposition of a medium with a large specific inductive capacity. 76. Two Parallel Plates separated by a Di- electric. Let us take the case of two parallel plates 1 28 SPECIFIC INDUCTIVE CAPACITY. [CH. IV. separated by an insulating medium whose specific in- ductive capacity is K. Let V be the potential difference between the plates, a the surface density of the electrifi- cation on the positive plate, then — cr will be that on the negative. Let R be the electric intensity between the plates, d the distance by which they are separated; then by Art. 71 4770- = KR _KV ~ d ' The force on one of the plates per unit area is ^Ra- ~ K ' that is if the charges are given the force between the plates is inversely proportional to the specific inductive capacity of the medium separating them. Again, since we see that if the potentials of the plates be given the attraction between them is directly proportional to the specific inductive capacity. This result is an example of the following more general one which we leave to the reader to work out; if in a system of conductors main- tained at given potentials and originally separated from each other by air we replace the air by a dielectric whose specific inductive capacity is K, keeping the position of the conductors and their potentials the same as before, the forces between the conductors will be increased K times. 77] SPECIFIC INDUCTIVE CAPACITY. 129 Thus for example if we fill the space between the needles and the quadrants of an electrometer with a fluid whose specific inductive capacity is K, keeping the potentials of the needles and quadrants constant, the couple on the needle will be increased K times by the introduction of the fluid. Thus if we measure the couples before and after the introduction- of the fluid the ratio of the two will give us the specific inductive capacity of the fluid. This method has been applied to measure the specific inductive capacity of liquids, especially those such as water or alcohol, which are not sufficiently good insulators to allow the method described in Art. 81 to be applied. 77. The next case we shall consider is when a slab of dielectric is placed between two infinite parallel conducting planes, the faces of the slab being parallel to the planes. yin///niiiiiiiiii/i/iH////////i////////n//i/i/i\ Fig. 41. Let d be the distance between the planes, t the thickness of the slab, h the distance between the upper face of the slab and the upper plane. The Faraday tubes will go straight across from plane to plane, so that the polarization will be everywhere normal to the conducting planes and to the planes separating the slab of dielectric from the air. T. E. > 9 130 SPECIFIC INDUCTIVE CAPACITY. [CH. IV We saw in Art. 73 that the normal polarization does not change as we cross from one medium to another, and as the tubes are straight the polarization will not change as long as we remain in one medium. Thus the polariza- tion which we shall denote by P is constant between the planes. In air the electric intensity is 47rP, in a dielectric of specific inductive capacity, K the electric intensity is equal to ^irPjE. Thus between A and B the electric intensity is 47rP, B and G — ,^- , Candi) 47rP. The difference of potential between the plates is the work done on unit charge when it is taken from one plate to the other. Now when unit charge is taken across the space AB, the work done on it is 47rP X h ; when it is taken across the plate of dielectric the work done is 47rP when it is taken across CD the work done is 47rP [d - {h + t)}. Hence V the excess of the potential of the plate A above that of D is equal to 47rP 4>7rPh+^t + 47rP [d - (h + t)] = 47rP U-i(^- ^ 77] SPECIFIC INDUCTIVE CAPACITY. 131 If a is the surface density of the electricity on the positive plate, (t = P, so that V = 47r<7 ['-'^i) w- Hence the capacity per unit area of the plate, i.e. the value of a when F= 1, is {'-'^¥) i.e. it is the same as if the plate of dielectric were re- placed by a plate of air whose thickness was tjK. The presence of the dielectric increases the capacity of the condenser. The alteration in the capacity does not depend upon the position of the slab of dielectric between the parallel plates. Let us now consider the force between the plates ; the force per unit area where R is the electric intensity at the surface of the plate; but since the surface of the plate is in contact with air R = 47ro-, thus the force per unit area = 27ro-'^ ; hence if the charges on the plates are given, the attraction between them is not affected by the interposition of the plate of dielectric. Next let the potentials be given; we see from equa- tion (1) that V ^ = 7 fT' 4>7rld-t-{-^] 9—2 132 SPECIFIC INDUCTIVE CAPACITY. [CH. IV hence 27ra^ the force per unit area is equal to V Sir (d — t + -^ The force between the plates when there is nothing but air between them is Sird' Now since K is greater than 1, d — t-\- tjK is less than d, so that l/(c^ - t + tjKy is greater than lld\ Thus when the potentials are given the force between the plates is increased by the interposition of the dielectric. If K be very great tjK is very small, thus d — t-\- t/K is very nearly equal to d — t, and the effect of the inter- position of the slab of dielectric both on the capacity and on the force between the plates is approximately the same as if the plates had been pushed towards each other through a distance equal to the thickness of the slab, the dielectric between the plates being now supposed to be air. This result, which is approximately true whenever the specific inductive capacity of the slab is very large, is rigorously true when the slab is made of a conducting material. Effect of the Slab of Dielectric on the Potential Energy. The potential energy is by Art. 23 equal to thus the energy corresponding to the charge on each unit of area of the plates is equal to 77] SPECIFIC INDUCTIVE CAPACITY. 133 this by equation (1) is equal to 27ro-M(^ -'('-i)i. it is thus less than ^ira^d, the value of the energy for the same charges when no slab of dielectric is interposed. The interposition of the slab thus lowers the potential energy. We can easily see why this is the case : when the charges are given the number of Faraday tubes is given : and when the plate of dielectric is interposed the Faraday tubes in part of their journey between the plates are in the dielectric instead of air, and we know from Art. 72 that when the Faraday tubes are in the dielectric their energy is less than when they are in air. Since the potential energy of a system always tends to get as small as possible, there will be a tendency to drag as much as possible of the slab of dielectric between the plates of the condenser. Thus if a slab of dielectric projected on one side beyond the plates it would be sucked between the plates until as much of its area as possible was within the region between the plates. Energy expressed in terms of difference of potential. The energy per unit area of the plates is as we have seen equal to this by equation (1) is equal to 1 V^ «'^U-^ri-' Kl) 134 SPECIFIC INDUCTIVE CAPACITY. [CH. IV If the potentials are given the energy when no slab is interposed is so that when the potentials are kept constant the electric energy is increased by the interposition of the slab. 78. Capacity of two concentric spheres with a shell of dielectric interposed between them. If we have two concentric conducting spheres with a concentric shell of dielectric between them, then if e is the charge on the inner sphere, a the radius of this sphere ; b, c the radii of the inner and outer surfaces of the dielectric shell, d the inner radius of the outer conducting sphere, V is the difference of potential between the conducting spheres, and K the specific inductive capacity of the shell, we may easily prove that ^~^\a b'^KKb cj'^c d Thus the capacity of the system is equal to 1 a d V Kl\b cj 79. Two coaxial cylinders. As another example we shall take the case of two coaxial cylinders with a co- axial cylindric shell of a dielectric whose specific inductive capacity is K placed between them. If V is the difference of potential between the two conducting cylinders, U the charge per unit length of the cylinder, a the radius of the inner cylinder, b and c the radii of the inner and outer 80] SPECIFIC INDUCTIVE CAPACITY. 135 surfaces of the dielectric shell, d the inner radius of the outer cylinder, we easily find by the aid of Art. 58 that F=2^{log^4log| + log|, SO that the capacity per unit length of this system is 1 ,6 1 , c . d 80. Force on a piece of dielectric placed in an electric field. If a piece of dielectric such as sulphur or glass is placed in the electric field, then when the Faraday tubes traverse the dielectric there is, Art. 72, less energy per unit volume than when the same number of Faraday tubes pass through air. Thus, as we see in Fig. 39, the Faraday tubes tend to run through the dielectric, because by so doing the electric energy is decreased. If the di- electric is free to move then it can still further decrease the energy by moving from its original position to one where the tubes are more thickly congregated, because the more tubes which get through the dielectric the greater the decrease in the electric energy. The body will tend to move so as to make the decrease in the energy as great as possible, thus it will tend to move so as to enclose as great a number of Faraday tubes as possible. It will therefore be urged towards the part of the field where the Faraday tubes are densest, i.e. to the strongest parts of the field. There will thus be a force on a piece of dielectric tending to make it move from the weak to the strong parts of the field. The dielectric will not move except in a variable field where it can get more Faraday tubes by its change of position. In a uniform field such 136 SPECIFIC INDUCTIVE CAPACITY. [CH. IV as that between two parallel infinite plates the dielectric would have no tendency to move. The force acting upon the dielectric differs in another respect from that acting on a charged body, inasmuch as it would not be altered if the direction of the electric intensity at each part in the field were reversed without altering its magnitude. 81. Measurement of Specific inductive capacity. The specific inductive capacity of a slab of dielectric can be measured in the following way, provided we have a parallel plate condenser one plate of which can be moved by means of a screw through a distance which can be accurately measured. To avoid the disturbance due to the irregular distribution of the charge near the edges of the plate (see Art. 57) care must be taken that the distance between the plates never exceeds a small fraction of the diameter of the plates. Let us call this parallel plate con- denser A ; to use the method described in Art. 62, first take the condenser A and before inserting the slab of dielectric adjust the other variable condenser used in that method until there is no deflection of the electrometer. If the slab of dielectric be now inserted between the plates of A its capacity will be increased, A will no longer be balanced by the other condensers and the electrometer will be deflected. The capacity of A can be diminished by screwing the plates further apart, and by moving the plates through a certain distance the diminution in the capacity due to the increase in the distance between the plates will balance the in- crease due to the insertion of the slab of dielectric ; the stage when this occurs will be indicated by there being again no deflection of the electrometer. Suppose that 81] SPECIFIC INDUCTIVE CAPACITY. 137 when the deflection of the electrometer is zero before the slab is inserted the distance between the plates of the condenser is d, while the distance after the slab is inserted when the electrometer is again in equili- brium is d'. Then the capacity of A in these two cases is the same. But if A is the area of the plate of A the capacity before the slab is inserted is A ^ird' If t is the thickness of the slab and K its specific inductive capacity the capacity after the insertion of the slab is (see Art. 77) equal to A but since the capacities are equal t d = d -i + ^, so that d' — d= t ll — j^[ . But d' — d is the distance through which the plate has been moved, so that if we know this distance and t we can determine K the specific inductive capacity of the slab. It should be noticed that this method does not require the knowledge of the initial or final distances between the plates, but only the difference of these quantities, and this can be measured with great accuracy by the screw attached to the mov-eable plate. CHAPTER V. Electrical Images and Inversion. 82. We shall now proceed to discuss some geometrical methods by which we can find the distribution of electricity in several very important cases. We shall illustrate the first method by considering a very simple example ; that of a very small charged body placed in front of an infinite conducting plane maintained at potential zero. Let P, Fig. 42, be the charged hody, AB the conducting plane. Any solution of the problem must in the region to the right of the plane AB, Fig. 42, satisfy the following con- ditions : (a) it must make the potential zero over the plane B Fig. 42. AB, and (l3) it must make the total normal induction taken over any closed surface enclosing P equal to 47re, where e is the charge at P, while if the closed surface does CH. V. 82] ELECTRICAL IMAGES AND INVERSION. 139 not include P the total normal induction over it must vanish. We shall now prove that there is only one solution which satisfies these conditions. Suppose there were two different solutions, which we shall call (1) and (2). Take the solution corresponding to (2) and reverse the sign of all the charges of electricity in the field including that at P ; this new solution which we shall denote by (— 2) will correspond to a field in which the electric intensity at any point is equal and opposite to that due to the solution (2) at the same point. The solution (—2) corresponds to a field in which the electric potential is zero over AB and at any point at an infinite distance from P ; it also makes the total normal induction over any closed surface enclos- ing P equal to — 47re, that is equal and opposite to the total induction over the same surface due to the solution (1) ; and the total induction over any other closed surface in the region to the right of J. 5 zero. Now consider the field got by superposing the solutions (1) and (— 2) : it will have the following properties ; the potential over AB will be zero and the total normal induction over any closed surface in the region to the right oi AB will vanish. Since the normal induction vanishes over all closed surfaces in this region, there will in the field correspond- ing to this solution be no charge of electricity. We may regard the region as the inside of a closed surface at zero- potential (bounded by the plane AB and an equipotential surface at an infinite distance): by Art. 18, however, the electric intensity must vanish throughout this region as there is no charge inside it. Thus the electric intensity in the field corresponding to the superposition of the solutions (1) and (— 2) is zero : that is, the electric intensity in the solution (1) is equal and opposite to that 140 ELECTRICAL IMAGES AND INVERSION. [CH. V in (— 2), but the electric intensity in (—2) is equal and opposite to that in (2). Hence the electric intensity in (1) is at all points the same as (2), in other words, the solutions give identical electric fields. Hence if we get in any way a solution satisfying the conditions (a) and (^) it must be the only solution of the problem. 83. Take a point P' on the prolongation of the per- pendicular PN let fall from P on the plane, such that P'N = PN, and place at P' a charge equal to — e. Con- sider the properties, in the region to the right of AB, of the field due to the charge e at P and the charge —e at P'. The potential due to — e at P' and + e at P at a point Q on the plane AB is equal to but since AB bisects PP' at right angles PQ = P'Q, thus the potential at Q vanishes. Again, any closed surface drawn in the region to the right of the plane AB does not enclose P\ so that the charge at P' is without effect upon the total induction over any such surface. The total induction over such a surface is zero or 47re according as the closed surface does not or does include P. In the region to the right of AB the electric field due to e at P and — e at P' thus satisfies the conditions (a) and (/S) and therefore represents the state of the electric field. Thus the electrical effect of the electricity induced on the conducting plane AB will for points to the right oi AB be the same as that of the charge — e at P'. This charge at P' is called the electrical image of the charge P in the plane. 83] ELECTRICAL IMAGES AND INVERSION. 141 The attraction on P towards the plane will be the same as the attraction between the charges e at P, and — e at P', that is e" 1 ^ {2PNY~ 4>~PN~^' Thus the attraction on P varies inversely as the square of the distance of the charged body from the plane. To find the surface density of the electricity induced on the plane AB we require the electric intensity at right angles to the plane. The electric intensity at right angles to the plane J. 5 at a point Q due to the charge e at P is equal to e PJ[ PQ^'PQ' and acts from right to left. The electric intensity due to — e at P' in the same direction is e FN P'Q' ' P'Q ' hence since PQ = P'Q and PN^P'N the resultant normal electric intensity at Q _ 2ePN This by Coulomb's law is equal to 47ro-, if a is the surface density of the electricity at Q, hence - ± ?K "" 27rP(^' or the surface density varies inversely as the cube of the distance from P. The total charge of electricity on the plane is — e, as all the tubes which start from P end on the plane. 142 ELECTRICAL IMAGES AND INVERSION. [CH. V The electrical energy is equal to JS^F, so that if the small body at P is a sphere radius a, the energy in the field is equal to le^_l ^ The dielectric in this case is supposed to be air. The electric intensity vanishes in the region to the left of AB. 84. Electrical Images for spherical conductors. In applying the method of images to spherical conductors we make great use of the following theorem due to Apol- lonius. If >S> is a point on a sphere whose centre is and radius a, and P and Q two fixed points on a straight line passing through 0, such that OP . (g = a^ then QSjPS is constant wherever 8 may be on the sphere. For consider the triangles QOS, POS. Since OQ OS OQ.OP = OS', OS OP' Fig. 43. hence these triangles have the angle at common and the sides about this angle proportional; they are therefore similar triangles, so that QS_PS OQ'OS' 85] ELECTRICAL IMAGES AND INVERSION. 143 QS_OQ_qS ^^ PS~OS OP' Hence QS/PS is constant whatever may be the position of ,Sf. .85. Now suppose that we have a spherical shell (Fig. 43) at potential zero whose centre is at and that a small body with a charge e of electricity is placed at P and we wish to find the electric field outside the sphere. There is no field inside the sphere, as the sphere is an equi- potential surface with no charge inside it. Let OP =/, OS = a. Consider the field due to a charge e at P, and e' at Q where OQ . OP = a\ The potential at a point S on the sphere due to this field is but by Art. 84, PS'^QS' QS^PS.j thus the potential Sit 8 = \e + e-l 1 PS' Hence if e = — eajf, the potential is zero over the surface. Thus under these circumstances the field satisfies condition (a) of Art. 82, and it obviously satisfies the condition that the total normal induction over any closed surface not enclosing the sphere is zero or ^ire according as the surface does not or does enclose P, so that by Art. 82 this is the actual field due to the sphere and charged body. Hence the effect at a point outside the sphere of the electricity induced on the sphere by the charge at P 144 ELECTRICAL IMAGES AND INVERSION. [CH. V is the same as that of a charge - ea/f at Q. This charge at Q is called the electrical image of P in the sphere. Since this charge produces the same effect as the electri- fication on the sphere the total charge on the sphere must equal the charge at Q, i.e. it must be equal to — ea/f (compare Art. 30). So that of the Faraday tubes which start from P the fraction a// falls on the sphere. The force on P is an attraction towards the sphere and is equal to a e^ a ^ _a e^ e^fa ■ ~fPQ'^ /{OP^^roQY ~J lf_ ^Y - O - <^'f ■ We see from this result that when the distance of P from the centre of the sphere is large compared with the radius the force varies inversely as the cube of the distance from the centre of the sphere: while when P is close to the surface of the sphere the force varies inversely as the square of the distance from the nearest point on the surface of the sphere. When P is very near to the surface of the sphere the problem becomes identical with that of a charge placed in front of a plane at potential zero. We shall leave it as an exercise for the student to deduce the solution for the plane as the limit of that of the sphere. If the body at P is a small sphere of radius 6, then since the electric energy is equal to ^EV, it is in this case 1 |e ea \ \'\h~~fpq = ^e" 1 a 2 [h p - a' 86] ELECTRICAL IMAGES AND INVERSION. 145 86. To find the surface density at a point S on the surface of the sphere, we must find the electric intensity along the normal. The electric intensity at S due to the charge e at P can by the triangle of forces be resolved into the two com- ponents ^"^ TS' PS ^^^^^ ^^' p OP (/3) ;^,^ along PO, while the electric intensity due to the charge —ea/fsit Q can be resolved into the components Hence the components of the resultant intensity are a 4- 7 along the normal OS, and 13 + B along PO. Now the resultant intensity is along the normal, so that the component /B -{-B must vanish, and the resultant intensity along the normal must be equal to a + 7, i.e. to e . Ob |p^^^3 ^ ^j^^^3 Him- PS' fQS' e^S PS' Since PS/QS is constant the quantity inside the bracket is constant. T. E. 10 47ro-=: .^ 1 146 ELECTRICAL IMAGES AND INVERSION. [CH. V If o- is the surface density of the electrification at S, then by Coulomb's law FS' [ f\QSJ\ FS'i «'r so that the surface density of the electrification varies inversely as the cube of the distance from P, and is, since yis greater than a, everywhere negative. 87. If the sphere is insulated instead of being at zero- potential, the conditions are that the potential over the sphere should be constant and that the charge on the sphere should be zero. The charge on the sphere in the last case was — ea/f. Hence if we superpose on the last solution the field due to a quantity of electricity equal to ea/f placed at the centre of the sphere, which will give rise to a uniform potential over the sphere, the resulting field will have the following properties; (1) the potential over the sphere is constant, (2) the total charge on the sphere is zero, (8) the total normal induction over any closed surface is equal to Atire if the surface encloses P and is zero if it does not; hence it is the solution in the region outside the sphere when a charge e placed at P in front of an insulated conducting sphere. Thus out- side the insulated sphere the electric field is the same as that due to the three charges, e at P, — ea/f at Q, ea/f at 0. Let us consider the potential of the sphere : the charges at P and Q together produce zero-potential over the sphere, so that the potential will be that due to the charge ea/f at ; this charge produces at any point on the sphere a potential equal to e/f so that by the presence of P the potential of the sphere is raised by e/f This result was proved by a different method in Art. 29. 87] ELECTRICAL IMAGES AND INVERSION. 147 The force on P in this case is an attraction equal to PQ'f f-f e^ a f f 1 / [{f-o^r f- _eKa? I f - g' SO that in this case when y is very large compared with a the force varies inversely as the fifth power of the distance. When the point is very close to the surface of the sphere the force is the same as if the sphere were at zero-po- tential. The potential energy, ^^EV is, if the body at P is a small sphere of radius 6, equal to 1 ie ea ea 1 {e ea^ 2 [b fHp-a^) The surface density at S will have on the value given in Art. 86 thus 6(1 the uniform density ^^~- — - superposed, •— S(5-')*.> "' At R the point on the sphere nearest to P, 10—2 148 ELECTRICAL IMAGES AND INVERSION. [CH. V SO that the surface density is equal to 47ral(/-ay / e (3/- g) 47r/(/-ay At J?' the point on the sphere most remote from P, and the surface density is equal to e_ (3/+ a) ^-rr fif+ay Since the total charge on the sphere is zero the surface density of the electricity must be negative on one part of the sphere, positive on another part. The two parts will be separated by a line on the sphere along which there is no electrification. To find the position of this line put a- equal to zero in equation (1), we get if S is a point on this line P,S3 = (/^-o,V=/^f/-^" / = OP' X PQ, hence the points at which the electrification vanishes will be at a distance {OP'' x PQf from P. The parts of the surface of the sphere whose distances from P are less than this value are charged with electricity of the opposite sign to that at P, the other parts of the sphere are charged with electricity of the same sign as that at P. 88] ELECTRICAL IMAGES AND INVERSION. 149 88. If the sphere instead of being insulated and with- out charge is insulated and has a charge E, we can deduce the solution by superposing on the field discussed in Art. 87 that due to a charge E uniformly distributed over the surface of the sphere ; this at a point outside the sphere is the same as that due to a charge E at 0. So that the field in this case outside the sphere is the same as that due to charges ^+^atO, -yatQ, eat P. The repulsive force acting on P is equal to {e-\- ea\ e e^a fir f^pQ" _Ee _e^ (2/^-a^) When the point is very near the sphere we may put /= a + w, where x is small, then the repulsion is approxi- mately equal to Ee_^ a^ 4a-2 ' and this is negative, i.e. the force is attractive unless Thus when the charges are given, and when P gets within a certain distance of the sphere, P will be attracted towards the sphere even though the sphere is charged with electricity of the same sign as that on P. When we recede from the sphere we reach a place where the attrac- tion changes to repulsion, and at this point there is no force on P. So that if P is placed at this point it will be in 150 ELECTRICAL IMAGES AND INVERSION. [CH. V equilibrium. The equilibrium will however be unstable, for if we displace P towards the sphere the force on it becomes attractive and so tends to bring P still nearer to the sphere, that is to increase its displacement, while if we displace P away from the sphere the force on it becomes repulsive and tends to push P still further away from the sphere, thus again increasing the displacemeut. This is an example of a more general theorem due to Earnshaw that no charged body (whether charged by induction or other- wise) can under the influence of electric forces alone be in stable equilibrium in the electrostatic field. 89. If the potential of the sphere is given instead of the charge, we can still use a similar method to find the field round the sphere. Thus if the potential of the sphere is F, then the effect outside the sphere is the same as that due to a charge Va at 0, — eajf at Q, and e at P. 90. Sphere placed in a uniform field. As the point P moves further and further away from the Faraday tubes due to the charge at P get to be in the neighbourhood of the sphere more and more nearly parallel to OP, thus when P is at a very great distance from the sphere the problems we have just considered become in the limit problems relating to the distribution of electricity on a sphere placed in a uniform electric field. Suppose that as the charged body P travels away from the sphere the charge e increases, in such a way that the electric intensity at the centre of the sphere due to this charge remains finite and equal to F, we have thus 91] ELECTRICAL IMAGES AND INVERSION. 151 Now consider the problem of an insulated sphere without charge placed in this uniform field. We see by Art. 87 that the electrification on the sphere produces the same effect at points outside the sphere as would be pro- duced by two charges, one equal to eajf placed at the centre 0, the other equal to - ea//at Qthe image of P. If we express these charges in terms of P we see that they are equal respectively to ±Faf; when/ is infinite they are also infinite. Since OQ=a^/f the distance between these charges diminishes indefinitely as / increases, and we see that the product of either of the charges into the distance between them is equal to Fa^ and is finite. The electrification over the surface of the sphere when placed in a uniform field produces the same effect therefore as an electrical system consisting of two oppositely charged bodies, placed at a very short distance apart, the charges on the bodies being equal in magnitude and so large that the product of either of the charges into the distance between them is finite. Such a system is called an electrical doublet and the product of either of the charges into the distance between them is called the moment of the doublet. 91. Electric Field due to a doublet. Let A, B be the two charged bodies, let e be the charge dX A,—e 152 ELECTRICAL IMAGES AND INVERSION. [CH. V that at B\ let be the middle point of AB, M the moment of the doublet. Let (7 be a point at which the electric intensity is required, and let the angle A0G = 6. The intensity at right angles to OC is equal to ^2 sin AGO+^^ sin BCO = ^,AOsmO + -^^BOsme approximately since ^0 is very small compared with 0(7, _ if sin (9 The intensity in the direction 00 due to the doublet is equal to 6 6 -r-^ COS AGO — -7T7^„ cos BOO, AO^ BO^ but we have approximately AO=0O-A0cosd, BO = 00 -^ BO cos 6. Hence the electric intensity along 00 is approximately e f^ 2A0 .\ e /, 250 cos (9\ ^.(l + ^^cos^j-^,^1 -^-j 2e AB cos (9 00' 2McosO 00' 92] ELECTRICAL IMAGES AND INVERSION. 153 92. Let us now return to the case of the sphere placed in the uniform field : the moment of the doublet which represents the effect of the electrification over the sphere is Fa^. Hence when the sphere is placed in a uniform field F parallel to PO, the intensity at a point G is the resultant of electric intensities, F parallel to PO, FaHmOjOG^ at right angles to OG, and 2Fa^ cos 6/ OG^ along GO ; denotes the angle POG. At the surface of the sphere where OG = a, the result- ant intensity along the outward drawn normal is -i^cos^-2i^cos^, or - SF cos 6 ; but by Coulomb's law, if a is the surface density of the electrification on the sphere, 47ro- = -3i^cos6>, 3 or a' = '--j—Fcosd. 4f7r Hence we see that when an insulated conducting sphere is placed in a uniform field the surface density at any point on the sphere is proportional to the distance of that point from a plane through the centre of the sphere at right angles to the electric intensity in the uniform field. The concentration of the Faraday tubes on the sphere produces a field where the maximum intensity is three times the intensity in the uniform field. 93. We have hitherto supposed the electrified body to be outside the sphere, but we can apply the same method when it is inside. Thus if we have a charge e 154 ELECTRICAL IMAGES AND INVERSION. [CH. V at a point Q inside a spherical surface maintained at zero potential then the effect of the electricity induced on the Fig. 45. sphere will, inside the sphere, be the same as that due to a charge - e . a/OQ at Q where OP . OQ = a\ The charge on the sphere is — e, since all the tubes which start from Q end on the sphere. If the sphere is insulated, then the charge on the inside of the sphere and the force inside are the same as when it was at potential zero; the only difference is that on the outside of the sphere there is a charge equal to e uniformly distributed over the sphere, and the field outside is the same as that due to a charge e at the centre. Again, if there is a charge E on the sphere, the effect inside is the same as in the two previous cases, only now there is a charge E-\-e uniformly distributed over the surface of the sphere raising its potential to {E + e)la. In all these cases the surface density of the electrifica- tion at any point on the inner surface of the sphere varies inversely as the cube of the distance of that point from P. 94] ELECTKICAL IMAGES AND INVERSION. 155 94. Case of two spheres intersecting at right angles and maintained at unit potential. Let the figure represent the section of the spheres, A and B being their centres, and G a point on the circle in which they Fig. 46. intersect, CD a part of the chord common to the two circles, then since the spheres intersect at right angles AOB is a right angle and CD is the perpendicular let fall from C on AB. Then we have by Geometry AD,AB = AC\ DB.AB = BG\ Thus D and B are inverse points with regard to the sphere with centre A, and A and D are inverse points with regard to the sphere whose centre is B. Let ^C = a, BG = 6, then GD .AB = AG .BG, so that ab GD = J a' + ¥ 156 ELECTRICAL IMAGES AND INVERSION. [CH. V Consider the effect of putting a positive charge at A numerically equal to the radius AC, s. positive charge at B equal to BC, and a negative charge at D equal to CD. The charges at A and D will together, by Art. 85, produce zero potential over the sphere with centre B. For A and D are inverse points with respect to this sphere, and the charge at D is to the charge at A as — CD is to AC, i.e. as —BC is to AB, so that the ratio of the charges is the same as that of those on a point and its image, which together produce zero potential at the sphere. Thus the value of the potential over the surface of this sphere is that due to the charge at B, but the charge is equal to the radius of the sphere, so that the potential at the surface being equal to the charge divided by the radius is equal to unity. Thus these three charges produce unit potential over the sphere with centre B ; we can in a similar way show that they give unit potential over the sphere with centre A. The two spheres then are an equipotential surface for the three charges, and the electric effect of the conductor formed by the two spheres when maintained at unit po- tential is at a point outside the sphere the same as that due to the three charges. Capacity of the system. The charge on the system is equal to the sum of the charges on the points inside it which produce the same effect, thus the charge on the system which, since the potential is unity, is equal to the capacity is equal to ah a + h J a'' -4- ¥ 95] ELECTRICAL IMAGES AND INVERSION. 157 95. If b is very small compared with a, the system becomes a small hemispherical boss on a large sphere as shewn in Fig. 47. The capacity is equal to a-\-h- . J a' + b' (, b bf, b'- or ail -h 1 + ( a a\ and as in this case, b/a is very small, the capacity is approximately equal to ( b b (^ U^ a\\ + 1 -- — , 1 6^ _ volume of boss 2 a' volume of big sphere * Thus we have, since a is the capacity of the large sphere without the boss, increase in capacity due to boss _ volume of boss capacity of sphere ~ volume of sphere * 158 ELECTRICAL IMAGES AND INVERSION. [CH. V 96. To compare the charges on the surface of the two spheres. The charge on the sphere EFG (Fig. 46) is by Coulomb's law equal to l/47r of the total normal induction over EFG. Now the total normal induction is the sum of the total normal inductions due to the charges at J. , J?, D. Since B is the centre of the sphere CFE the total normal induction due to ^ over CFE bears the same ratio to 47r6 (the total normal intensity over the whole sphere) as the area of the surface CFE does to the area of the sphere. But the area of the surface of a sphere included between two parallel planes is proportional to the distance between the planes, thus areaof J^i^C ^ h + BD area of sphere 26 ' hence the total normal induction due to the charge at B over CFE = 27r{b + BD). The total normal induction due to the charge A over the closed surface CFEL is zero, therefore the total normal induction due to A over CFE is equal in magnitude and opposite in sign to the total normal induction over CLE, that is, it is equal to the total normal induction over CLE reckoned outwards from the side J.. But CLE is a portion of a sphere of which A is the centre, therefore the induc- tion over CLE is to 47ra the induction over the whole sphere with centre A, as the area of CLE is to the area of the sphere, that is as DL : 2a. Thus the induction due to A over CFE is equal to 27rDL. Now consider the induction over CFE due to the charge at i) : it is by the same reasoning as above equal 97] ELECTRICAL IMAGES AND INVERSION. 159 to the induction over CLE measured outwards from D. Now of the tubes starting from B as many would go to the right as to the left if it were alone in the field, so that the induction over CLE will be half that due to D over a closed surface entirely surrounding it; the latter induction is equal to 47r times the charge at D, i.e. to — 47r . CD, hence the induction due to D over the surface CLE is - 27r . CD. Thus the total induction over CLE due to the three charges is 27r{h^-BD + DL-CD), and the charge on CFE is therefore equal to The charge on CGE can be got by interchanging a and h in this expression, it is thus equal to J a+ y +6- . , - . ^ .-•(2). 97. In the case of a hemispherical boss on a large sphere, h is very small compared with a, in this case when h is small compared with a the expression (1) becomes approximately 6^ /, , 6^ i6+-4-a-a(l-i-)-6 362 4a' 160 ELECTRICAL IMAGES AND INVERSION. [CH. V This is equal to the charge on the boss. The mean density on the boss is this expression divided by 27r¥ the area of the surface of the boss, and is therefore 3 When h/a is very small the expression (2) is approxi- mately equal to a, thus the charge on the sphere is a and the mean density is got by dividing a by 47ra^ the area of the sphere, thus the mean density on the sphere is 1 47ra* Hence the mean density on the boss is to the mean density on the sphere as 3:2. 98. Since a plane may be regarded as a sphere of infinite radius this applies to a hemispherical boss of any radius on a plane surface. It thus applies to the case Fig. 48. shown in the figure. Since the mean density over the boss is 3/2 that over the plane, and the area of the boss is twice the area of its base ; there is three times as much electricity on the part occupied by the boss as there is on the average on an area of the plane equal to the base of the boss. 99] ELECTRICAL IMAGES AND INVERSION. 161 99. When b is very small compared with a, the points B and D, Fig. 46, are close together, the distance between them being approximately h^/a ; the charge on B is b, that on D is ab and this when b is very small compared with a is approximately equal to — b. Thus the charges at B and D form a doublet whose moment is ¥la. The point A is very far away and the force at J5 or Z) due to its charge is 1/a. Thus the moment of the doublet is ¥ times this force. This as far as the sphere is concerned is exactly the case considered in Art. 92. Hence if F is the force at the boss due to the charge A alone, the surface density at or* a point P on the boss is -r— cos 6, where 6 is the angle OP makes with the axis of the doublet. Now if o-q is the surface density on the plane at some distance from the boss F — ^TTo-Q, Hence the surface density at P a point on the boss is equal to ScTo cos 6, where 6 is the angle OP makes with the normal to the plane. The electric intensity at Q a point on the plane due to the doublet is (Art. 91) equal to the moment of the doublet divided by B(^ and is at right angles to the plane, thus the normal electric intensity at Q is and a the surface density at Q is given by the equation T. E. 11 162 ELECTRICAL IMAGES AND INVERSION. [CH. V We have thus found the distribution of electricity on a charged infinite plane with a hemispherical boss on it. 100. In the general case when the two spheres are of any size the surface density on the conductor can be got by calculating the normal electric intensity due to the three charges. We shall leave this as an example for the student, remarking that since the potential of the con- ductor is the highest in the field there can be no negative electrification over the surface and that the electrification vanishes along the intersection of the two spheres. 101. Effect of Dielectrics. We have hitherto only considered the case when the field due to the charge at P was disturbed by the presence of conductors, but by applying the principle that a solution which satisfies the electric conditions is the only solution, we can find the electric field in some simple cases when dielectrics are present. 102. The first case we shall consider is that of a small charged body placed in front of an infinite slab of uniform B Fm. 49. dielectric bounded by a plane face. Let P be the charged body, AB the plane separating the dielectric from air, the medium to the right of AB being air, that to the left a 102] ELECTRICAL IMAGES AND INVERSION. 163 dielectric whose specific inductive capacity is K. From P draw PN perpendicular to AB ; produce PN to P', so that PN=:P'N. Then we shall show that the field to the right o^ AB can be regarded as due to e at P, a charge e at P', and that to the left of J. 5 as due to e" at P. These charges being supposed to produce the same field as if there was nothing but air in the field. In the first place this field satisfies the condition that the potential at an infinite distance is zero, also that the induction over any closed surface surrounding P is 47re, while the induction over any closed surface not enclosing P is zero. This is obvious if the surface is drawn entirely to the left or entirely to the right of AB. If it crosses this plane it can be regarded as two surfaces, one entirely to the left bounded by the portion of the surface to the left and the portion of the plane AB intersected by the surface, the other entirely to the right bounded by the same portion of the plane and the part of the surface to the right. The only other conditions we have to satisfy are that along the plane AB the electric intensity parallel to the surface is the same in the air as in the dielectric, and that over this plane the normal polarization is the same in the air as in the dielectric. At a point Q m AB the electric intensity parallel to AB is in the air e QN e' QN PQ^ PQ'^ P'Q' PQ- This, since PQ = P'Q, is equal to 11—2 164 ELECTRICAL IMAGES AND INVERSION. [CH. V The electric intensity parallel to J.jB in the dielectric is this is equal to that in air if e + e' = e" (1). Again, the polarization at Q at right angles to AB reckoned from right to left is in air 1 , ..pjsr that in the dielectric these are equal if e-e' = Ke" (2). Hence both the conditions are satisfied if e and e" satisfy (1) and (2), i.e. if (^-1) Thus the attraction of P towards the plane is ee' _ K-1 e^ (2Pi\^)2~ir+14Pi\^2' if K is infinite this equals which is the same as when the dielectric to the left of AB is replaced by a conductor. 103] ELECTRICAL IMAGES AND INVERSION. 165 Thus if -K" = 10, as it does for some kinds of heavy- glass, the force on P when placed in front of the glass would be about 9/11 of the attraction when P is placed in front of a conducting plate. Inside the conducting plate the tubes are straight and pass through P; the effect of the dielectric is, while not affecting the direction of the electric intensity, to reduce the magnitude of it to 2/(1 + ^) of its value in air. The lines of force when K=1'7 are shown in Fig. 50. Fig. 50. 103. Case of a dielectric sphere placed in an electric field. We have seen that when a conducting sphere is placed in a uniform field the effect of the electricity induced on the surface of the sphere can be represented at points outside the sphere by a doublet (see Art. 91) placed at the centre of the sphere. Since 166 ELECTRICAL IMAGES AND INVERSION. [CH. V we have seen that the effects of a dielectric are similar in kind though different in degree to those due to a conductor, we are led to try if the disturbance produced by the presence of the sphere cannot be represented at a point outside the sphere by a doublet placed at its centre. With regard to the field inside the sphere we have as a guide the result obtained in the last article, that in the case when the radius of the sphere is infinitely large the field inside the dielectric is not altered in direction but only in magnitude by the dielectric. We therefore try if we can satisfy the conditions which must hold when a sphere is placed in a uniform electric field by supposing the field inside the sphere to be uniform. Let the uniform field before the insertion of the sphere be one where the electric intensity is horizontal and equal to H. After the insertion of the sphere let the field outside consist of this horizontal force plus the field due to a doublet whose moment is M placed at the centre of the sphere. Inside the sphere let the intensity be horizontal and equal to H'. We shall see that it is possible to satisfy the con- ditions of the problem by a proper choice of M and H'. The field due to the doublet is by Art. 91 equivalent at P to an intensity jyp^ cos 6 along OP, and an intensity -^p^ sin 6 at right angles to it where 6 is the angle OP 103] ELECTRICAL IMAGES AND INVERSION. 167 makes with the direction of the uniform electric intensity. Thus at a point Q just outside the sphere the intensity tangential to the sphere is equal to M Hsm6 -sin 6, where a is the radius of the sphere. The intensity in the same direction at a point close to Q but just inside the sphere is The normal intensity at Q outside the sphere is Hcos6-\ — - cos 6, that inside the sphere is H' cos 0, The conditions which must be satisfied are that the tangential intensity at the surface of the sphere must be the same in the air as in the dielectric, this will be true if we have M . Hbir 6 sin =H' sin 0, a? M or H--^ = ir (1). Again, the normal polarization at the surface of the sphere must be the same in the air as in the dielectric, thus -— ^5^cos^+-^cos^^ = :r-H' COS $, 4>7r [ a^ J 47r 9M or H+^=KE (2). 168 ELECTRICAL IMAGES AND INVERSION. [CH. V Equations (1) and (2) will be satisfied, if H' 2 + K and if M= — j^ — —^ a\ Thus since if H' and M have these values the con- ditions are satisfied ; this will be the solution of the problem. We see that the intensity inside the sphere is 3/(2 + K) of that in the original field, so that the intensity of the field is less inside the sphere than out, on the other hand the number of Faraday tubes which pass through unit area inside the sphere is SK/{K + 2) times the number passing through unit area in the uniform field outside. When K is very great SK/(K + 2) is approximately equal to 3, so that the Faraday tubes in this case will be 3 times as dense inside the sphere as they are at a great distance away from it. This illus- trates the crowding of the Faraday tubes to the sphere. The diagram of the lines of force for this case was given in Fig. 39. Method of Inversion. 104. This is a method by which when we have ob- tained the solution of any problem in electrostatics we can by a geometrical process obtain the solution of another. Definition of inverse points. If is a fixed point, P a variable one, then if we take P' on OP, so that OP,OP'=k\ 105] ELECTRICAL IMAGES AND INVERSION. 169 where k is a constant, P' is defined to be the inverse point of P with regard to 0. And is called the centre of inversion, k the radius of inversion. If the point P moves about so as to trace out a surface, then P' will trace out another surface which is called the inverse surface to that traced out by P. We shall now proceed to prove some geometrical pro- positions about inversion. 105. The inverse surface of a sphere is another sphere. Let be the centre of inversion, P a point Fig. 51. on the sphere to be inverted, G the centre of this sphere. Let the chord OP cut the sphere again in P\ let Q be the inverse point to P, Q the inverse point to P\ R the radius of the sphere to be inverted, then OP.OQ = k^ but OP. OF = 00' -R'', thus 0(3 = ^^0P- similarly ' OQ' =^ ^^^—^^OP, 170 ELECTKICAL IMAGES AND INVERSION. [CH. V thus OQ,Oq = ^^^^-^^OP.OP' 1^ Thus OQ bears a constant ratio to OF'; hence the locus of Q is similar to the locus of P\ and is therefore a sphere. Thus a sphere inverts into a sphere. If the sphere inverts into itself. To find the centre of the inverted sphere let the dia- meter OG cut the sphere to be inverted in A and B. Let A', B' be the points inverse to A and B respectively, 0' the centre of the inverted sphere, 00'=\{OA' + OB') ~2[0G-R'^ OG + r) OG OG^-R'' If D is the point where the chord of contact of tangents from to the sphere cuts OG, then ^^ OG ' Hence D inverts into the centre of the sphere. The radius of the inverted sphere = i(OA'-OB') R = k'' OG'-R' 106] ELECTRICAL IMAGES AND INVERSION. 171 106. Since a plane is a particular case of a sphere a plane will invert into a sphere ; this can be proved independently in the following way : Fig. 52. Let AB be the plane to be inverted, P a point on that plane, N the foot of the perpendicular let fall from on the plane, Q and N' the points inverse to P and N respectively; then since OQ.OP = ON' .ON ON' ON OP' thus the two triangles QON\ PON have the angle at common and the sides about this angle proportional, they are therefore similar, and the angle OQN' is equal to the angle ONP. Hence OQN' is a right angle. And the locus of Q is a sphere on ON' as diameter. 172 ELECTRICAL IMAGES AND INVERSION. [CH. V 107. Let be the centre of inversion, PQ two points, P'Q the corresponding inverse points. OQ' ~ OP ' Then Fig. 53. thus the triangles POQ, POQ' are similar, so that PQ^P^ OP oq ' If we have a charge e at Q, and a charge e' at Q', then if Vp is the potential at P due to the charge at Q, and F V the potential at P' due to the charge at Q', ^^- ^ ^'-PQ • FQ'-OP ' OQ" Take e : e' = OQ : A; (1), k then F p. = ^p ^/ • If we have any number of charged bodies at different points and take the inverse of these points and place there charges given by the expression (1) then if Vp be the po- tential at a pointP due to the assemblage of charged bodies, Vp> the potential at P^ (the point inverse to P) due to the charges on the inverted figure, OP'' Vp>=Vp 108] ELECTRICAL IMAGES AND INVERSION. 173 thus if the assemblage of points produces a constant poten- tial V over a surface S, the inverted system will produce a Vk . . potential jyp' ^^ ^ point P' on the inverse of >Si. Hence if we add to the inverted system a charge - A;F at the centre of inversion the potential over the inverse of S will be zero. If the charges on the original system are distributed over a surface instead of being concentrated at a point the charges on the inverted system will also be distributed over a surface. Let a be the surface density at Q, a place on the original system, a' the surface density at Q', the corre- sponding place on the inverted system, a a small area at Q, a the area into which it inverts ; then by (1) (TOL : a' a! = OQ : k and since a and a are similar figures, a : a' = OQ' : OQ'^ .-. a : a' = OQ'' : kOQ kOQ k^ ... ^=^-OQ^=^W^ (^^- This expression gives the surface density of the inverted figure in terms of that at the corresponding point of the original figure. 108. As an example of the use of the method of inversion let us invert the system consisting of a sphere with a uniform distribution of electricity over it, the surface density being F/47ra; where V is the potential and a the radius of the sphere. We know in this case that the potential is constant over the sphere and equal to V. Take the point of inversion outside the sphere and choose the constant of inversion so that the sphere inverts into itself. Then if to the inverted system we add a charge 174 ELECTRICAL IMAGES AND INVERSION. [CH. V — kVsit the origin the inverted system will be at poten- tial zero. By equation (I) or' the surface density at the inverted system at Q' is given by the equation "^ ~ 47ra • Oq' ' if we put e = — A;F, this equals -e k^ ^ -e.{OG'-a') 47ra • OQ'' ~ 4>7ra . OQ' ' where G is the centre of the sphere. Thus a charge e at induces on the sphere at zero potential a distribution of electricity such that the surface density varies inversely as the cube of the distance from 0. Thus in this way we get by inversion the solution of the problem which we solved in Art. 86 by the method of images. 109. As another example illustrating the uses of the method of inversion as well as that of images, let us consider the solution by the method of images of a charged body placed between two infinite conducting planes main- tained at potential zero. Let P be the charged point, AB and CD the two planes at potential zero, e the charge at P. Then if we place a charge —e at P' where P' is the image oi P in AB the potential over AB will be zero, it will not however be zero over CD; to make the potential over CD zero we must place a charge —eatQ, the image of P in CD, and a charge e at Qi, the image of P' in CD. These two charges will however disturb the potential of AB ; to restore zero potential to AB we must introduce a charge +e at Pj the image of Q in AB, and a charge — e at P" the image 109] ELECTRICAL IMAGES AND INVERSION. 175 of Qi in AB. The charges at Pi and P" will disturb the potential over the plane CD ; to restore it to zero we must place a charge — e at Q' the image of Pi in CD and a charge +e at Q2 the image of P'' in CD, and so on ; we get in this way an infinite series of images to the right of AB and to the left of GB. The images to the right of AB are (1) charges — e, at F, P", P'"... ; and (2) charges +e, at Pi, P„P,.... Now P" is the image in AB of Q^ which is the image of P' in CD ; hence FP" = PQi = FE+ EP' = 2FE + FP\ thus FP"-FP' = PT" = 2FE=2c if c is the distance between the plates. D A F p B Fig. 54. Similarly P'P" = P"P"' = . . . = 2c and we can show in a similar way that PPi = P1P2 = P^Pz^ . . . = 2c. Thus on the right of ^-B we have an infinite series of charges equal to — e at the distance 2c apart, beginning at P the image of 176 ELECTRICAL IMAGES AND INVERSION. [CH. V P in AB, and a series of positive images at the same dis- tance 2c apart, beginning at Pj, a point distant 2c from P. Similarly to the left of CD we have an infinite series of images with the charge — e at the distance 2c apart, beginning at Q, the image of P in CD, and an infinite series of images each with the charge + e, at points at a distance 2c apart, beginning at Qi a point distant 2c from P. Now invert this system with respect to P. The two planes invert into two spheres touching each other at P, and maintained at a potential — e/k, the images to the right oi AB invert into a series of charged points inside the sphere to the right of P, the images to the left of CD invert into a system of charged points inside the sphere to the left of P. The system of charged points inside the spheres will produce a constant potential — e/k over the surface of the spheres, and therefore at a point outside the spheres the electric field due to the two spheres in contact will be the same as that due to the system of the electrified points. If a, h are the radii of the spheres into which the planes AB, CD invert, and if PF= d. Then k^ , (k^ ¥ 26 = — 3 ' c — d Consider now the series of images to the right of AB. The series of positive charges at the distance 2c apart invert into a series of charges inside the sphere whose radius is a, of magnitude ek ek ek 2c' 4c' '^c '" ' 109] ELECTRICAL IMAGES AND INVERSION. 177 since charge on inverted system charge on original system k ~ distance of original system from centre of inversion * The series of negative images at the distance 2c apart invert into a series of negative charges eh ek ek ~2d' ~ 2c + 2cZ ' ~4c + 2c?"*' Similarly inside the sphere into which the plane GB inverts we have a series of positive charges ek ek ek 2^' 4^' Qc'"' and a series of negative ones ek ek ek ~ 1{c-dy ~'ic^^d' ~6c-2d' "* Thus ^1 the sum of the charges on the points inside the first sphere is given by the equation ^-^^{(i^c+^c-- the charge E2 inside the second sphere is given by the equation + :cAir. + -]\ (2). ,2c-2d 4c-2d T. E. 12 178 ELECTRICAL IMAGES AND INVERSION. [CH. V Rearranging the terms we may write E--lek[^---^ ^ ^—- \ '~ 2 (d c{c-\-d) 2c{2c-^d) Sc(Sc + d) '"y _1^JJ^ 1 1 I ^'~ 2^" c jc-cZ"^2(2c-rf)'^3(3c-(^)"^-j • Expanding the expressions for E^ and E^ in powers of djc we get 1 , /I d r, d' d' ^'=-I^K5-'^^^4^-^^-) (3)' ^ 1 , d f ci d CI d^ CI d^ n 2 c-\ c C c^ o 1 1 1 1 where ^n=j^+^+^n + ^i+"' The values of 8n are given in De Morgan's Differential and Integral Calculus, p. 554, ^2 = 1-645, S,= l'0S7, ^3 = 1-202, >Sf6= 1-017, ^4=1-082, Sj = 1-008. Since E^^ can be got from E^ by writing c — dfovd we get E., = -lek^-^{s, + '-^8. + ^^S....) (4). Now the total charge spread over the surface of the first sphere is equal to the sum of the charges on the images inside the sphere as these produce the same effect at external points as the electrification over the surface of the sphere: thus Ej^ will be the charge on the first sphere, E^ that on the second. If V is the potential of the spheres 109] ELECTRICAL IMAGES AND INVERSION. 179 Substituting for e, c, d their values in terms of V, a, b we get from (1), (2), (3) and (4) E,=:V (^+2^ + 3^ + 4^ + -) A a b a b a b / 1 111 l^lv' + 2 + 3-^4 + a b (-5), i\ b a b a b a / -rT(i+i+l+i+-)t ^'^' Let us now consider some special cases. The first case we shall take is when a = b', then from equation (5) = Va log 2, the logarithm being the Napierian logarithm. Since log 2 = -693 ^, = •693 Fa; the charge on the second sphere is also E^; thus the charge on the two spheres is 1-386 Fa. 12-^2 180 ELECTRICAL IMAGES AND INVERSION. [CH. V When V=l the charge on the two spheres is equal to the capacity of the system; hence the capacity of two equal spheres in contact is l'386a. If the spheres had been a great distance apart the capacity of the two would have been 2a; if there had only been one sphere the capacity would have been a. We can find from this the work done on an uncharged sphere when it moves under the attraction of a charged sphere of equal radius from an infinite distance into con- tact with the charged sphere. Let a be the radius of the spheres, e the charge on the charged sphere ; then when the spheres are at an infinite distance apart the poten- tial energy is e^/2a, when the spheres are in contact the potential energy is e'^/2 x l"386a, hence the work done by the electric field when the uncharged sphere falls from an infinite distance into contact with the charged sphere is 2a\ 1-3^ , - 14 1-386J If one sphere had a charge E, the other the charge e, then when they are an infinite distance apart, the potential energy is 2^{^' + ^')- When the spheres are in contact the potential energy Hence the potential energy is less in the second case than the first by 109] ELECTRICAL IMAGES AND INVERSION. 181 li F=e this is equal to This is the work required to push the spheres together against the repulsions exerted by their like charges. The expression (9) vanishes when E/e is approximately 5 or 1/5; in this case the potential energy is the same when the spheres are in contact as when they are an infinite distance apart; thus no work is spent or gained in bringing them together. The attraction due to the induced electrification on the average balances the re- pulsion due to the like charges. The next case we shall consider is where one sphere is very large compared with the other. Let h be very large compared with a. Then by (8) we have approximately or approximately, ^1 = -^^'. = V = 1-645 b ^' ~b 6 b Interchanging a and b in (7) we get approximately 182 ELECTRICAL IMAGES AND INVERSION. [CH. V or approximately = 7(5-5 f) = F(6-1.645f). The mean surface density over the small sphere is _Zl^==JLZ-^=X 1-645 47ra2 47r 6 6 4>7rb The mean surface density over the large sphere is approximately E, ^1 V 47r62 47r h ' hence the mean surface density on the small sphere is TT^B or 1*645 times that on the large sphere. We saw in Art. 97 that when a small hemisphere was placed on a large sphere the mean density on the hemisphere was 1*5 times that on the sphere. Since a plane may be regarded as a sphere of infinite radius we see that if a sphere of any size is placed on a conducting plane the mean surface density of the elec- tricity on the sphere is ir^jQ of that on the plane. We have z=Vh\l + 2-404 y^ I approximately. Thus the capacity of the system of two spheres is approximately 6|l + 2-404p 109] ELECTRICAL IMAGES AND INVERSION. 183 We have thus Increase of capacity due to small sphere Capacity of large sphere — 9'/ia± ^^1^^® ^f small sphere " volume of large sphere * Thus in this case as in that discussed in Art. 95 the increase of capacity due to the small sphere is proportional to the volume of the sphere. From this result we can deduce the work done on a small uncharged sphere of radius a when it moves from an infinite distance up to a large sphere of radius b with a charge E. For the potential energy when they are at an infinite distance apart is equal to 2 b ' when the spheres are in contact the potential energy is 1 E^ 2 , f. w 6|l + 2-404g| The work done on the small sphere is the difference between these expressions, or approximately, En'202^. b^ CHAPTER VL MAGNETISM. 110. A MINERAL called ' lodestone ' or magnetic oxide of iron, which is a compound of iron and oxygen, is often found in a state in which it possesses the power of at- tracting small pieces of iron such as iron filings ; if the lodestone is dipped into a mass of iron filings and then withdrawn, some of the iron filings will cling to the lode- stone, collecting in tufts over its surface. The behaviour of the lodestone is thus in some respects analogous to that of the rubbed sealing-wax in the experiment described in Art. 1. There are however many well-marked differences between the two cases; thus the rubbed sealing-wax attracts all light bodies indifferently, while the lodestone does not show any appreciable attraction for anything except iron and, to a much smaller extent, nickel and cobalt. If a long steel needle is stroked with a piece of lode- stone, it will acquire the power possessed by the lodestone of attracting iron filings; in this case the iron filings will congregate chiefly at two places, one at each end of the needle, which are called the poles of the needle. The piece of lodestone and the needle are said to be magnetized ; the attraction of the iron filings is an example of a large class of phenomena known as magnetic. Bodies which exhibit the properties of the lodestone or the needle CH. VI. Ill] MAGNETISM. 185 are called magnets, and the region around them is called the magnetic field. The property of the lodestone was known to the ancients, and is frequently referred to by Pliny and Lucretius. The science of Magnetism is indeed one of the oldest of the sciences and attained considerable develop- ment long before the closely allied science of Electricity : this was chiefly due to Gilbert of Colchester, who in his work De Magnete published in 1600 laid down in an admirable manner the cardinal principles of the science. 111. Forces between magnets. If we take a needle which has been stroked by a lodestone and suspend it by a thread attached to its centre it will set itself so as to point in a direction which is not very far from north and south. Let us call the end of the needle which points to the north, the north end, that which points to the south, the south end, and let us when the needle is suspended mark the end which is to the north ; let us take another needle, rub it with the lodestone, suspend it by its centre and again mark the end which goes to the north. Now bring the needles together; they will be found to exert forces on each other, and the two ends of a needle will be found to possess sharply contrasted properties. Thus if we place the magnets so that the two marked ends are close together while their unmarked ends are at a much greater distance apart, the marked ends will be repelled from each other ; again, if we place the magnets so that the two unmarked ends are close together while the marked ends are at a much greater distance apart, the unmarked ends will be found to be repelled from each other ; while if we place the two magnets so that the 186 MAGNETISM. [CH. VI marked end of one is close to the unmarked end of the other, while the other ends are much further apart, the two ends which are near each other will be found to be attracted towards each other. We see then that poles of the same kind are repelled from each other, while poles of opposite kinds are attracted towards each other. Thus the two ends of a magnet possess properties analogous to those shown by the two kinds of electricity. 112. We shall find it conduces to brevity in the statement of the laws of magnetism to introduce the term charge of magnetism, and to express the property possessed by the ends of the needles in the preceding experiment by saying that they are charged with magnetism, one end of the needle being charged with positive magnetism, the other end with negative. We regard the end of the needle which points to the north as having a charge of positive magnetism, the end which points to the south as having a charge of negative magnetism. It will be seen from the preceding experiment that two charges of magnetism are repelled from or attracted towards each according as the two charges are of the same or opposite signs. It must be distinctly understood that this method of regarding the magnets and the magnetic field is only introduced as affording a convenient method of describing briefly the phenomena in that field and not as having any significance with respect to the constitution of magnets or the mechanism by which the forces are produced : we saw for example that the same terminology afforded a convenient method of describing the electric field, though we ascribe the action in that field to effects taking place in the dielectric between the charged bodies rather than in the charged bodies themselves. 114] MAGNETISM. 187 113. Unit Charge of Magnetism, often called pole of unit strength. Take two very long, thin, uniformly magnetized needles, equal to each other in every respect (we can test the equality of their magnetic properties by observing the forces they exert on a third magnet), let A be one end of one of the magnets, B the like end of the other magnet, place A and B at unit distance apart in air, the other ends of the magnets being so far away that they exert no appreciable effect in the region about A and B : then each of the ends A and B is said to have a unit charge of magnetism or to be a pole of unit strength when A is repelled from B with the unit force. If the units of length, mass and time are re- spectively the centimetre, gramme and second the force between the unit poles is one d3Tie. A charge of magnetism equal to 2, or a pole of strength 2, is one which would be repelled with the force of two dynes from unit charge placed at unit distance in air. If m and mf are the charges on two ends of two magnets (or the strengths of the two poles), the distance between the charges being the unit distance, the repulsion between the charges is mm' dynes. If the charges are of opposite signs mm' is negative : we interpret a negative repulsion to mean an attraction. 114. Coulomb by means of the torsion balance suc- ceeded in proving that the repulsion between like charges of magnetism varied inversely as the square of the dis- tance between them. We shall discuss in Art. 131 a more delicate and convenient method of proving this result. Since the forces between charges of magnetism obey 188 MAGNETISM. [CH. VI the same laws as those between electric charges we can apply to the magnetic field the theorems which we proved in Chap. ii. for the electric field. 115. The Magnetic Force at any point is the force which would act on unit charge if placed at this point, the introduction of this charge being supposed not to influence the magnets in the field. 116. Magnetic Potential. The magnetic potential at a point P is the work which would be done on unit charge by the magnetic forces if it were taken from P to an infinite distance. We can prove as in Art. 17 that the magnetic potential due to a charge m at a distance r from the charge is equal to mjr. 117. The total charge of Magnetism on any magnet is zero. This is proved by the fact that if a magnet is placed in a uniform field the resultant force upon it vanishes. The earth itself is a magnet and produces a magnetic field which may be regarded as uniform over a space enclosed by the room in which the experiments are made. To show the absence of any horizontal resultant force on a magnet, we may mount the magnet on a piece of wood and let this float on a basin of water, then though the magnet will set so as to point in a definite direction, there will be no tendency for the magnet to move towards one side of the basin. There is a couple acting on the magnet tending to twist it so that the magnet sets in the direction of the magnetic force in the field, but there is no resultant horizontal force on the magnet. The absence of any vertical force is shown by the fact that the process of magnetization has no influence upon the 118] MAGNETISM. 189 weight of a body. Either of these results shows that the total charge on the body is zero. For let rrii, m^, m^ &c. be the magnetic charges on the body, F the external magnetic force, then the total force acting on the body in the direction of F is l^Fm. This, since the field is uniform, is equal to FXm. As this vanishes Sm = 0, i.e. the total charge on the body is zero. Hence on any magnet the positive charge is always equal to the negative one. When considering electric phenomena we saw that it was impossible to get a charge of positive electricity with- out at the same time getting an equal charge of negative electricity. It is also impossible to get a charge of posi- tive magnetism without at the same time getting an equal charge of negative magnetism ; but whereas in the electrical case all the positive electricity might be on one body and all the negative on another, in the magnetic case if a charge of positive magnetism appears on a body an equal charge of negative magnetism must appear on the same body. This difference between the two cases would disappear if we regarded the dielectric in the electrical case as analogous to the magnets; the various charged bodies in the electrical field being regarded as portions of the surface of the dielectric. 118. Poles of a Magnet. In the case of very long and thin uniformly magnetized pieces of iron and steel we approximate to a state of things in which the magnetic charges can be regarded as concentrated at the ends of the magnet, which are then called its poles ; the positive 190 MAGNETISM. [CH. VI magnetism being concentrated at the end which points to the north, which is called the positive pole, the negative charge at the other end, called the negative pole. In general however the magnetic charges are not localized to such an extent as in the previous case, they exist more or less over the whole surface of the magnet ; to meet these cases we require a more extended definition of ' the pole of a magnet.' Suppose the magnet placed in a uniform field, then the forces acting on the positive charges will be a series of parallel forces all acting in the same direction, these by statics may be replaced by a single force acting at a point P called the centre of parallel forces for this system of forces. This point P is called the positive pole of the magnet. Similarly the forces acting on the negative charges may be replaced by a single force acting at a point Q. This point Q is then called the negative pole of the magnet. The resultant force acting at P is by statics the same as if the whole positive charge were concentrated at P; this resultant is equal and opposite to that acting at Q. 119. Axis of a Magnet. The axis of a magnet is the direction of the line joining its poles, the line being drawn from the negative to the positive pole. 120. Magnetic Moment of a Magnet is the pro- duct of the charge of positive magnetism multiplied by the distance between the poles. It is thus equal to the couple acting on the magnet when placed in a uniform magnetic field where the intensity of the magnetic force is unity, the axis of the magnet being at right angles to the direction of the magnetic force in the uniform field. MAGNETISM. 191 121. The Intensity of Magnetization is the mag- netic moment of a magnet per unit volume. It is to be regarded as having direction as well as magnitude, its direction being that of the axis of the magnet. 122. Magnetic Potential due to a Small Mag- net. Let A and B, Fig. 55, represent the poles of a small magnet, m the charge of magnetism Sit B, —m that Fig. 55. at A. Let be the middle point of AB. Consider the magnetic potential at P due to the magnet AB. The 772< magnetic potential at P due to m at -S is ^p , that due m to — m at J. is — ^p , hence the magnetic potential at P due to the magnet is m m WAP' From A and B let fall perpendiculars AM and BN on OP: since the angles BFO, APO are very small and the angles at M and N are right angles, the angles 192 MAGNETISM. [CH. VI PBN and PAM will be very nearly right angles, so that approximately BP = PN=PO-ON, AP = PM=PO-^OM = PO-{- OK Then BP AP PO-ON PO + ON 2m. ON ~0F- ON' ' and this, since ON is very small compared with OP, is approximately equal to 2m. ON OP' mAB cos 6 OP' ' where 6 is the angle ^POB. If if ie 3 the magnetic moment of the M = mAB, magnet hence the potential due to the magnet is equal to M cos 6 OP' ' 123. Resolution of Small Magnets^ We shall first prove that the moment of a small magnet may be resolved like a force, i.e. if the moment of the magnet is M, and if a force M acting along the axis of the magnet be resolved into forces mj, m^, m^, &c. acting in directions OL^, OL^, OL^, &c., where is the point midway between the poles, then the magnetic action of the original magnet at a distant point is the same as the combined effects of the magnets whose moments are Mj, M^, M^, &c., and whose axes are along OXi, OL^, 0L„ &c. 124] MAGNETISM. 193 Now suppose a force M in the direction AB is the resultant of the forces M^, Mc^, M^ in the directions OB^, 0B„ OB,, &c., let 0B„ OB^, OB^ make angles 6^, 6,, 6, with OP, then if cos ^ = i/i cos ^1 + ifa cos 6>2 + ... , if cos _Mi cos 6i M2 cos 0^ and ~aP~"~aP~ OP'^ "^•••* Now ifi cos Oi/OP^ is the magnetic potential at F due to the magnet whose moment is ifi and whose axis is along 0^1, M2 cos OJOP'^ is the potential due to the magnet whose moment is M^ and whose axis is OB^, and so on ; hence we see that the original magnet may be replaced by a series of magnets, the original moment being the resultant of the moments of the magnets by which it is replaced. In other words, the moment of a small magnet may be resolved like a force. By the aid of this theorem the problem of finding the force due to a small magnet at any point may be reduced to that of finding the force due to a magnet at a point on its axis produced, and at a point on a Hue through its centre at right angles to its axis. 124. To find the magnetic force at a point on the axis produced. Let AB, Fig. 56, be the magnet, P the point at which the force is required. The magnetic force at P due to the charge m at B is equal to m {OP-OBf The magnetic force due to —mdXA is equal to m {OP-vOBf T. E. 13 194 MAGNETISM. [CH. VI The resultant magnetic force at P is equal to m m 4>m.0B.0P (OP - OBf (OP + OBf (OP" - OBJ _ 4^mOB . OP OP' approximately, since OB is small compared with OP. Fig. 56. If M is the moment of the magnet M=27nOB, thus the magnetic force at P is equal to 2M OP'' The direction of this force is along OP. 125. To find the magnetic force at a point Q on the line through O at right angles to AB. Since Q is equidistant from A and B, Fig. 56, the forces due to A and B are equal in magnitude ; the one being a 126] MAGNETISM. 195 repulsion, the other an attraction. The resultant of these forces is equal to 2m 0B_ M BQ' ' BQ'BQ^ _ M since BQ is approximately equal to OQ. The direction of this force is parallel to BA and at right angles to OQ. If Q, a point on the line through at right angles to AB, is the same distance from as P, a point on AB produced, we see from these results that the force at P is twice that at Q. This is the foundation of Gauss's method (see Art. 131) of proving that the force between two poles varies inversely as the square of the distance between them. 126. Magnetic force due to a small magnet at any point. Let AB, Fig. 57, represent the small magnet, Fig. 57. let M be its moment, its centre, P the point at which the force is required, let OP make an angle 6 with AB, the axis of the magnet. By Art. 123 the effect of M is 13—2 196 MAGNETISM. [CH. VI equivalent to that of two magnets, one having its axis along OP and its moment equal to ikf cos 0, the other having its axis at right angles to OP and its moment equal to M sin 6. Let OP = r. The force at P due to the first is, by Art. 124, along OP and equal to 2Mcosd/r^, the force at P due to the second magnet is at right angles to OP and equal to il/ sin O/r^, hence the force due to the magnet AB at P is equivalent to the forces 2Mcos6 , ^ „ ^5— along OP, and — — — at right angles to OP, Let the resultant magnetic force at P make an angle (f) with OP, then MsinO tan 6 = rrrrP rr = i tan 0. ^ 2M cos ^ Let the direction of the resultant force at P cut AB produced in T, draw TN at right angles to OP, then TN tan^ = ^^, and since tan = i tan 0, PN = 20 K Thus ON = I OP. Thus, to find the direction of the magnetic force at P, trisect OP in N, draw NT at right angles to OP to cut AB produced in T, then PT will be the direction of the force at P. 127] MAGNETISM. 197 The magnitude of the resultant force is — \/4cos2|9 + sin2^ = - Vl + 3 cos^^ ; for a given vahie of r it is greatest when ^ = or tt, i.e. at a point along the axis, and least when S = 7r/2 or 37r/2, i.e. at a point on the line at right angles to the axis. The maximum value is twice the minimum one. 127. Couples on a Magnet in a Uniform Mag- netic Field. If a magnet is placed in a uniform field the couple acting on the magnet, and tending to twist it about a line at right angles both to the axis of the magnet and the force in the external field, is where M is the moment of the magnet, H the force in the uniform field, and 6 the angle between the axis of the magnet and the direction of the force. Let AB hQ the magnet, the negative pole being at A, the positive one at B. Then if m is the strength of the pole at B, the forces on the magnet are a force mH at B in the direction of the external field and an equal and opposite force at A. These two forces are equivalent to a couple whose moment is HmNM where NM is the distance between the lines of action of the two forces. But NM=^ ABsmd, if 6 is the angle between AB and H\ hence the couple on the magnet is HmAB^me=HM^me. 128. Couples between two Small Magnets. Let AB, CD, Fig. 58, represent the two magnets, ilf, M' their moments; r the distance between their centres 198 MAGNETISM. [CH. VI 0, 0\ Let AB, CD make respectively the angles 6, 6' with 0, 0'. Fig. 58. Consider first the couple on the magnet CD. The magnetic forces due to AB are ^5— along 00 , if sin ^ X • 1 . 14. nn/ — — — at right angles to UU . These may be regarded as constant over the space occupied by the small magnet CD, The couple on CD tending to produce rotation in the direction of the hands of a watch, due to the first component is 2if cos e , ilf'sin<9', that due to the second is if sin 6 ,,, ^, — if cos 6 \ hence the total couple on CD is MM' (2 cos 6 sin 6' + sin 6 cos 6'). 128] MAGNETISM. 199 This vanishes if tan 0' = — ^ tan 6, i.e. if CD is along the line offeree due to AB, see Art. 126. We may show in a similar way that the couple on AB due to CD tending to produce rotation in the direction of the hands of a watch is — 3 (2 cos 6' sin 6 4- sin 6' cos 6). If both these couples vanish, ^ = or tt, ^' = or tt, or ^ = + — , 0' = ±-^ , so that the axes of the magnets must be parallel to each other, and either parallel or perpendicular to the line joining the centres of the two magnets. We shall find it convenient to consider four special positions of the two magnets as standard cases. Case I. m^ — D Fig. 59. ^ = 0, ^' = 0, couples vanish, equilibrium stable. Case II. n D I / o Fig. 60. 77" *7r ^ = - , 6' = - , couples vanish, equilibrium unstable. 200 MAGNETISM. [CH. VI Case III. A ^ " " ' ' ' ' ' ' ' Fig. 61. O U=0, 6 =^ , couple on CD= — - — , couple on ^5= -— — . When the magnets are arranged as in this case, AB is said to be 'end on' to CD, while CD is broadside on to AB. Case IV. O D Fig. 62. TT 6=^ , 6 =0, couple on ClJ= , couple on AB= — . In this case ^5 is broadside on to CD. We see that the couple exerted on CD by ^J5 is twice as great when the latter is end on as when it is broadside on. It will be noticed that the couples on ^B and CD are not in general equal and opposite; at first sight it might appear that this result would lead to the absurd conclusion that if two magnets were firmly fastened to a board, and the board floated on a vessel of water, the board would be set in rotation and would spin round with gradually increasing velocity. The paradox will however be explained if we consider the forces exerted by one magnet on the other. 129] MAGNETISM. 201 129. Forces between two Small Magnets. Let AB, CD (Fig. 58) represent the two magnets, 0, 0' the middle points of AB, CD respectively, 0, 6' the angles which AB, CD respectively make with 00\ Let (/> be the angle DOG', r= 00' :m, m' the strength of the poles of AB and CD. The force due to the magnet AB on the pole at D consists of the component ^-^ cos (I9-0), along OD, and 0^ sm (0-ct>) at right angles to OD. These are equivalent to a force equal to 2Mm' cos (6 — <^) cos Mm^ sin (6 — ) sin <^ OD" ^ 05^ ' along 00' , and a force equal to 2Mm cos {6 — ) sin Mmf sin (6 — ) cos OD" OD' ' acting upwards at right angles to 00\ Neglecting squares and higher powers of CD/00' we have COS 9 = 1, sin9 = -^smcr /^T^ l>^T-w /!/ 1 lo L/JJ ^, 202 MAGNETISM. [CH. VI Substituting these values we see that the force exerted hy AB on D is approximately equivalent to a component 2Mm'cose _ SMm' CD cos d cos 6' 3 if m^ OD sin (9 sin ^^ 7^ r^ "^2 r* ' along 00\ and a component Mm sine S Mm' CD sin 6 cos 6' S Mm' CD cos sin 6 " ^ +2 r* "^2 r^ ' acting upwards at right angles to 00\ We may show in a similar way that the force exerted by ^5 on C is equivalent to a component 2ifm^cos e SMm' CD cos 6 cos6' S Mm' C D sin $ sin 6' ^ ^ 2 r'' ' along 00', and a component itfm' sin ^ S Mm' CD sine cos 6 ' S Mm' CD cos e sin e' ^3+2 r* '^2' r*' acting upwards at right angles to 00'. Hence the force on the magnet CD, which is the resultant of the forces acting on the poles C, D, is equi- valent to a component - ^^ (2 cos e cos e' - sin (9 sin (9'), along OO'y and a component ~ (sin e cos e' + cos e sin e'), acting upwards at right angles to 00'. The force on the magnet AB is equal in magnitude and opposite in direction to that on CD. 130] MAGNETISM. 203 If we consider the two magnets as forming one system, the two forces at right angles to 00' are equivalent to a couple whose moment is — — — (cos ^ sm ^ + sin Q cos Q), this couple is equal in magnitude and opposite in direction to the algebraical sum of the couples on the magnets AB, CD found in Art. 128 : this result explains the paradox alluded to at the end of that article. 130. Force between the Magnets in the four standard positions. In the positions described in Art. 128, the forces between the magnets have the following values. Case I. Fig. (59). ^ = 0, 6' — 0. Force between magnets is an attraction along the line joining their centres equal to QMW r" * Case II. Fig. (60). — ^ , 6' = -. Force is a repulsion along the line joining the centres .equal to Case III. Fig. (61). ^ = 0, ^' = o • Force is at right angles to the line joining the centres and equal to 204 MAGNETISM. [CH. VI Case IV. Fig. (62). rrr = ^ , 6' = 0. Force is at right angles to the line joining the centres and equal to jA • The forces between the magnets vary inversely as the fourth power of the distance between their centres while the couples vary inversely as only the cube of this distance. The directive influence which the magnets exert on each other thus diminishes less quickly with the distance than the translatory forces, so that when the magnets are far apart the directive influence is much the more important of the two. 131. Gauss' proof that the force between two magnetic poles varies inversely as the square of the distance between them. We saw, Art. 128, that, the distance between the magnets remaining the same, the couple exerted by the first magnet on the second was twice as great when the first magnet was 'end on' to the second as wlien it was * broadside on.' This is equi- valent to the result proved in Art. 126, that when P and Q are two points at the same distance from the centre of the magnet, P being on the axis of the magnet and Q on the line through the centre at right angles to the axis, the magnetic force at P is twice that at Q. This result only holds when the force varies inversely as the square of the distance ; we shall proceed to show that if the force varied inversely as the pth power of the distance the magnetic force at P would be p times that at Q. 131] MAGNETISM. 205 If the magnetic force varies inversely as the pth. power of the distance, then if m is the strength of one of the poles of the magnet, the magnetic force at P, Fig. 56, due to the magnet AB is equal to m m ^ m^ ~ AFP ni m ~(OP-OB)P (OP+OBy . _ 2mp . OB <- r - Qpp+i > ,- p ^ approximately, if OB is very small compared with OP ; if M is the moment of AB this is equal to pM Opf^i' mi r ^ r\ ^'*' OB 771 OA The force at Q = ^^^ + -^^^ M 0PP+^' ^ approximately. Thus the magnetic force at P is ^ times that at Q. We see from this that if we have two small magnets the couple on the second when the first magnet is ' end on ' to it is p times the couple when the first magnet is * broadside on.' Hence by comparing the value of the couples in these positions we can determine the value of ^. This can be done by an arrangement of the following kind. Suspend the small magnet to be deflected so that it can turn freely about a vertical axis : a convenient way of doing this and one which enables the angular motion of the magnet to be accurately determined, is to place the 206 MAGNETISM. [CH. VI magnet at the back of a very light mirror and suspend the mirror by a silk fibre. When the deflecting magnet is far away the suspended magnet will under the influence of the earth's magnetic field point magnetic north and south. When this magnet is at rest bring the deflecting magnet into the field and place it so that its centre is due east or west of the centre of the deflected magnet, the axis of the deflecting magnet passing through this centre. The couple due to the deflecting magnet will make the sus- pended magnet swing from the north and south position until the couple with which the earth's magnetic force tends to bring the magnet back to its original position just balances the deflecting couple. Let H be the magnetic force in the horizontal plane due to the earth's magnetic field. Then when the deflected magnet has twisted through an angle 6 the couple due to the earth's magnetic field is, see Art. 127, equal to HM' sin (9, where M' is the moment of the deflected magnet. The other magnet may be regarded as producing a field such that the magnetic force at the centre of the deflected magnet is east and west and equal to Mp where M is the moment of the deflecting magnet, r the distance between the centres of the deflected and deflect- ing magnets. Thus the couple on the deflected magnet due to this magnet is MMp cos 6 131] MAGNETISM. 207 The suspended magnet will take up the position in which the two couples balance : when this is the case t"^"^"^ w- Now place the deflecting magnet so that its centre is north or south of that of the suspended magnet, and at the same distance from it as in the last experiment, the axis of the deflecting magnet being again east and west. Let the suspended magnet be in equilibrium when it has twisted through an angle 6'. The couple due to the earth's magnetic field is HM' sin e\ The couple due to the deflecting magnet is MM' cos d ' Since the suspended magnet is in equilibrium these couples must be equal, hence MM' cos 6' EM' sin 0' M hence tan6>'= ^=—-^ (2). ,f.p+i Thus tan^ _ Hence if we measure 6 and 6' we can determine p. By experiments of this kind Gauss showed that p = 2, i.e. that the force between two poles varies inversely as the square of the distance between them. If we place the deflecting magnet at different dis- tances from the deflected we find that tan 6 and tan 6' vary as l/?-^, and thus obtain another proof that p = 2. 208 MAGNETISM. [CH. VI 132. Determination of the Moment of a Small Magnet and of the horizontal component of the Earth's Magnetic force. Suspend a small auxiliary magnet in the same way as the deflected magnet in the experiment described in the last example and place the magnet A whose moment is to be determined so that its centre is due east or west of the centre of the auxiliary magnet, the axis of the magnet A passing through the centre of the suspended magnet. Let 6 be the deflection of the suspended magnet, H the horizontal component of the earth's magnetic force, M the moment of ^ : we have, by Art. 131, equation (1), putting j9 = 2 M jg = ir^ tan 6; hence if we measure r and we can determine M/H. To determine MH suspend the magnet ^ by a vertical axis, about which it can rotate freely, passing through its centre, taking care that the magnetic axis of A is hori- zontal. When the magnet makes an angle 6 with the direction in which If acts, i.e. with the north and south line, the couple tending to bring it back to its position of equilibrium is equal to MHsmd. Hence if K is the moment of inertia of the magnet about the vertical axis the equation of motion of the magnet is K'^i-MHsme = 0, or if 6 is small dt K'^ + MHe^O, 133] MAGNETISM. 209 hence T, the time of a small oscillation, is given by or MH=-^^, hence if we know K and T we can determine MH ; knowing MjH from the preceding experiment we can find both M and H. The value oiH at Cambridge is about •18 c.G.S. units. 133. Magnetic Shell of Uniform Strength. A magnetic shell is a thin sheet of magnetizable substance magnetized at each point in the direction of the normal to the shell at that point. The strength of the shell at any point is the product of the intensity of magnetization into the thickness of the shell measured along the normal at that point, it is thus equal to the magnetic moment of unit area of the shell at the point. To tind the potential of a shell of uniform strength. Consider a small area a of the shell round the point Q, Fig. (63), let / be the intensity of magnetization of the shell Fig. 63. at Q, t the thickness of the shell at the same point. The moment of the small magnet whose area is a is IcLt, hence if is the angle which the direction of magnetization T. E. 14 210 MAGNETISM. [CH. VI makes with PQ, the potential of the small magnet at P is by Art. 122 equal to lat cos 6 If (p is the strength of the magnetic shell hence the potential at P is at P. This by Art. 10 is equal to &), where co is the area cut off from the surface of a sphere of unit radius with its centre at P by lines drawn from P to the boundary of the shell ; co is called the solid angle subtended by the shell at P ; it only depends on the shape of the boundary of the shell. If the shell is closed, then if P is outside the shell the potential at P is zero, since the total normal electric in- duction over a closed surface due to a charge at a point outside the surface is zero ; if the point P is inside the surface and the negative side of the shell is on the outside, then since the total normal electric induction over the shell due to a charge at P is 4>7r(f), the magnetic poten- tial at P is 47r(^ ; as this is constant throughout the shell, the magnetic force vanishes inside the space bounded by the shell. 133] MAGNETISM. 211 The signs to be ascribed to the solid angle bounded by the shell at various points are determined in the following way. Take a fixed point and with it as centre describe a sphere of unit radius, and let P be a point at which the magnetic potential of the shell is required. The contri- bution to the magnetic potential of any small area round a point Q on the shell, is the area cut off from the surface of the sphere of unit radius by the radii drawn from parallel to the radii drawn from P to the boundary of the area round Q ; the area traced out by the lines from is to be taken as positive or negative according as the lines drawn from P to Q strike first against the positive or negative side of the shell ; by the positive side of the shell we mean the side charged with positive magnetism, by the negative side the side charged with negative magnetism. With this convention with regard to the signs of the solid angle let us consider the relation between the poten- tials due to a shell at two points P and P' ; P being close V Fm. 64. to the shell on the positive side, P' close to P but on the negative side of the shell. Consider the areas traced out on the unit sphere by radii from parallel to those drawn from P and P'. The area corresponding to those drawn from P will be the shaded part of the sphere, let this area be co, the potential at P is a). The area 14—2 212 MAGNETISM. [CH. VI corresponding to the radii drawn from P' will be the unshaded portion of the sphere whose area is 47r — w, but inasmuch as the radii from P' strike first against the negative side of the shell the solid angle subtended at P' will be minus this area, i.e. «— 47r; hence the magnetic potential due to the shell at P' is (ft) — 47r). The potential at P thus exceeds that at P' by 47r(/). In spite of this finite increment in the potential in passing from P' to the adjacent point P, there will be continuity of potential in passing through the shell if we regard the potential as given in the shell by the same laws as outside. Consider the potential at a point Q in the shell, and divide the original shell into two, one on each side of Q. Fig. 65. Then as the whole shell is uniformly magnetized the strength of the shells will be proportional to their thick- nesses. Thus if {Vp-Vp)oi PP' But {Vp — Vp')IPP' is the magnetic force due to the external system along PP\ the normal to the shell. Let this force be denoted by —Hn, the force being taken as positive when it is in the direction of magnetization of the shell, i.e. when the magnetic force passes from the 214 MAGNETISM. [CH. VI negative to the positive side through the shell, then the mutual potential energy of the external system and the small magnet at Q is equal to Since the strength of the shell is uniform the mutual potential energy of the external system and the whole shell is equal to %HnOL being the sum of the products got by dividing the surface of the shell up into small areas, and multiplying each area by the component along its normal of the magnetic force due to the external system, this com- ponent being positive when it is in the direction of mag- netization of the shell. This quantity is often called the number of lines of magnetic force due to the external system which pass through the shell. It is analogous to the total normal electric induction over a surface in Electrostatics, see Art. 9. 135. Force acting on the shell when placed in a magnetic field. If X is the force acting on the shell in the direction x, and if the shell is displaced in this direction through a distance hx, then Xhx is the work done on the shell by the magnetic forces during the dis- placement ; hence by the principle of the Conservation of Energy, Xhx must equal the diminution in the energy due to the displacement. Suppose that J., Fig. Q^, repre- sents the position of the edge of the shell before, B its position after the displacement. The diminution in the energy due to the displacement is, by the last paragraph, equal to {N'-N) (1), 135] MAGNETISM. 215 where N and N' are the numbers of lines of magnetic force Fig. 66. which pass through A and B respectively. Consider the closed surface having the shell in its two positions A and B as ends, the sides of the surface being formed by the lines PP' &c. which join the original position P of a point to its displaced position. We see, as in Art. 10, that unless the closed surface contains an excess of mag- netism of one sign Sjff^a taken over its surface must vanish, Hn denoting the magnetic force along the normal to the surface drawn outwards. But SiT^a over the whole surface = N' —N+ %Hna taken over the sides, hence W-]^=-XH,,a (2); the right-hand side of this equation being taken over the sides. Consider a portion of the sides bounded by 216 • MAGNETISM. [CH. VI PQ, P'Q'\ P' > Q being the displaced positions of P and Q respectively. Since the area PQP'Q is equal to hx X PQ X sin 6, where is the angle between PQ and PP. If H is the magnetic force at P due to the external system, the value of HnOL for the element PQQ'P' is equal to Zx X PQ xsm6 X H cos %, where x is the angle which the outward-drawn normal to PQQ'P' makes with H. Hence since X8x = (i>{N'- N) we have by equation (2) Xhx = — (^ [thx X PQ X sin 6 xHcos x}, or since Bx is the same for all points on the shell X=-N, where N is the number of lines of magnetic force due to the external system which pass through the shell in the direction in which it is magnetized, i.e. which enter the shell on the side with the negative magnetic charge and leave it on the side with the positive charge : the shell will tend to move so as to make N as large as possible, for by so doing it makes the potential energy as small as possible. The force on each element of the boundary will therefore be in such a direction as to tend to move the element of the boundary so as to enclose a greater number of lines of magnetic force passing through the shell in the positive direction. Thus if the direction of the magnetic force at the 218 MAGNETISM. [CH. VI element PQ is in the direction FT in Fig. 67, the force + Fig. 67. on PQ will be outwards along PS as in the figure, for if PQ were to move in this direction the shell would catch more lines of force passing through it in the positive direction. Since X8x = {N'-N) ^ ,dN we get ^=*d.- This expression is often very useful for finding the total force on the shell in any direction. 136. Magnetic force due to the shell. Suppose that the external field is that due to a single unit pole at a point A, the result of the preceding article will give the force on the shell due to the pole, this must how- ever be equal and opposite to the force exerted by the shell on the pole. If however the field is due to a unit pole dX A, H the magnetic force due to the external system at an element PQ of the shell is equal to l/AP^ and acts along AP: hence by the last article the mag- netic force at A due to the shell is the same as if we supposed each unit of length of the boundary of the shell to exert a force equal to 138] MAGNETISM. 219 where 6 is the angle between AP and the tangent to the boundary at P, is the strength of the shell. This force acts along the line which is at right angles both to AP and the tangent to the boundary at P. The direction in which the force acts along this line may be found by the rule that it is opposite to the force acting on the element of the boundary at P arising from unit magnetic pole at A, this latter force may be found by the method given at the end of the preceding article. 137. If the external magnetic field in Art. 135 were due to a second magnetic shell, then the mutual potential energy of the two shells is equal to where (f> is the strength of the first shell, and N the number of lines of force which pass through the first shell, and are produced by the second. It is also equal to where ' is the strength of the second shell, and N' the number of lines of force which pass through the second shell and are produced by the first. Hence by making = (j> we see that, if we have two shells a and y8 of equal strengths, the number of lines of force which pass through a and are due to /3 is equal to the number of lines of force which pass through yS and are due to a. 138. Magnetic Field due to a uniformly mag- netized sphere. Let the sphere be magnetized parallel to X, and let / be the intensity of magnetization. We may regard the sphere as made up, as in Fig. 68, of a great number of uniformly magnetized bar magnets of uniform cross section a, the axes of these magnets being 220 MAGNETISM. [CH. VI parallel to the axis of x. On the ends of each of these magnets we have charges of magnetism equal to + Iol. Now consider a sphere whose radius is equal to that of the magnetized sphere and built up of bars in the same way, each of these bars being however wholly filled with positive magnetism whose volume density is p : consider also another equal sphere divided up into bars in the same way, each of these bars being however filled with negative magnetism whose volume density is — p ; suppose n: Fig. 68. that these spheres have their centres at 0' and 0, Fig. 69, two points very close together, 00' being parallel to the axis of X. Consider now the result of superposing these two spheres : take two corresponding bars ; the parts of the bars which coincide will neutralize each other's effects, but the negative bar will project a distance 00' to the left, and on this part of the bar there will be a charge of negative magnetism equal to 00' x a x p : the positive bar will project a distance 00' to the right, and on this part of the bar there will be a charge of positive magnetism 188] MAGNETISM. 221 equal to 00' x a x p. If 00' is very small we may regard these charges as concentrated at the ends of the bars, so that if 00' X p = I the case will coincide with that of the uniformly magnetized sphere. We can easily find the effects of the positive and negative spheres at any point either inside or outside. Let us first consider the effect at an external point P. The potential due to the positive sphere is equal to 4 Tra^p 3 or" if a is the radius of the sphere. The potential due to the negative sphere is equal to 4 Tra^p " 3 ~0F ' Fig. 69. Hence the potential due to the combination of the spheres is equal to 4 Tray 4 \0T of] 00' COS d OF' approximately, if 00' is very small, and is the angle which OF makes with 00'. 222 MAGNETISM. [CH. VI Now we have seen that this case coincides with that of the uniformly magnetized sphere if /? x 00' = 1, where / is the intensity of magnetization of the sphere ; hence the potential due to the uniformly magnetized sphere at an external point P is 4 ^ J. cos 6 where r = OP. Comparing this result with that given in Art. 122 we see that the uniformly magnetized sphere produces the same effect as a very small magnet placed at the centre of the sphere, the axis of the small magnet being parallel to the direction of magnetization of the sphere, while the moment of the magnet is equal to the intensity of magnetization multiplied by the volume of the sphere. The magnetic force inside the sphere is indefinite without further definition, since to measure the force on the unit pole, we have to make a cavity to receive the pole and the force on the pole depends on the shape of the cavity so made : this point is discussed at length in Chapter viii. For the sake of completing the solution of this case, we shall anticipate the results of that chapter and assume that the quantity which is defined as ' the magnetic force * inside the sphere is the force which would be exerted on the unit pole if the sphere were regarded as a spherical air cavity in a magnet, the surface of the cavity having spread over it the same distribution of magnetic charge as actually exists over the surface of the magnetized sphere. We may thus in calculating the effect of the 138] MAGNETISM. 223 charges on the surface suppose that they exert the same magnetic forces as they would in air. To find the magnetic force at an internal point Q, Fig. 69, we return to the case of the two uniformly charged spheres. The force due to the uniformly positively charged sphere at Q is equal to |tp . O'Q, and acts along O'Q ; the force due to the negative sphere is equal to i^p.OQ, and acts along QO. Fig. 70. 224 MAGNETISM. [CH. VI By the triangle of forces the resultant of the forces exerted by the positive and negative sphere is equal to i^rp . 00', and acts along 00'. We have seen that the case of the positive and negative spheres coincides with that of the uniformly magnetized sphere if 00' x p = I. Hence the force inside the uniformly magnetized sphere is uniform and parallel to the direction of magnetization of the sphere and equal to |,r/. The lines of force inside and outside the sphere are given in Fig. 70. CHAPTER VII. Terrestrial Magnetism. 139. The pointing of the compass in a definite direc- tion was at first ascribed to the special attraction for iron possessed by the pole star. Gilbert, however, in his work De Magnete, published in 1600, pointed out that it showed that the earth was itself a magnet. Since Gilbert's time the study of Terrestrial Magnetism, i.e. the state of the earth's magnetic field, has received a great deal of attention and forms one of the most important, and undoubtedly one of the most mysterious departments of Physical Science. 140. To fix the state of the earth's magnetic field we require to know the magnetic force over the whole of the surface of the earth ; the observations made at a number of magnetic observatories, scattered unfortunately somewhat irregularly at very wide intervals over the earth, give us an approximation to this. To determine the magnitude and direction of the earth's magnetic force we require to know three things : the three usually taken are (1) the magnitude of the horizontal component of the earth's magnetic force, usually called the earth's horizontal force ; (2) the angle which the direction of the horizontal force makes with the geographical meridian, this angle is called the declination, T. e. 15 226 TERRESTRIAL MAGNETISM. [CH. VII the vertical plane through the direction of the earth's horizontal force is called the magnetic meridian ; (3) the dip, that is the complement of the angle which a magnet, when suspended so as to be able to turn freely about an axis through its centre of gravity at right angles to the magnetic meridian, makes with the vertical. The fact that a compass needle when free to turn about a horizontal axis would not settle in a horizontal position, but 'dipped/ so that the north end pointed downwards, was discovered by Norman in 1576. For a full description of the methods and precautions which must be taken to determine accurately the values of the magnetic elements the student is referred to the article on Terrestrial Magnetism in the Encyclopcedia Britannica : we shall in what follows merely give a general account of these methods without entering into the details which must be attended to if the most accurate results are to be obtained. The method of determining the horizontal force has been described in Art. 132. 141. Declination. To determine the declination an instrument called a declinometer may be employed ; this instrument is represented in Fig. 71. The magnet — which is a hollow tube with a piece of plane glass with a scale engraved on it at the north end and a lens at the south end — is suspended by a single long silk thread from which the torsion has been removed by suspending from it a plummet of the same weight as the magnet: the suspension and the reading telescope can rotate about a vertical axis and the azimuth of the system determined by means of a scale engraved on the fixed horizontal base. 141] TERRESTRIAL MAGNETISM. 227 The observer looks through the telescope and observes the division on the scale at the end of the magnet with which a cross wire in the telescope coincides; the magnet is then turned upside down and resuspended and the division of the scale with which the cross wire coincides again noted ; this is done to correct for the error that would * Fig. 71. otherwise ensue if the magnetic axis of the cylinder did not coincide with the geometrical axis. The mean of the readings gives the position of the magnetic axis. If now we take the reading on the graduated circle and add to this the known value in terms of the graduations on this circle of the scale divisions seen through the telescope, we shall find the circle reading which corresponds to the magnetic meridian. Now remove the magnet and turn the telescope round until some distant object, whose 15—2 228 TERRESTRIAL MAGNETISM. [CH. VII azimuth is known, is in the field of view ; take the reading on the graduated circle, the difference between this and the previous reading will give us the angular distance of the magnetic meridian from a plane whose azimuth is known : in other words, it gives us the magnetic declina- tion. 142. Dip. The dip is determined by means of an instrument called the dip-circle, represented in Fig. 72. It Fig. 72. consists of a thin magnet with an axle of hard steel whose axis is at right angles to the plane of the magnet, and ought to pass through the centre of gravity of the needle ; this axle rests in a horizontal position on two agate 143] TERRESTRIAL MAGNETISM. 229 edges, and the angle the needle makes with the vertical is read off by means of the vertical circle. The needle and the vertical circle can turn about a vertical axis. To set the plane of motion of the needle in the magnetic meridian, the plane of the needle is turned about the vertical axis until the magnet stands exactly vertical ; when in this position the plane of the needle must be at right angles to the magnetic meridian. The instrument is then twisted through 90° (measured on the horizontal circle) and the magnet is then in the magnetic meridian ; the angle it makes with the horizontal in this position is the dip. To avoid the error arising from the axle of the needle not being coincident with the centre of the vertical circle the positions of the two ends of the needle are read ; to avoid the error due to the magnetic axis not being coincident with the line joining the ends of the needle, the needle is reversed so that the face which originally was to the east is now to the west and a fresh set of readings taken ; and to avoid the errors which would arise if the centre of gravity were not on the axle the needle is remagnetized so that the end which was previously north is now south and a fresh set of readings is taken. The mean of these readings gives the dip. 143. We can embody in the form of charts the deter- minations of these elements made at the various magnetic observatories : thus, for example, we can draw a series of lines over the map of the world such that all points on one of these lines have the same declination, these are called isogonic lines : we may also draw another set of lines so that all the places on a line have the same dip, these are called isoclinic lines. The lines however which give the 230 TERRESTRIAL MAGNETISM. [CH. VII best general idea of the distribution of magnetic force over the earth's surface are the lines of horizontal magnetic 140 160 ISO 140 160 180 160 140 120 100 80 100 120 140 IBO East VariatiiiQ Fig. 73. West Variation Fig. 74. 144] TERRESTRIAL MAGNETISM. 231 force on the earth's surface, i.e. the lines which would be traced out by a traveller starting from any point and always travelling in the direction in which the compass pointed ; they were first used by Duperrey in 1836. The isoclinic lines, the isogenic lines and Duperrey 's lines for the Northern and Southern Hemispheres for 1876 are shown in Figs. 73, 74, 75, and 76 respectively. 144. The points to which Duperrey's lines of force converge are called 'poles,' they are places where the horizontal force vanishes, that is where the needle if freely suspended would place itself in a vertical position. ¥ia. 11 Gauss by a very thorough and laborious reduction of magnetic observations gave as the position in 1836, of the pole in the Northern Hemisphere, latitude 70° 35', 232 TERRESTRIAL MAGNETISM. [CH. VII longitude 262'^ 1' E., and of the pole in the Southern Hemisphere, latitude 78° 35', longitude 150^0' E. The poles are thus not nearly at opposite ends of a diameter of the earth. 145. An approximation, though only a very rough one, to the state of the earth's magnetic sphere, may be got by regarding the earth as a uniformly magnetized sphere. On this supposition we should have by Art. 138 that if 6 is the dip at any place, i.e. the complement of the angle between the magnetic force and the line joining the place to the centre of the earth, I the magnetic latitude, i.e. the complement of the angle this line makes with the direction of magnetization of the sphere, tan 6=2 tan I, while the resultant magnetic force would vary as {l + 3sin^Z}l 146] TERRESTRIAL MAGNETISM. 233 These are only very rough approximations to the truth but are sometimes useful when more accurate knowledge of the magnetic elements is not available. 146. Variations in the Magnetic Elements. During the time within which observations of the magne- tic elements have been carried on the declination at London has changed from pointing 11" 15' to the East of North as it did in 1580 to 24° 38' 25" to the West of North as it did in 1818. It is now going back again to the East, but still points between 17° and 18° to the West. The variations in the declination and dip in London are shown in the following table. Date Declination Dip 1576 71° 50' 1580 iri5'E. 1600 72° 0' 1622 6° O'E. 1634 4° 6'E. 1657 0° O'E. 1665 r22'W. 1672 2°30'W. 1676 73° 30' 1692 6° 30' W. 1723 14°17'W. 74° 42' 1748 17°40'W. 1773 21° 9'W. 72° 19' 1787 23°19'W. 72° 8' 1795 23° 57' W. 1802 24° 6'W. 70° 36' 1820 24° 34' 30" W. 70° 3' 1830 24° 69° 38' 234 TERRESTRIAL MAGNETISM. [CH. VII Date Declination Dip 1838 69° 17' 1860 21° 39' 51" 68°19'-29 1870 20° 18' 52" 67° 57'-98 1880 18° 57' 59" 1893 17° 27' 67° 30' This slow change in the magnetic elements is often called the secular variation. The points of zero declina- tion seem to travel westward. 147. Besides these slow changes in the earth's mag- netic force, there are other changes which take place with much greater rapidity. Diurnal Variation. A freely suspended magnetic needle does not point continually in one direction during the whole of the day. In England in the night from about 7 p.m. to 10 a.m. it points to the East of magnetic North and South (i.e. to the East of the mean position of the needle), and during the day from 10 a.m. to 7 p.m. to the West of magnetic North and South. It reaches the westerly limit about 2 in the afternoon, its easterly one about 8 in the morning, the arc travelled over by the compass being about 10 minutes. This arc varies however with the time of the year, being greatest at midsummer and least at midwinter. The diurnal variation changes very much from one place to another, it is exceedingly small at Trevandrum, a place near the equator. In the Southern Hemisphere the diurnal variation is of the opposite kind to that in the Northern, i.e. the easterly limit in the Southern Hemisphere is reached in the afternoon, the westerly in the morning. 147] TERRESTRIAL MAGNETISM. 235 In the following diagram, due to Prof. Lloyd, the radius vector represents the disturbing force acting on the magnet at different times of the day in Dublin, the PM 5 6 7 4^ ^ N~~" '^ ,/ 3/^ 1 MID.N V 5^ 1 Cam w/ 3> E 1 PM v 1 > -^ Cij r °v M^ S S^ y / AMID Fig. 77. forces at any hour are the average of those at that hour for the year. The curve would be different for different seasons of the year. There is also a diurnal variation in the vertical com- ponent of the earth's magnetic force. In England the vertical force is least between 10 and 11 a.m., greatest at about 6 p.m. The extent of the diurnal variation depends upon the condition of the sun's surface, being greater when there are many sun spots. As the state of the sun with regard to 236 TERRESTRIAL MAGNETISM. [CH. VII sun spots is periodic, going through a cycle in about eleven years, there is an eleven-yearly period in the mag- nitude of the diurnal variation. 148. Effect of the Moon. The magnetic declina- tion shows a variation depending on the position of the moon with respect to the meridian, the nature of this variation varies very much in different localities. 149. Magnetic Disturbances. In addition to the periodic and regular disturbances previously described, rapid and irregular changes in the earth's magnetic field, called magnetic storms, frequently take place ; these often occur simultaneously over a large portion of the earth's surface. Aurorge are mostly accompanied by magnetic storms, and there is very strong evidence that a magnetic storm accompanies the sudden formation of a sun spot. 150. Very important evidence as to the locality of the origin of the earth's magnetic field, or of its variations, is afforded by a method due to Gauss which enables us to determine whether the earth's magnetic field arises from a magnetic system above or below the surface of the earth. The complete discussion of this method requires the use of Spherical Harmonic Analyses. The principle underlying it however can be illustrated by considering a simple case, that of a uniformly magnetized sphere. Let PQ be two points on a spherical surface con- centric with the sphere, then by observation of the hori- zontal force at a series of stations between P and Q, we can determine the difference between the magnetic potential at P and Q. If Up and Hq are the magnetic potentials 150] TERRESTRIAL MAGNETISM. 237 at P and Q respectively these observations will give us Up - IIq. By Art. 138 if 0, , 6., are the angles OP and OQ make with the direction of magnetization of the sphere M flp — Hq = — (cos 6i — cos 62) (1 ), where M is the magnetic moment of the sphere and r^OP^OQ, where is the centre of the sphere. If Zp, Zq are the vertical components of the earth's magnetic force, i.e. the forces in the direction OP, and OQ respectively, then „ 2if COS^i .(2). Zp and Zq can of course be determined by observations made at P and Q. By equations (1) and (2), we have np-nQ = i{Zp^ZQ)r (3), hence if the field over the surface of the sphere through P and Q were due to an internal uniformly magnetized sphere, the relation (3) would exist between the horizontal and vertical components of the earth's magnetic force. Now suppose that P and Q are points inside a uniformly magnetized sphere, the force inside the sphere is uniform and parallel to the direction of magnetization, let H be the value of this force, then in this case Up - Hq = Hr (cos 62 — cos ^1), Z[. = H cose,, Zq = H cos O2, 238 TERRESTRIAL MAGNETISM. [CH. VII hence in this case np-nQ = -r(Zp-z^) (4). Thus if the magnetic system were above the places at which the elements of the magnetic field were determined, the relation (4) would exist between the horizontal and vertical components of the earth's magnetic force. Con- versely if we found that relation (3) existed between these components we should conclude that the magnets pro- ducing the field were below the surface of the earth, while if relation (4) existed we should conclude the magnets were above the surface of the earth; if neither of these relations was true we should conclude that the magnets were partly above and partly below the surface of the earth. Gauss showed that no appreciable part of the mean values of the magnetic elements was due to causes above the surface of the earth. Schuster has however recently shown by the application of the same method that the diurnal variation must be largely due to such causes. Balfour Stewart had previously suggested as the probable causes of this variation the magnetic action of electric currents flowing through rarified air in the upper regions of the earth's atmosphere. CHAPTER VIII. Magnetic Induction. 151. When a piece of unmagnetized iron is placed in a magnetic field it becomes a magnet, and is able to attract iron filings ; it is then said to be magnetized by induction. Thus if a piece of soft iron (a common nail for example) is placed against a magnet it becomes magnet- ized by induction and is able to support another nail, while this nail can support another one, and so on until a long string of nails may be supported by the magnet. If the positive pole of a bar magnet be brought near to one end of a piece of soft iron, that end will become charged with negative magnetism, while the remote end of the piece of iron will be charged with positive magnetism. Thus the opposite poles of these two magnets are nearest each other, and there will therefore be an attraction between them, so that the piece of iron, if free to move, will move towards the inducing magnet, i.e. it will move from the weak to the strong parts of the magnetic field due to this magnet. If, instead of iron, pieces of nickel or cobalt are used they will tend to move in the same way as the iron, though not to so great an extent. If however we use bismuth instead of iron we shall find that the bismuth is repelled from the magnet, instead of being attracted towards it, the bismuth tending to move from the strong 240 MAGNETIC INDUCTION. [CH. Vlll to the weak parts of the field; the effect is however very small compared with that exhibited by iron ; and to make the repulsion evident it is necessary to use a strong electromagnet. When the positive pole of a magnet is brought near a bar of bismuth the end of the bar next the positive pole becomes itself a positive pole, while the further end of the bar becomes a negative pole. Substances which behave like iron, i.e. which move from the weak to the strong parts of the magnetic field, are called paramagnetic substances, while those which behave like bismuth and tend to move from the strong to the weak parts of the field are called diamagnetic substances. When tested in very strong fields all substances are found to be para- or dia-magnetic to some degree, though the extent to which iron transcends all other substances is very remarkable. 152. Magnetic Force and Magnetic Induction. The magnetic force at any point in air is defined to be the force on unit pole placed at that point, or — what is equivalent to this — the couple on a magnet of unit moment placed with its axis at right angles to the magnetic force. When however we wish to measure the magnetic force inside a magnetizable substance we have to make a cavity in the substance in which to place the magnet used in measuring the force. The walls of the cavity will however become magnetized by induction, and this magnetization will affect the force inside the cavity. The magnetic force thus depends upon the shape of the cavity, and this shape must be specified if the expression magnetic force is to have a definite meaning. 152] MAGNETIC INDUCTION. 241 Let P be a point in a piece of iron or other magnet- izable substance, and let us form about P a cylindrical cavity, the axis of the cylinder being parallel to the direction of magnetization at P. Let us first take the case when the cylinder is a very long and narrow one. Then in consequence of the magnetization at P, there will be a distribution of positive magnetism over one end of the cylinder, and a distribution of negative magnetism over the other. Let / be the intensity of the magnetization at P, reckoned positive when the axis of the magnet is drawn from left to right, then when the cylindrical cavity has been formed round P there will be, if a is the cross section of the cavity, a charge la of magnetism on the end to the left, and a charge — la on the end to the right. If 21, the length of the cylinder, is very great compared with the diameter, then the force on unit pole at the middle of the cylinder due to the magnetism at the ends of the cylinder will be 2Ia/l^ and will be indefinitely small if the breadth of the cylinder is indefinitely small compared with its length. In this case the force on unit pole in the cavity is independent of the intensity of magnetization at P. The force in this cavity is defined to be * the magnetic force at P' Let us denote it by H. Let us now take another co-axial cylindrical cavity, but in this case make the length of the cylinder very small compared with its diameter so that the shape of the cavity is that of a narrow crevasse. On the left end of this crevasse there is a charge of magnetism of surface density /, and on the right end of the crevasse, a charge of magnetism of surface density — /. If a unit pole be placed inside the crevasse the force on it due to this distribution of magnetism will be the same as the force T. E. 16 242 MAGNETIC INDUCTION. [CH. VIII on unit charge of electricity placed between two infinite plates charged with electricity of surface density + / and — I respectively, i.e. by Art. 14, the force on the unit pole in this case will be 47r/. Thus in a crevasse the total force on the unit pole at P, will be the resultant of the magnetic force at P and a magnetic force 47r/ in the direction of the magnetization at P. The force on the unit pole in the crevasse is defined to be the ' magnetic induction' at P, we shall denote it by B. If we had taken a cavity of any other shape the force due to the magneti- zation at P would have been intermediate in value be- tween zero for the long cylinder and 47r/ for the crevasse ; thus if the cavity had been spherical the force due to the magnetization would (Art. 138) have been 47r//3. The m^neiic intiuction is not necessarily in the same direction as the magnetic force, it will only be so when the magnetization at P is parallel to the magnetic force. 153. Tubes of Magnetic Induction. A curve drawn such that its tangent at any point is parallel to the magnetic induction at that point is called a line of magnetic induction : in non-magnetizable substances the lines of magnetic induction coincide with the lines of magnetic force. We can also draw tubes of magnetic induction just as we draw tubes of magnetic force. We shall choose the unit tube so that the magnetic induction at any place whether in the air or iron is equal to the number of tubes of induction which cross a unit area at right angles to the induction. Let us consider the case of a small bar magnet, the magnetism being yjjjj;gj^ at its ends. Suppose A and B are the ends of the magnet, A being the negative, B the 153] MAGNETIC INDUCTION. 243 positive end, then in the air the lines of magnetic in- duction coincide with those of magnetic force and go A CD B Fia. 78. from B to A. To find the lines of magnetic induction at a point P inside the magnet, imagine the magnet cut by a plane at right angles to the axis and the two portions separated by a short distance, the lines of magnetic force in this short air space will be the lines of magnetic in- duction in the section through P. If the magnet is cut as in the figure then the end G will be a positive pole of the same strength as A, the end D a negative pole of the same strength as B. Thus through the short air space between G and D tubes of induction will pass running in the direction AB. Draw a closed surface passing through the gap between G and D and enclosing AG or DB, The magnetic force at any point on this surface is equal to the magnetic induction at the same point due to the undivided magnet. Since this surface encloses as much positive as negative magnetism, we see as in Art. 10 that the total magnetic force over its surface vanishes. Hence we see that the tubes of induction inside the magnet are equal in number at each cross-section and this number is the 16—2 244- MAGNETIC INDUCTION. [CH. VIII same as the number of those which leave the pole B and enter A. In fact the lines of magnetic induction due to the magnet form a series of closed curves all passing through the magnet and then spreading out in the air, the lines running from 5 to J. in the air and from A to B in the magnet. Thus w^e may regard any small magnet, whose in- tensity is / and area of cross-section a. as the origin of a bundle of closed tubes of induction, the number of tubes being 47r/a ; every tube in this bundle passes through the magnet; running through the magnet in the direction of the magnetization. It is instructive to compare the differences between the properties of the tubes of electric polarization in electrostatics and those of magnetic induction in mag- netism : the most striking difference is that whereas in electrostatics the tubes are not closed but begin at posi- tive electrification and end on negative, in magnetism the tubes of induction always form closed curves and have neither beginning nor end. A surface charged with electricity of surface density a is the origin of ^ t e i r °i\' 1 ■>*\ 2 1 *.p8 2p22-| 2,C 00, *.SfP = /then IBH is represented on the diagram by the area SPQT. Thus the total work done by the magnet when the field is increased from OK to OL is represented by the area CKLDE. Similarly the work done on the magnet when the field is diminished from OL to OK is represented by the area DFGKL. Thus the excess of the work done on the magnet over that done by the magnet, when the magnetic force goes through a complete cycle, is repre- sented by the area of the loop CEDFG. The area of the loop thus represents the excess of the work spent over that obtained : but since the magnetic force and magnet- ization at the end of the cycle are the same as at the begin- ning, this work must have been dissipated and converted into heat. The mechanical equivalent of the amount of heat produced in each unit volume of the iron is repre- sented by the area of the loop. 254 MAGNETIC INDUCTION. [CH. VIII If instead of a curve showing the relation between / and H we use one showing the relation between B and H, there will be similar loops in this second curve and the area of these loops will be 47r times the area of the corresponding loops on the / and H curve. For the area of a loop on the first curve is -SIdH, this is equal to -l\iB-HHH = -l\BdH, since JHdH=0, as the initial and final values of H are equal. The area of a loop on the B and H curve is however equal to -JBdH. Hence we see that this area is 47r times the area of the corresponding loop on the / and H curve. 157. Conditions which must hold at the boun- dary of two substances. At the surface separating two media the magnetic field must satisfy the following conditions. 1. The magnetic force parallel to the surface must be the same in the two media. 2. The magnetic induction at right angles to the surface must be the same in the two media. To prove the first condition, let P and Q be two points close to the surface of separation, Q being in the air, P in 157] MAGNETIC INDUCTION. 255 the iron. Now the magnetic force at P is by definition (see Art. 152) the force on a unit pole placed in a cavity round P, when the magnetism on the walls of the cavity can be neglected: hence since this magnetism is to be dis- regarded the only difference between the magnetic forces at P and Q must arise from the magnetism on the surface between P and Q : but though the forces at right angles to this portion of the surface due to its mag- netism are different at P and Q, the forces parallel to the surface are the same. Hence we see that the tangential magnetic forces will be the same at P as at Q. We shall now show that the normal magnetic induction is continuous. All the tubes of magnetic induction form closed curves. Hence any tube must cut a closed surface an even number of times; half these times it will be entering the surface, half leaving it. The contributions of each tube to the total normal magnetic induction will be the same in amount but opposite in sign when it enters and when it leaves the surface. Hence the total contribution of each tube is zero, and thus the total normal magnetic induction over any closed surface vanishes. Consider the surface of a very short cylinder whose sides are parallel to the normal at P, one end being in the medium (1), the other in (2). The total normal induction over this surface is zero, but as the area of the sides is negligible compared with that of the ends, this implies that the total normal induction across the end in (1) is equal to that across the end in (2), or that, since the areas of these ends are equal, the induction parallel to the normal in (1) is the same as that in the same direction in (2). This is always true whether the magnet is permanently magnetized or only magnetized by induction. 256 MAGNETIC INDUCTION. [CH. VIII In Art. 73 we proved that the conditions satisfied at the boundary of two dielectrics are 1. The tangential electric intensity must be the same in both media. 2. When there is no free electricity on the surface the normal electric polarization must be the same in both. That is, if F, F' are the normal electric intensities in the media whose specific inductive capacities are re- spectively K and K\ KF=K'F'. If we compare these with the conditions satisfied at the boundary of two media in the magnetic field and with the condition that when the magnetization is induced, the magnetic induction is equal to fju times the magnetic force, we see that we have complete analogy between the disturbance of an electric field produced by the presence of uncharged dielectrics and the disturbance in a magnetic field produced by para- or dia-magnetic bodies in which the magnetism is entirely induced. Fig. 83. 158] MAGNETIC INDUCTION. 257 Hence from the solution of any electrical problem we can deduce that of the corresponding magnetic one by writing magnetic force for electric intensity, and fi for K. We can prove, as in Art. 74, that if O^ is the angle which the direction of the magnetic force in air makes with the normal at a point P on a surface, 6^ the angle which the magnetic force in the magnetizable substance makes with the normal at the same point, then yu, tan 6 1 — tan 6^. Thus when the lines of force go from air to a para- magnetic substance they are bent away from the normal in the substance, since in this case //, is greater than 1 ; when they go from air to a diamagnetic substance they are bent towards the normal, since in this case yu, is less than 1. The effects produced when paramagnetic and diamag- netic spheres are placed in a uniform field of force are shown in Figs. 39 and 83. 158. If iM is infinite tan 6^ vanishes, and then the lines of force in air are at right angles to the surface, so that the surface of a substance of infinite permeability is a surface of equi-magnetic potential. The surface of such a substance corresponds to the surface of an insulated conductor without charge in electrostatics, and any problem relating to such conductors can be at once applied to the corresponding case in magnetism. In particular we can apply the principle of images (Chap. V.) to find the effect produced by any distribution of magnetic poles in presence of a sphere of infinite magnetic permeability. T. E. 17 258 MAGNETIC INDUCTION. [CH. VIII 159. Sphere in uniform field. We showed in Art. 103 that if a sphere, whose radius is a, and whose specific inductive capacity is K, is placed in a uniform electric field, and if H is the electric intensity before the introduction of the sphere, then the field when the sphere is present will at a point P outside the sphere consist of -ff and an electric intensity whose component along PO is equal to 2±/-^-77 — ^cos^— , K + 2 r^ ' and whose component at right angles to PO in the direction tending to diminish is a ^ — ^ sm ^ — ; K+2 r^ ' in these expressions OP = r, 6 is the angle OP makes with the direction of H, is the centre of the sphere. Inside the sphere the electric intensity is constant, parallel to H and equal to 3 K+2 H. If we write fi for K the preceding expressions will give us the magnetic force when a sphere of magnetic permea- bility fi is placed in a uniform magnetic field where the magnetic force is H. A very important special case is when fi is very large compared with unity. In this case the magnetic forces due to the sphere are approximately 2H'^cose 161] MAGNETIC INDUCTION. 259 along PO, and ^— sin ^ at right angles to it. Inside the sphere the magnetic force is and is very small compared with that outside. The mag- netic induction inside the sphere is 3^. Thus through any area in the sphere at right angles to the magnetic force, three times as many tubes of induction pass as through an equal and parallel area at an infinite distance from the sphere. The resultant magnetic force in air vanishes round the equator of the sphere. 160. Magnetic Shielding. Just as a conductor is able to shield oif the electric disturbance which one electrical system would produce on another, so masses of magnetizable material, for which fi has a large value, will shield off from one system magnetic forces due to another. Inasmuch however as /a has a finite value for all sub- stances the magnetic shielding will not be so complete as the electrical. 161. Iron Shell. We shall consider the protection I afforded by a spherical iron shell against a uniform mag- pnetic field. We saw in Art. 159 that, when a solid iron sphere is placed in a uniform magnetic field, the magnetism induced on the sphere produces outside it a radial mag- netic force proportional to 2 cos ^/?*^, and a tangential force 17—2 260 MAGNETIC INDUCTION. [CH. VIII proportional to sm^/7^, and a constant force inside the sphere. We shall now proceed to show that we can satisfy the conditions of the problem of the spherical iron shell by supposing each of the distributions of magnetism induced on the two surfaces of the shell to give rise to forces of this character. Let a be the radius of the inner surface of the shell, h that of the outer surface. Let H be the force in the uniform field before the shell was introduced. Let the magnetic forces due to the magnetism on the outer surface of the shell consist, at a point P outside the sphere, of a radial force 2ifi cos e a tangential force M^ sin 6 where r=OP and 6 is the angle OP makes with the direction of H. The magnetic force due to this distribu- tion of magnetism will be uniform inside the sphere whose radius is h, it will act in the direction of H and be equal to — Mi/b^ Let the magnetization on the inner surface of the shell give rise to magnetic forces given by similar ex- pressions with M2 written for ifc/j and a for b. This system of forces, w^hatever be the values of Ml and M2, satisfies the condition that as we cross the surfaces of the shell the tangential components of the 161] MAGNETIC INDUCTION. 261 magnetic force are continuous. We must now see if we can choose M^, M^ so as to make the normal magnetic induction continuous. The normal magnetic induction (reckoned positive along the outward drawn normal) in the air just outside the outer shell is equal to „ . 2ifi . 2if2 ^ iZ cos ^ + -, ;- COS V + -^r— COS 0, the normal magnetic induction in the iron just inside the outer surface of the shell is equal to { ^r /I ^1 /I 2il/2 ^\ fi ill C08 6 — jY cos 6 -{- -jy- cos6 ] . These are equal if or, if i-^-±3^^-:pt^3E^^^,_^)H (1). The normal magnetic induction in the iron just outside the inner surface of the shell is fjL ill cos 6 - -jj COS 6 -\ f cos6] , the normal magnetic induction in the air just inside the shell is equal to Hcosd- -J .^ cos ^ ^ cos 0: these are equal if M M (m-1)j;-(2/.+ 1)^ = (^-1)// (2). 262 MAGNETIC INDUCTION. [CH. VIII Equations (1) and (2) are satisfied if (2/.+ l)(M + 2)-2(^-iyg M, = -(^,-l)H — (2/. + l)(^ + 2)-2(;.-l)=g The magnetic force in the hollow cavity is equal to I? a' ' Substituting the values of M^ and 31^ we see that this is equal to 9/. + 2(/.-iy(i-g If fi is very large compared with unity this is approxi- mately equal to H 1 + H'-i) The denominator may be written in the form 2 volume of shell '^^ ■ volume of outer sphere ' Hence the force inside the shell will not be greatly less than the force outside unless fi is greater than the ratio of the volume of the outer sphere to that of the shell. In the cases where fi = 1000 and //, = 100, the ratio oiH', the force inside the sphere, to H for different values of a/b is given in the following table. 162] MAGNETIC INDUCTION. 263 H'lH H'lH ajb /A=:1000 /i=:100 •99 3/23 9/15 •9 1/67 1/7 •8 1/109 1/12 •7 1/146 1/15 •6 1/175 1/18 •5 1/195 1/20 '4 1/209 1/22 •3 1/216 1/22 •2 1/221 1/23 •1 1/223 1/23 •0 1/223 1/23 Galvanometers which have to be used in places exposed to the action of extraneous magnets are sometimes pro- tected by surrounding them with a thick-walled tube made of soft iron. We may regard the shielding effect of the shell as an example of the tendency of the tubes of magnetic induction to run as much as possible through iron; to do this they leave the hollow and crowd into the shell. 162. Expression for the energy in the magnetic field. We shall suppose that the field contains per- manent magnets as well as pieces of magnetizable substances magnetized by induction. If the distribution of the permanent magnets is given, the magnetic field will be quite determinate. The forces between magnetic charges follow the same laws as those between electrical ones. Hence the energy due to any system of magnetized bodies will, if the magnetization due to induction is proportional to the magnetic force, i.e. if ^m is constant, be equal to the sum of one half the product of the 264 MAGNETIC INDUCTION. [CH. VIII strength of each permanent pole into the magnetic potential at that pole. Thus if Q is the potential energy of the magnetic field, where m is the strength of the permanent pole and fl the magnetic potential at that pole. Let us divide each of the permanent magnets up into little magnets and con- sider the contribution of one of these to the energy. Let 7o be the intensity of the permanent magnetization, and a the area of the cross section : then the magnet has a pole of permanent magnetism of strength /„« at A, another pole of strength —lod at B. If fl^, fl^ are the values of the magnetic potentials at A and B, the contribution of this magnet to the energy is therefore equal to Now the magnet may be regarded as the origin of 47r/oa tubes of magnetic induction forming closed curves running through the magnet, leaving it at A and entering it at 5 ; if ds is an element of one of these tubes, and R the resultant magnetic force which acts along this element, then n^ — n^= Rds, J A the integration being extended over the part of the tube outside the magnet. Hence the contribution of this magnet to the energy is the same as it would be if each tube of which it is the origin had per unit length at P an amount of energy equal to I/Stt of the resultant magnetic force at P. The portion of the tube inside the little magnet in which it has its origin, must not be taken into account. 162] MAGNETIC INDUCTION. 265 Now let us consider any small element of volume in the magnetic field, let us take it as cylindrical in shape, the axis of the cylinder being parallel to the resultant magnetic force R at the element. Let I be the length of this cylinder, co the area of its cross section. Now each of the tubes of magnetic induction which pass through the element and have not their origin within it, con- tributes R/Stt units of energy for each unit of length of the tube. Let /q be the intensity of the permanent magnetization of the element, /the induced magnetization, then the number of tubes of induction which pass through unit area of the base of the cylinder is equal to the value of the magnetic induction, i.e. it is equal to i? + 4,r(/ + /„); fl but of these, 47r/o have their origin in the element, and hence the number of tubes per unit area which contribute to the energy is equal to E+47r/, and since /= kR and yu. = 1 + 4f7rk, this is equal to therefore the number passing through the base of the cylinder is equal to fiRco. The energy of the portion of each of the tubes within the element is equal to RI/Stt, hence the energy contributed by the element is thus the energy per unit volume is equal to fiR^/STr. We may then regard the energy of the magnetized system as 266 MAGNETIC INDUCTION. [CH. VIII distributed throughout the magnetic field, there being fiB^lS-rr units of energy in each unit volume of the field. 163. When a tube of induction enters a paramag- netic substance from air the resultant magnetic force is — when the magnetization is entirely induced — less in the paramagnetic substance than in air, the energy per unit length will be less in the magnetic substance than in the air since the energy per unit length of a tube of induction is proportional to the resultant magnetic force along it. Thus in accordance with the principle that when a system is in equilibrium the potential energy is a minimum, the tubes of induction will tend to leave the air and crowd into the magnet, when this act does not involve so great an increase in their length in the air as to neutralize the diminution of the energy due to the parts passing through the magnet. Again, when a tube of induction enters a diamagnetic substance the magnetic force inside this substance is greater than it is in the air just outside, the tubes of induction will therefore tend to avoid the diamagnetic substance. Examples of this and the previous effect are seen in Figs. 83 and 39. A small piece of iron placed in a magnetic field where the force is not uniform will tend to move from the weak to the strong parts of the field, since by doing so it encloses a greater number of tubes of induction and thus produces a greater decrease in the energy. The direction of the force tending to move the iron is in the direction along which the rate of increase of R^ is greatest. This is not in general the direction of the magnetic force. Thus in the case of a bar magnet AB, the greatest 164] MAGNETIC INDUCTION. 267 rate of increase in R^ Sit G Si point equidistant from A and B is along the perpendicular let fall from G on AB, and this is the direction in which a small sphere placed at G will tend to move ; it is however at right angles to the direction of the magnetic force at G. There will be no force tending to move a piece of soft iron placed in a uniform magnetic field. A diamagnetic substance will tend to move from the strong to the weak parts of the field, since by so doing it will diminish the number of tubes of magnetic induc- tion enclosed by it and hence also the energy, for the tubes of induction have more energy per unit length when they are in the diamagnetic substance than when they are in air. 164. Ellipsoids. We have hitherto only considered the case of spheres placed in a uniform field. Bodies which are much longer in one direction than another have very interesting properties which are conveniently studied by investigating the behaviour of ellipsoids placed in a uniform magnetic field. We saw in Art. 138 that the magnetic field, due to a sphere uniformly magnetized in the direction of the axis of 00, might be regarded as due to two spheres, one of uniform density p with its centre at 0', the other of uniform density —p with its centre at 0, the points and 0' being very close together and 00' parallel to the axis of oc: the distance 00' is given by the condition that pOO' is equal to the intensity of magnetization of the sphere. An exactly similar proof will show that if we have a body of any shape uniformly magnetized, the magnetic potential due to it is the same as that due to 268 MAGNETIC INDUCTION. [CH. VIII two bodies of the shape and size of the magnet, one having the density p, the other the density — p, and so placed that if the negative body is displaced through the distance f in the direction of magnetization, it will coincide with the positive body if p^ = A, A being the intensity of magnetization of the body. Let us suppose that the body is uniformly magnetized with intensity A in the direction of the axis of x, and let pfl be the potential of the positive body at the point P, then the potential of the negative body at P will be equal to — pfl', where pfl' is the potential of the positive body at P', if PP' is parallel to the axis of x and equal to f . But since P^P is small, The potential of the negative body is therefore -'("-^s)- Thus the potential of the positive and negative bodies together, and therefore of the magnetized body, will be .dn -~^dx' since p^ = A. If the body instead of being magnetized parallel to x is uniformly magnetized so that the components of the intensity parallel to x, y, z are respectively A, B, C, the magnetic potential is -(a~ 7?— ^,dSl\ \ dx dy dz ) '' 164] MAGNETIC INDUCTION. 269 We shall now show that if an ellipsoid is placed in a uniform magnetic field it will be uniformly magnetized by induction. To prove this it will be sufficient to show that if we superpose on to the uniform field, the field due to a uniformly magnetized ellipsoid, it is possible to choose the intensity of magnetization so as to satisfy the two conditions, (1) that the tangential magnetic force and (2) that the normal magnetic induction, are continuous at the surface of the ellipsoid. The first of these con- ditions is evidently satisfied whatever the intensity of magnetization may be : we proceed to discuss the second condition. The forces due to the attraction of an ellipsoid of uniform density, parallel to the axes of x, y, z (these are taken along the axes of the ellipsoid) are, see Routh's Statics, vol. II. p. 112, equal to Lx, My, Nz respectively, where L, if, N are constant as long as the point whose coordinates are x, y, z is inside the ellipsoid. Hence by (1) since -;— = — LX, &C., ax the magnetic potential inside the ellipsoid due to its magnetization will be {ALx + BMy + GNz), so that the magnetic forces parallel to the axes of x, y, z due to the magnetization of the ellipsoid will be -AL, -BM, -CN respectively. Hence if N^ is the component of these forces along the outward drawn normal to the surface of the ellipsoid, N, = - {All + BMm + CNn), 270 MAGNETIC INDUCTION. [CH. VIII where I, m, n are the direction cosines of the outward drawn normal. If N.^ is the force due to the magnetization on the ellipsoid in the same direction just outside the ellipsoid, then N, = N^ + ^ir (I A + 7nB + n(7) = I A (47r -L)-\-mB (47r - M) + nC (47r - N). Let X, Y, Z be the components of the force due to the uniform field. Then N^\ the total force inside the ellipsoid along the outward drawn normal, will be given by the equation N^==lX + mY^nZ + N„ and if N^ is the total force just outside the ellipsoid along the outward drawn normal N^=lX^mY+nZ^-N,. If fjL is the magnetic permeability of the ellipsoid, the normal magnetic induction will be continuous if that is if I {iiX - fiAL) + m (fiY- fiBM) + n {fxZ - fiGN) = qZ + ^ (47r - L)} + m{Y+B(4>7r-M)] + n{^+0(47r-i\^)). But this condition will be satisfied if (/x-l)X (2). {f^-l)Z 164] MAGNETIC INDUCTION. 271 These equations give the intensity of magnetization of an ellipsoid placed in a uniform magnetic field. The force inside the ellipsoid due to its magnetization has —AL,— BM, — ON for components parallel to the axes of X, y, z respectively ; these components act in the opposite direction to the external field and the force of which these are the components is called the demagnetiz- ing force. We see from equations (2) that the com- ponents of the demagnetizing force are (/x -\)LX • 47r -I- X (/^ - 1) ' 47r+il/(/i-l)' 47r + iV(/4-l)' We shall now consider some special cases in detail. Let us take the case of an infinitely long elliptic cylinder, let the infinite axis be parallel to z, let 2a, 26 be the axes in the direction of x and y\ then (Routh's Analytical Statics, vol. II. p. 112) Z = 47r^, ,ilf=47r ~^, i\^ = 0. a + h a4-6 Thus A = B {^.-l)X hX 4.^1 + (^-l)— I 1+(M-1)^-^ {fji-l)Y kY where k is the magnetic susceptibility. 272 MAGNETIC INDUCTION. [CH. VIII We see from this equation that A/X is approximately- equal to k when (//. — 1) 6/(a + b) is very small, but only then. A very common way of measuring k is to measure A/X in the case of an elongated solid, magnetized along the long axis ; but we see that in the case of an elongated cylinder this will be equal to k only when (/j, — 1) 6/(a + b) is very small. Now for some kinds of iron fi is as gi-eat as 1000, hence if this method were to give in this case results correct to one per cent., the long axis would have to be 100,000 times as long as the short one. This extreme case will show the importance of using very elongated figures when experimenting with substances of great permeability. Unless this precaution is taken the ex- periments really determine the value of a/b and not any magnetic property of the body. When the body is an elongated ellipsoid of revolution the ratio of the long to the short axis need not be so enormous as in the case of the cylinder, but it must still be very considerable. If the axis of x is the axis of revolution, then by Routh's Analytical Statics, vol. Ii. p. 112, we have approximately X = 47r^|log^-l|, M = ]Sr=27r. Thus ^ ^ Thus if /JL were 1000, the ratio oi a to b would have to be about 900 to 1 in order that the assumption A/X = k should be correct to one per cent. 165] MAGNETIC INDUCTION. 273 165. Couple acting on the Ellipsoid. The mo- ment of the couple tending to twist the ellipsoid round the axis of z, in the direction from x to y, is equal to (volume of ellipsoid) {YA - XB) ahc(fM-iyXY(M-L) -^'^ (477-1- (/.- 1) L] {47r + (y^ _ l)i/} • If the magnetic force in the external field is parallel to the plane ocy and is equal to H and makes an angle 6 with the axis of x, X =11 cos e, Y=H sine, and the couple is equal to sin e cos dj/Ji-iy (M - L) ^'^^ {47r-f(/.-l)Z}{47r + (/x-l)ilf)' If (X > 6, ilf is greater than L. Thus the couple tends to make the long axis coincide in direction with the external force, so that the ellipsoid, if free to turn, will set with its long axis in the direction of the external force. This will be the case whether fi is greater or less than unity, i.a w^hether the substance is paramagnetic or diamagnetic, so that in a uniform field both paramagnetic and dia- magnetic needles point along the lines of force. It generally happens that a diamagnetic substance places itself athwart the lines of magnetic force, this is due to the want of uniformity in the field, in consequence of which the diamagnetic substance tries to get as much of itself as possible in the weakest part of the field. This tendency varies as (/a— 1) ; the couple we are investigating in this Article varies as (yu, — Vf, and as (//, — 1) is exceed- ingly small for bismuth, this couple will be overpowered unless the field is exceptionally uniform. T.E. 18 274 MAGNETIC INDUCTION. [CH. VIII 166. Ellipsoid in Electric Field. The investiga- tion of Art. 164 enables us to find the distribution of electrification induced on a conducting ellipsoid when placed in a uniform electric field. To do this we must make ^ infinite in the expressions of Art. 164. The quantity lA + ynB -\- nC which occurs in the magnetic problem corresponds to cr. Putting /^ = oo in equations (2) we find IX mY nZ L^ M'^ N If the force in the electric field is parallel to the axis of x IX Thus when the electric field is parallel to one of the axes of the ellipsoid, the density of the electrification is, as in the case of a sphere, proportional to the cosine of the angle which the normal to the surface makes with the direction of the electric intensity in the undisturbed field. By Coulomb's law the normal electric intensity at the surface of the ellipsoid is equal to 47ro-, i.e. to ^irlX L ' Thus the electric intensity at the surface of the ellipsoid is 47r/i times the electric intensity in the same direction in the undisturbed field. If the ellipsoid is a very elongated one with its longer axis in the direction of the electric force, then by Art. 164 47r_ a" 1 2^ 1 166] MAGNETIC INDUCTION. 275 Thus, when ajh is large, ^irjL is a large quantity, and the electric intensity at the surface of the ellipsoid is very large compared with the intensity in the undisturbed field. Thus if ajh = 100, the electric intensity at the surface is about 2500 times that in the undisturbed field. This result explains the power of sharply pointed con- ductors in discharging an electric field, for when these are placed in even a moderate field the electric intensity at the surface of the conductor is great enough to overcome the insulating power of the air, see Art. 37, and the electrification escapes. , 18—2 CHAPTER IX. Electric Currents. 167. Let two conductors A and B be at different potentials, A being at the higher potential and having a charge of positive electricity, while 5 is at a lower potential and has a charge of negative electricity ; then if A is connected to B by a metallic wire the potential of A will begin to diminish and A will lose some of its positive charge, the potential of B will increase and B will lose some of its negative charge, so that in a short time the potentials of A and B will be equalized. During the time in which the potentials of A and B are changing the following phenomena will occur: the wire connecting A and B will be heated and a magnetic field will be produced which is most intense near the wire. If^ and B are merely charged conductors, their potentials are equalized so rapidly, and the thermal and magnetic effects are in consequence so transient, that it is some- what difficult to observe them. If, however, we maintain A and B at constant potentials by connecting them with the terminals of a voltaic battery the thermal and magnetic effects will persist as long as the connection with the battery is maintained, and are then easily observed. CH. IX. 167] ELECTRIC CURRENTS. 277 The wire connecting the two bodies A and B at different potentials is said to be conveying a current of electricity, and when A is losing its positive charge and B its negative charge the current is said to flow from A \, ^2, ^3--- respectively. Then if v^, v^ are the potentials of A and B respectively, we have by Ohm's Law V^-^ZJ = ^2^2, Now ^ = ^l + ^2 + 4+ ... But if R is the resistance of the system of conductors, then by Ohm's Law, hence comparing this expression with the preceding one we see that 1111 -^ = - + - + - + ..., it n 7\ n or the reciprocal of the resistance of a number of con- ductors in parallel is equal to the sum of the reciprocals of the individual resistances. The reciprocal of the resist- ance of a conductor is called its conductivity, hence we see that we may express the result of this investigation by saying that the conductivity of a number of conductors in parallel is equal to the sum of the conductivities of the individual conductors. In the special case when all the wires connected up in multiple arc have the same resistance, and if there are n wires, their resistance when in multiple arc is 1/n of the resistance of one of the individual wires. 286 ELECTRIC CURRENTS. [CH. IX 176. Specific resistance of a substance. If we have a wire whose length is I and whose cross section is uniform and of area a, we may regard it as built up of cubes whose edges are of unit length, in the following way ; take a wire formed by placing I of these cubes in series, and then place a of these filaments in parallel ; the resistance of this system is evidently the same as that of the wire under consideration. If a is the resistance of one of the cubes the resistance of the filament formed by placing I such cubes in series is la, and when a of these filaments are placed in parallel the resistance of the system is lajoL ; hence the resistance of the wire is la- a ' Since a- only depends on the material of which the wire is made we see that the resistance of a wire of uniform cross section is proportional to the length and inversely proportional to the area of the cross section. The quantity denoted by a in the preceding expression is called the specific resistance of the substance of which the wire is made ; it is the resistance of a cube of the substance of which the edge is equal to the unit of length, the current passing through the cube parallel to one of its edges. 177. Heat generated by the passage of a cur- rent through a conductor. Let A and B be two points connected by a conductor, let E be the electromotive force from A to B. By the definition of electromotive force, work equal to E is done on unit positive charge when it goes from A to B, and on unit negative charge 177] ELECTRIC CURRENTS. 287 when it goes from B to A; hence if in unit time N units of positive charge go from A to B and N' units of nega- tive charge from B to A, the work done is E {N + N'). But N-\- N' is equal to C, the strength of the current flowing from A to B, thus the work done is equal to EG. If R is the resistance of the conductor between A and B, E = RG ; thus the work done in unit time is equal to RG\ We see that the same amount of work would be spent in driving a current of the same intensity in the reverse direction, viz. from B to A. This work, by the principle of the Conservation of Energy, cannot be lost ; the work spent by the electric forces in driving the current must give rise to an equiva- lent amount of energy of some kind or other. The passage of the current heats the conductor, but if the heat is caused to leave the conductor as soon as produced the state of the conductor is not altered by the passage of the current. The mechanical equivalent of the heat pro- duced in the conductor was shown by Joule to be equal to the work spent in driving the current through the con- ductor, so that the work done in driving the current is in this case entirely converted into heat. Thus if H is the mechanical equivalent of the heat produced in time t, H = RGH. The law expressed by this equation is called Joule's Law. It states that the heat produced in a given time is pro- portional to the square of the strength of the current. Since by Ohm's Law E = RG, the heat produced in the time t is also equal to E^ ^t==EGt 288 ELECTRIC CURRENTS. [CH. IX 178. Voltaic Cell. We have seen that in an electric field due to any distribution of positive and negative electricity, the work done when unit charge is taken round a closed circuit vanishes ; the electric intensity due to such a field tending in some parts of the circuit to stop the unit charge in some parts of its course and to help it on in others. Hence such a field cannot produce a steady current round a closed circuit. To maintain such a current work must be done; this work may be supplied from chemical sources, as in the voltaic battery, from thermal sources, as in the thermoelectric circuit, or by mechanical means, as when currents are produced by dynamos. We shall consider here the case of the voltaic circuit. Let us consider the simple form of battery consisting of two plates, one of zinc, the other of copper, dipping into a vessel containing dilute sulphuric acid. If the zinc and copper plates are connected by a wire, a current will flow round the circuit, flowing from the zinc to the copper through the acid, and from the copper to the zinc through the wire. When the current flows round the circuit the zinc is attacked by the acid and zinc sulphate is formed. For each unit of electricity that flows round the circuit one electro-chemical equivalent of zinc and sulphuric acid disappears and equivalent amounts of zinc sulphate and hydrogen are formed. Now if a piece of pure zinc is placed in dilute acid very little chemical action goes on, but if a piece of copper is attached to the zinc the latter is immediately attacked by the acid and zinc sulphate and hydrogen are produced ; this action is accompanied by a considerable heating eff'ect, and we find that for each gramme of zinc consumed a definite amount of heat is produced. Now let us consider two 179] ELECTRIC CURRENTS. 289 vessels (a) and (ff), such that in (a) the zinc and copper form the plates of a battery, while in (^) the zinc has merely got a bit of copper fastened to it: let a definite amount of zinc be consumed in the latter and then let the current run through the battery until the same amount of zinc has been consumed in (a) as in (/8). The same amount of chemical combination has gone on in the two cells, hence the loss of chemical energy is the same in (a) as in (0). This energy has been converted into heat in both cases, the difference being that in the cell (y8) the heat is produced close to the zinc plate, while in (a) the places where heat is produced are distributed through the whole of the circuit, and if the wire connecting the plates has a much greater resistance than the liquid between them, by far the greater portion of the heat is produced in the wire, and not in the liquid in the neighbourhood of the zinc. Though the distribution • of the places in which the heat is produced is different in the two cases, yet, since the same changes have gone on in the two cells, it follows from the principle of the Conservation of Energy, that the total amount of heat produced in the two cases must be the same. Thus the total amount of heat produced by the battery cell (a) must be equivalent to that developed by the combination of the amount of zinc consumed in the cell while the current is passing with the equivalent amount of sulphuric acid. 179. Electromotive Force of a Cell. If C is the current, R the resistance of the wire between the plates, r that of the liquid between the plates, t the time the current has been flowing, then by Joule s law the mechanical equivalent of the heat generated in the wire is RC% that of the heat generated in the liquid is rCH. We shall T. E. 19 290 ELECTKIC CURRENTS. [CH. IX see in Chapter XIII, that when a current flows across the junction of two different metals, heat is produced or absorbed at the junction; this effect is called the Peltier effect. The laws governing the thermal effects at the junction of two metals differ very materially from Joule's Law. The heat developed in accordance with Joule's Law in a conductor AB is, as long as the strength of the current remains unaltered, the same whether the current flows from ^ to 5 or from ^ to -4. The thermal effects at the junction of two metals C and D depend upon the direction of the current ; thus if there is a development of heat when the current flows across the junction from C to D there will be an absorption of heat at the junction if the current flows from I) to C. These heat effects which change sign are called reversible heat effects. The heat developed at the junction of two substances in unit time is directly proportional to the strength of the current and not to its square. In the case of the voltaic cell formed of dilute acid and zinc and copper plates, the current passes across the junction of the zinc and acid, of the acid and copper as well as across the metallic junctions which occur in the wire used to connect up the two plates. Let F be the heat developed at all these junctions when traversed by unit current for unit time. Then the total amount of heat developed in the voltaic cell is RCH + rCH + PGt Since a current G has passed through the cell for a time t, the number of units of electricity which have passed through the cell is Gt, hence, if e is the electro- chemical equivalent of zinc, eGt grammes of zinc have been 179] ELECTRIC CURRENTS. 291 converted into zinc sulphate. Let w be the mechanical equivalent of the heat produced when one gramme of zinc is turned into zinc sulphate, then the mechanical equi- valent of the heat which would be developed by the chemical action which has taken place in the cell is eCtw ; but this must be equal to the mechanical equivalent of the heat developed in the cell, and hence we have RCH + rCH + PGt = eCtw, or {R-\-r)C=ew-P. The quantity on the right-hand side is called the electro- motive force of the cell. We see that it is equal to the sum of the products of the current through the external circuit and its resistance and the current through the battery and its resistance. We shall now prove that if the zinc and copper plates instead of being connected by a wire are connected to the plates of a condenser, then if these plates are made of the same material, they will be at different potentials and the difference between their potentials will equal the electromotive force of the battery. For if the system has got into a state of equilibrium, then when any change is made in the electrical conditions, the increase in the electrical energy must equal the energy lost in making the change. Suppose that the potential of the plate of the condenser in connection with the copper plate in the battery exceeds by E the potential of the other plate of the condenser in connection with the zinc plate of the battery, and suppose now that the electrical state is altered by a quantity of electricity equal to BQ passing from the plate of the condenser at low potential to the 19—2 292 ELECTRIC CURRENTS. [CH. IX plate at high potential through the battery from the zinc to the copper. The electrical energy of the condenser is increased by ^8Q, while the passage of this quantity of electricity will develop at the junctions of the different substances in the cell a quantity of heat whose mechanical equivalent is equal to PBQ. If t were the time this charge took to pass from the one plate to the other the average current would be equal to BQ/t, hence the heat developed in accordance with Joule's law would be pro- portional to (SQ/ty X ^ or to (SQy/t ; by making SQ small enough we can make this excessively small compared with either ESQ or P8Q which depend on the first powers of SQ. The loss of chemical energy is eSQ x w, and this must be equal to the heat produced plus the increase in the electrical energy, hence we have EBQ 4- PBQ = eSQ x w, or E = ew — P, that is, the difference of potential between the plates of the condenser is equal to the electromotive force of the battery. Hence we can determine this electromotive force by measuring the difference of potential. The simple form of voltaic cell just described does not give a constant E. M. F., as the hydrogen produced by the chemical action does not all escape from the cell ; some of it adheres to the copper plate, forming a gaseous film which increases the resistance and diminishes the electromotive force of the cell. The copper plate with the hydrogen adhering to it is said to be polarized and to be the seat of a back electro- motive force which makes the electromotive force of the battery less than its maximum theoretical value. We 179] ELECTRIC CURRENTS. 293 shall perhaps get a clearer view of the condition of the copper plate with its film of hydrogen from the following considerations. The hydrogen in an electrolyte follows the current and thus behaves as if it had a positive charge of electricity ; if now the atoms of hydrogen when they come up to the copper plate do not at once give up their charges to the plate but remain charged at a small distance from it, then we shall have what is equivalent to a charged parallel plate condenser at the copper plate, the positively charged hydrogen atoms corresponding to the positive plate of the condenser, and the copper to the negative plate. The condenser will tend to discharge itself through the cell in the direction of the arrow (Fig. 87), Zn Copper Fig. 87. i.e. in the opposite direction to that of the current through the cell ; the difference of potential between the plates of this condenser corresponds to the back electromotive force due to the polarization of the copper plate. Another cause of inconstancy is that the zinc sulphate formed acts as an electrolyte and carries some of the current ; the zinc, travelling with the current, is deposited against the copper plate and alters the electromotive force of the cell. The deposition of hydrogen against the positive plate 294 ELECTRIC CURRENTS. [CH. IX of the battery, and its liberation as free hydrogen can be avoided in several ways ; in the Bichromate Battery the copper plate is replaced by carbon, and potassium bichro- mate is added to the sulphuric acid ; as the bichromate is an active oxidising agent it oxidises the hydrogen as soon as it is formed and thus prevents its accumulation on the positive plate. 180. DanielPs Cell. In Daniell's cell, the zinc and sulphuric acid are enclosed in a porous pot (Fig. 88) made ZINC ROD SULPHURIC ACID SOL. POROUS POT CORPER SULPHATE SOL COPPER CYLINDER Fig. 88. of unglazed earthenware; the copper electrode usually takes the shape of a cylindrical copper vessel, in which the porous pot is placed. The space between the porous pot and the copper is filled with a saturated solution of copper sulphate in which crystals of copper sulphate are placed to replace the copper sulphate used up during the working of the cell. When the sulphuric acid acts upon the zinc, zinc sulphate is formed and hydrogen gas libe- rated ; the hydrogen following the current, travels through the porous pot, where it meets with the copper sulphate, 181] ELECTRIC CURRENTS. 295 chemical action takes place and sulphuric acid is formed and copper set free. This copper travels to the copper cylinder and is there deposited. Thus in this cell instead of hydrogen being deposited on the copper, we have copper deposited, so that no change takes place in the condition of the positive pole and there is no polarization. 181. Calculation of E. M. F. of DanielPs Cell. The chemical energy lost in the cell during the passage of one unit of electricity may be calculated as follows : in the porous pot we have one electro-chemical equivalent of zinc sulphate formed while one equivalent of sulphuric acid disappears ; in the fluid outside this pot one equiva- lent of sulphuric acid is formed and one equivalent of copper sulphate disappears, thus the chemical energy lost is that which is lost when the copper in one electro- chemical equivalent of copper sulphate is replaced by the equivalent quantity of zinc. Now the electro-chemical equivalent of copper is '003261 grammes, and when 1 gramme of copper is dissolved in sulphuric acid the heat given out is 909'5 thermal units or 909*5 x 4*2 x 10' mechanical units, since the mechanical equivalent of heat on the c. G. s. system is 4'2 X 10'. Thus the heat given out when one electro- chemical equivalent of copper is dissolved in sulphuric acid is '003261 x 909*5 x 42 x 10'= 1*245 x 10^ in me- chanical units. The electro-chemical equivalent of zinc is '003364 grammes, and the heat developed when 1 gramme of zinc is dissolved in sulphuric acid is 1670 x 4*2 x 10' in mechanical units. Hence the heat developed when one electro-chemical equivalent of zinc is dissolved in sulphuric 296 ELECTRIC CURRENTS. [CH. IX acid is -003364 x 1670 x 4*2 x 107 = 2-359 x 10« mechanical units. Thus the loss of chemical energy in the porous pot is 2*359 X 10^ while the gain in the copper sulphate is 1-245 X 10«, thus the total loss is 1-114 x lO^. Thus ew in Art. 179=1-114x10^. The electromotive force of a Daniell's cell is about 1-028 x 10^ We see from the near agreement of these values that the reversible ther- mal effects (see Art. 179) are of relatively small importance, though if we ascribe the difference between the two num- bers to this cause these effects would be much greater than those observed when a current flows across the junction of two metals. 182. In Grove's cell the hydrogen at the positive pole is got rid of by oxidising it by strong nitric acid. The zinc and sulphuric acid are placed in a porous pot, and this is placed in a larger cell of glazed earthenware containing nitric acid ; the positive pole is a strip of platinum foil dipping into the nitric acid. This cell has a large electro- motive force, viz. 1*97 x 10^ Bunsen's cell is a modification of Grove's, in which the platinum is replaced by hard gas carbon. 183. Clark's cell, which on account of its constancy has been legalized as the standard of electromotive force, is made as follows. The outer vessel (Fig. 89) is a small test tube containing a glass tube down which a platinum wire passes ; a quantity of pure redistilled mercury suffi- cient to cover the end of this wire is then poured into the tube ; on the mercury rests a paste made by mixing mercurous sulphate, saturated zinc sulphate and a little 184] ELECTRIC CURRENTS. 297 zinc oxide to neutralize it ; a rod of pure zinc dips into the paste and is held in position by passing through a MARINE GLUE ZINC SULPHATE SOLUTION ZINC SULPHATE CRYSTALS MERCUROUS SULPHATE PLATINUM WIRE Fig. 89. cork in the mouth of the test tube. The electromotive force of this cell is 1'4!34 x 10^ at 15° Centigrade. 184. Polarization. When two platinum plates are immersed in a cell containing acidulated water, and a current from a battery is sent from one plate to the other through the water, we find that the current for some time after it begins to flow is not steady but keeps diminishing. If we observe the condition of the plates, we shall find that oxygen adheres to the plate A, at which the current enters the cell, while hydrogen adheres to the other plate B, by which the current leaves the cell. If these plates are now disconnected from the battery and connected by a wire a current will flow round the circuit so formed, the current going from the plate B to the plate A through the electrolyte and from A to B through the wire. This current is thus in the opposite direction to that which originally passed through the cell. The 298 ELECTRIC CURRENTS. [CH. IX plates are said to be polarized, and the e.m.f. round the circuit, when they are first connected by the wire, is called the electromotive force of polarization. When the plates are disconnected from the battery and connected by the wire the hydrogen and oxygen gradually disappear from the plates as the current passes. In fact we may regard the polarized plates as forming a voltaic battery, in which the chemical action maintaining the current is the com- bination of hydrogen and oxygen to form water. Though hydrogen and oxygen do not combine at ordinary tem- peratures if merely mixed together, yet the oxygen and hydrogen condensed on the platinum plates combine readily as soon as these plates are connected by a wire so as to make the oxygen and hydrogen parts of a closed electrical circuit. There are numerous other examples of the way in which the formation of such a circuit facilitates chemical combination. 185. A Finite Electromotive Force is required to liberate the Ions from an electrolyte. This follows at once by the principle of the Conservation of Energy if we assume the truth of Faraday's Law of Electrolysis. Thus suppose for example that we have a single Daniell's cell placed in series with an electrolytic cell containing acidulated water; then if this arrangement could produce a current which would liberate hydrogen and oxygen from the electrolytic cell, for each electro-chemical equivalent of zinc consumed in the battery an electro-chemical equiva- lent of water would be decomposed in the electrolytic cell. Now when one electro-chemical equivalent of hydrogen combines with oxygen to form water, 1*4<7 x 10^ mechanical units of heat are produced, and the decomposition of one 185] ELECTRIC CURRENTS. 299 electro-chemical equivalent of water into free hydrogen and oxygen would therefore correspond to the gain of this amount of energy. But for each electro-chemical equi- valent of zinc consumed in the battery the chemical energy lost is (Art. 181) equal to 1*114 x 10^ mechanical units. Hence we see that if the water in the electrolytic cell were decomposed, 3'56 x 10'' units of energy would be gained for each unit of electricity that passed through the cell : as this is not in accordance with the principle of the Conservation of Energy the decomposition of the water cannot go on. We see that electrolytic decom- position can only go on when the loss of energy in the battery is greater than the gain of energy in the electro- lytic cell. If we attempt to decompose an electrolyte, acidulated water for example, by an insufficient electromotive force the following phenomena occur. When the battery is first connected to the cell a current of electricity runs through the cell, hydrogen travelling with the current to the plate where the current leaves the cell, oxygen travelling up against the current to the other plate. Neither the hydrogen nor the oxygen, however, is libe- rated at the plates, but adheres to the plates, polarizing them and producing a back E. M. F. which tends to stop the current ; as the current continues to flow the amount of gas against the plates increases, and with it the polari- zation, until the E. M. F. of the polarization equals that of the battery, when the current sinks to an excessively small fraction of its original value. The current does not stop entirely, a very small current continues to flow through the cell. This current has however been shown by V. Helm hoi tz to be due to hydrogen and oxygen 300 ELECTRIC CURRENTS. [CH. IX dissolved in the electrolytic cell and does not involve any separation of water into free hydrogen and oxygen. The way in which the residual current is carried is somewhat as follows. Suppose that the battery with its small e.m.f. has caused the current to flow through the cell until the polarization of the plates is just sufficient to balance the E. M. F. of the battery ; the oxygen dissolved in the water near the hydrogen coated plate will attack the hydrogen on this plate, combining with it to form water, and will, by removing some of the hydrogen, reduce the polarization of the plate ; similarly the hydrogen dissolved in the water or it may be absorbed in the plate, will attack the oxygen on the oxygen coated plate and reduce its polarization. The e. m. f. of the polarization being reduced in this way no longer balances the E. M. F. of the battery ; a current therefore flows through the cell until the polarization is again restored to its original value, to be again reduced by the action of the dissolved gases. Thus in consequence of the depolarizing action of the dissolved gases there will be a continual current tending to keep the E. M. F. of the polarization equal to that of the battery; this current however is not accompanied by the liberation of free hydrogen and oxygen and its production does not violate the principle of the Conservation of Energy. 186. Cells in Series. When a series of voltaic cells, Daniell's cells for example, are connected so that the zinc pole of the first is joined up to the copper pole of the second, the zinc pole of the second to the copper pole of the third, and so on, the cells are said to be connected up in series. In this case the total electromotive force of the cells so connected up is equal to the sum of the 187] ELECTRIC CURRENTS. 301 electromotive forces of the individual cells. We can see this at once if we remember (see Art. 179) that the electro- motive force of any system is equal to the difference be- tween the chemical energy lost, when unit of electricity passes through the system, and the mechanical equivalent of the reversible heat generated at junctions of different substances: when the cells are connected in series the same chemical changes and reversible heat effects go on in each cell when unity of electricity passes through as when the same quantity of electricity passes through the cell by itself, hence the e.m.f. of the cells in series is the sum of the e.m.f.'s of the individual cells. The resistance of the cells when in series is the sum of their resistances when separate. Thus if E is the E. M. F. and r the resistance of a cell, the e.m.f. and resistance of n such cells arranged in series are respectively nE and nr. 187. Cells in parallel. If we have n similar cells and connect all the copper terminals together for a new terminal and all the zincs together for the other terminal the cells are said to be arranged in parallel. In this case we form what is equivalent to a large cell whose e.m.f. is equal to E, that of any one of the cells but whose resistance is only rjn. 188. Suppose that we have N equal cells and wish to arrange them so as to get the greatest current through a given external resistance R. Let the cells be divided into m sets, each of these sets consisting of n cells in series, and let these m sets be connected up in parallel. The E. m. f. of the battery thus formed will be nE, its resistance nrjin, where E and r are respectively the e.m.f. and 302 ELECTRIC CURRENTS. [CH. IX resistance of one of the cells. The current through the external resistance R will be equal to 7iE E -r, nr R r ' R+— -+- m 71 m Now nm — N, hence the denominator of this expression is the sum of two terms whose product is given, it will therefore be least when the terms are equal, i.e. when n m* or R = — r. m Since the denominator in this case is as small as possible the current will have its maximum value. Since nr/m is the resistance of the battery we see that we must arrange the battery so as to make, if possible, the resistance of the battery equal to the given external resistance. This arrangement, though it gives the largest current, is not economical, for as much heat is wasted in the battery as is produced in the external circuit. 189. Distribution of a steady current in a System of Conductors. KirchhofT's Laws. The distribution of a steady current in a network of linear conductors can be readily determined by means of the following laws which were formulated by Kirchhoff. 1. The algebraical sum of the currents which meet at any point is zero. 190] ELECTRIC CURRENTS. . 303 2. If we take any closed circuit the algebraical sum of the products of the current and resistance in each of the conductors in the circuit is equal to the electromotive force in the circuit. The first of these laws expresses that electricity is not accumulating at any point in the system of conductors; this must be true if the system is in a steady state. The second follows at once from the relation (see Art. 179) RI-\-rI = E, where R is the external resistance, r the resistance of the battery whose E. M. F. is E and / the current through the battery. For RI is the difference of potential between the terminals of the battery, and by Ohm's law this is equal to the sum of the products of the strength of the current and the resistance for a series of conductors forming a continuous link between the terminals of the battery. 190. Wheatstone's Bridge. We shall illustrate these laws by applying them to a very important case of a network of conductors, the system known as the Fig. 90. Wheatstone's Bridge. In this system a battery is placed in a conductor AB, and five other conductors A C, BG, AD, BD, CD are connected up in the way shown in Fig. 90. 304 ELECTRIC CURRENTS. [CH. IX Let E be the electromotive force of the battery, B the resistance of the battery circuit AB, i.e. the resistance of the battery itself plus the resistance of the wires con- necting its plates to A and B. Let G be the resistance of CD, and 6, a, a, jS the resistances of AC, BG, AD, BD respectively. Let x be the current through the battery, y the current through AG, z that through GD. By Kirchhoff's first law the current through AD will be x — y, that through GB y — z, and that through DB X — y ^ z. Since there is no electromotive force in the circuit AGD we have by Kirchhoff's second law, hy ■\- Gz — a{x — y) -^\ the negative sign is given to the last term because travelling round the circuit in the direction AGD the current x-y flows in the direction opposite to that in which we are moving; rearranging the terms we get (6 + a)7/+(T^- a« = (1). Since there is no electromotive force in the circuit GDB, we have Gz ^- ^ {x - y -^ z) - a{y - z) = 0, or -(a + /3)2/ + (G^ + c^ + ^)^ + ^^=0 (2). From (1) and (2) we get x_ ^ y 6^ (a 4- 6 + a + /3) + (6 + a) (a + ^e) (? (a + ^) + a (a + /S) "iS^^P ^*'^^' Since the electromotive force round the circuit AGB is E, we have Bx -{■ by ■\- a {y — z) — E\ 190] ELECTRIC CURRENTS. 305 hence by (3), we have x={G(a + hi-a + l3) + (h-\-a)(a + l3)] E\ 3/={G^(a + /3) + a(a+/3)) E F a;-y={G{a + h) + b(a + ^)} y-,= {G(a + ^) + ^(a-^h)]^ y + z = {G (a + h) + a {h + a)] E A E A E A y.-w, where A=^BG{a + h + a + ^) + B(h + a)(a + ff) + G(a-h b){a + 0) -{-oL(a + I3){a -\-h)-a(aoL- hjS) = BG(a + h^a + 0)-\-B(b + a)(a-h0) -\-G(a + h)(oL + /3) + ahoL + a6/3 + aa/3 + hajS, A is the sum of the products of the six resistances B, G, a, 6, a, /3, taken three at a time, omitting the product of any three which meet in a point. In the expressions given in equations (4) for the currents through the various branches of the network of resistances, we see that the multiplier of E/A in the expression for the current through an arm (P) (other than CD) is the sum of the products of the resistances other than the battery resistance and the resistance of P taken two and two, omitting the product of any two which meet at either of the extremities of the battery arm or at either of the extremities of the arm P. T. K. 20 306 ELECTRIC CURRENTS. [CH. IX From these expressions we see at once that if we keep all the resistances the same then the current in one arm (A) due to an electromotive force E in another arm (B), is equal to the current in (B) when the electromotive E is placed in the arm A. This reciprocal relation is not confined to the case of six conductors, but is true what- ever the number of conductors may be. We may write the expression for x given by equation (4) in the form E ^~ B-^R' where p _ G^ (g + 6) (ct + yg) + aoL^ 4- aoih + a/36 + a^h 6^(a + 6 + a+;8) + (6 + a)(a + ^) ' R is the resistance, between A and B, of the crossed quadrilateral AGBD. We see that R = (sum of products of the 5 resistances of this quadrilateral taken 3 at a time : leaving out the product of any three that meet in a point) : divided by the sum of the products of the same resistances taken two at a time, leaving out the product of any pair that meet in A or B. 191. Conjugate Conductors. The current through CD will vanish if aa = 6/3 ; in this case ^5 and CD are said to be conjugate to each other, they are so related that an electromotive force in AB does not produce any current in CD: it follows from the reciprocal relation that when this is the case an electromotive force in CD will not produce any current in AB. 191] ELECTRIC CURRENTS. 307 The condition that CD should be conjugate to AB may be got very simply in the following way. If no current flows down CD, G and D must be at the same potential ; hence since <2 = 0, we have by Ohm's Law hy = a{x- y), since the difference of potential between A and G is equal to that between A and D. Since the difference of potential between G and B is equal to that between D and B, we have ay = ^{x-y)\ hence eliminating y owdi x — y, we get h_a or 6/3 = aa. When this relation holds we may easily prove that - (a + ^)(a+J)| ■^ a + 6 + aH-/3j ' which we may write as where S is the resistance of ADB, AGB placed in series, P the resistance of the same conductors when in parallel, and P' the resistance of GAD, GBD in parallel. When AB is conjugate to CD, then in whatever part of the network an electromotive force is placed, the current through one of these arms is independent of the resistance in the other. We may deduce this from the preceding expressions for the currents in various 20—2 308 ELECTRIC CURRENTS. [CH. IX arms of the circuit ; it can also be proved in the following way, which is applicable to any number of conductors. Suppose that an electromotive force in some branch of the system produces a current through AB, then we may introduce any e.m.f. we please into AB without altering the current through its conjugate CD. We may in par- ticular introduce such an electromotive force as would make the current through AB vanish, without altering the current in CD, but the effect of making the current in AB vanish would be the same as supposing AB to have an infinite resistance ; hence we may make the resistance oi AB infinite without altering the current through CD. 192. We may use Wheatstone's Bridge to get a differ- ence of potential which is a very small fraction of that of the battery in the Bridge. The difference of potential between C and D is equal to Gz, i.e. to G{aa-b^)E A it thus bears to E the ratio of G (aa — h^) to A. By making aa. — bjS small we can without using either very small or very large resistances make the ratio of the poten- tial difference between C and D to E exceedingly small; for example, let a =101, a =99, b = 13=100, B = G=1. Thus we find that this ratio is nearly equal to 1/4 x 10^, or the potential difference between C and D is only about one four- millionth part of the E.M.F. of the battery. 193. Heat produced in the System of Con- ductors. Assuming Joule's law (see Art. 177) we shall show that for all possible distributions consistent with 193] ELECTRIC CURRENTS. 309 Kirchhoff's first law, the one that gives the minimum rate of heat production is that which obeys the second law. For, consider any closed circuit in a network of con- ductors, let u, V, w ... he the currents through the arms of this circuit as determined by Kirchhoff's laws, and rj, Ta, ... the corresponding resistances. The rate of heat production in this closed circuit is by Joule's law equal to r^u^ + r^v^ + (1). Now suppose that the currents in this circuit are altered in the most general way possible consistent with leaving the currents in the conductors not in the closed circuit unaltered, and consistent also with the condition that the algebraical sum of the currents flowing into any point should vanish : we see that these conditions require that all the currents in the closed circuit should be increased or diminished by the same amount. Let them all be increased by f ; the rate of heat production in the circuit is now by Joule's law n (^ +?)'+ ^^2 (v + ?)'+... = r^ii^ -{- Vo^ -{-... + 2^ {rill -\- i\v + . . .) -^ {i\ + n^ r^-\- ...) p. Now since the currents u, v, w are supposed to be determined by Kirchhoff's laws i\u + ^2?; + . . . = 0, if there is no electromotive force in the closed circuit. Hence the rate of heat production is equal to n^t24-r2v' + ...+(rl + r2 + ?^3+...)f2 (2). Of the two expressions (1) and (2) for the rate of heat production (2) is always the greater ; hence we see that any deviation of the currents from the values determined by Kirchhoff's law would involve an increase in the rate of heat production. 310 ELECTRIC CURRENTS. [CH. IX 194. Use of the Dissipation Function. We may often conveniently deduce the actual distribution of the currents by writing down F the expression for the rate of heat production and making it a minimum, subject to the condition that the algebraical sum of the currents which meet in a point is zero. Or we may by the aid of this condition express as in the example of the Wheatstone's Bridge, the current, through the various arms in terms of a small number of currents x, y, z, then express the rate of heat production in terms of x, y, z. F is often called the Dissipation Function. When there are electromotive forces Ep, Eg in the arms through which currents iip, Uq are flowing respec- tively, then the actual distribution of current is that which makes F-2{EpUp + EgUq-\-...) a minimum. Thus in the case of the Wheatstone's Bridge (Art. 190) F = Bx'' + by^ + a(y- zf- -\-Gz''+a(x- yy + p{x-y-\- zf, and equations (4) of Art. 190 are equivalent to ^(F-2Ex) = {), dy^ ' ^^iF-2Ex) = 0, i(F-2Ex) = 0^ which are the conditions that F — 2Ex should be a minimum. A very important example of the principle that steady currents distribute themselves so as to make the 196] ELECTRIC CURRENTS. 311 rate of heat production as small as possible, is that of the flow of a steady current through a uniform wire ; in this case the rate of heat production is a minimum when the current is uniformly distributed over the cross section of the wire. 195. It follows from Art. 193 that if two electrodes are connected by any network of conductors, the equivalent resistance is in general increased, and is never diminished, by an increase in the resistance of any arm of this net- work. If R is the resistance between the electrodes, i the current flowing in at one electrode and out at the other, then Ri^ is the rate of heat production. Let A and B respectively denote the network before and after the increase in resistance in one or more of its arms. By suitable constraints we can make the distribution of currents through A the same as that actually existing in B. The rate of heat production in the constrained system is however greater than that in A. Now take this con- strained system and without altering the currents suppose that the resistances are increased until they are the same as in B. But since the resistances are increased without altering the currents the rate of heat production is in- creased, so that as this rate was greater than in A before the resistances were increased it will a fortiori be greater afterwards. But after the resistances were increased the currents and resistances are the same as B, hence the rate of heat production and therefore the resistance of B is greater than that of A. 196. Distribution of Current through an infinite Conductor. We shall now consider the case when the 312 ELECTRIC CURRENTS. [CH. IX currents instead of being constrained to flow along wires are free to distribute themselves through an unlimited conductor whose conductivity is constant throughout its volume. We shall suppose that the current is introduced into this conductor by means of perfectly conducting electrodes, i.e. electrodes made of a material whose specific resistance vanishes. The currents will enter and leave these electrodes at right angles, for a tangential current in the conductor would correspond to a finite tangential electric intensity in the conductor and therefore in the electrode, but in the perfectly conducting electrode a finite electric intensity would correspond to an infinite current. Let A and B be the electrodes, i the current which enters at A and leaves at B ; then we shall prove that the current at any point P in the conductor is in the same direction as, and numerically equal to, the electric intensity at the same point, if we suppose the conducting material between the electrodes to be replaced by air, and the electrodes A and B to have charges of electricity equal to ij^ir and — ij^TT respectively. For the current is determined by the conditions that it is at right angles to the surfaces A and B, and that since the current is steady, and there is no accumulation of electricity at any part of the con- ductor, the quantity of electricity which flows into any region equals the quantity which flows out. Hence we see that the outward flow over any closed surface enclos- ing A and not B is equal to i, over any closed surface enclosing B and not A is equal to — i, and over any closed surface enclosing neither or both of these surfaces is zero. But the electric intensity, when the conductor is replaced by air and A has a charge ij^ir of positive electri- city, while B has an equal charge of negative electricity. 197] ELECTRIC CURRENTS. 313 satisfies exactly the same conditions, which are sufficient to determine it without ambiguity ; hence the current in the conductor is equal to the electric intensity in the air and is in the same direction. A line such that the tangent to it at any point is in the direction of the current at that point is called a stream-line. The stream-lines coincide with the lines of force in the electrostatic problem. 197. If q is the intensity of the current at any point P (i.e. the current flowing through unit area at right angles to the stream-line at P), a the specific resistance of the conductor, ds an element of the stream-line, then by Ohm's law the E.M.F. between the electrodes A and B is equal to jaqds, the integral being extended from the surface of A to that of B. As (7 is constant, this is equal to ajqds. If F is the electric intensity at P in the electrostatic problem, since F=q, the E.M.F. between A and B is equal to portion of the surface of separation of two media, o-i the specific resistance of the upper medium, o-^ that of the lower, let 6 and <^ be the angles which the directions of the current in the upper and lower media respectively make with the normal to the surface. Let q^, q^ be the intensities of the currents in the two media, i.e. the amount of current flowing across unit areas drawn at right angles to the direction of flow. Then since, when things are in a steady state, there is no increase or decrease in the electricity at the junction of the two media, the currents along the normal must be the same in the two media. Thus qi cos 6 = q2 cos (f) (1). Again, the electric intensity parallel to the surface must be the same in the two media, and since the elec- tric intensity in any direction is equal to the specific 318 ELECTRIC CURRENTS. [CH. IX resistance of the medium multiplied by the intensity of the current in that direction, we have o-i^i sin 9 = o-25'2 sin (^ (2), hence from (1) and (2) we have o-j tan 6 = (Ti tan <^. This relation between the directions of the currents in the two media is identical in form with that given in Arts. 74 and 157, for the relation between the direc- tions of the lines of electric intensity and of magnetic force when these lines pass from one medium to another. We see that if o-j is greater than o-g, then (f> is greater than 6\ hence when the current flows from a poor con- ductor into a better one the current is bent away from the normal. The bending of the current as it flows from one medium into another is illustrated in Fig. 92, which is Copper taken from a paper by Quincke. The figure represents the current lines in a circular lamina, one half of which 198] ELECTRIC CURRENTS. 319 is lead, the other half copper, the electrodes being placed on the circumference. It shows how the currents in going from the worse conductor (the lead) to the better one (the copper) get bent away from the normal to the surface of separation. The electric intensity parallel to the normal in the medium whose specific resistance is (r^ is (Tiqi cos 6, that in the medium whose specific resistance is a^ is 0-2^2 cos 7r times the current enclosed by the circle. Hence we have 27rOP . T=4<7r (current enclosed by the circle with centre and radius OP). If the current is all outside this circle, the right-hand side of this equation vanishes: hence Evanishes and there is no magnetic force. Thus there is no magnetic force in the interior of a cylindrical tube conveying a current. If the current is uniformly distributed over the cross section, and i is the total current flowing through the cylinder whose radius we shall denote by a, the current through the circle whose radius is OP is equal to .OP' 330 MAGNETIC FORCE DUE TO CURRENTS. [CH. X Hence Thus when the current is uniformly distributed, the magnetic force inside the cylinder varies directly as the distance from the axis ; outside the cylinder it varies in- versely as this distance. 204. The total normal magnetic induction through any cylindric surface passing through two lines which inter- sect a plane at right angles to the current in the points A and B, Fig. 95, is the same whatever be the shape of the surface connecting these lines : this follows at once from the principle that the total magnetic induction over any closed surface is zero. Let us take the cylindrical surface such that if B is the point nearest to 0, the normal section Fig. 95. of the surface is the circular arc BG and the radial portion GA. Since the magnetic force is everywhere tangential 205] MAGNETIC FORCE DUE TO CURRENTS. 331 to BC no tube of force passes through the portion corre- sponding to BC ; if r is the distance of any point P on GA from 0, the magnetic force at P is 2i hence the number of tubes of magnetic force passing through the portion corresponding to AC is /, o^2i. ^., OA , —dr=2i log Y^ oc r ^ DC and this represents the number passing through each unit of length of any cylindric surface passing through A and B. 205. Two infinitely long^ straight parallel cur- rents flowing in opposite directions. Let A and B, Fig. 96, be the points where the axes of the currents intersect a plane drawn at right angles to the direction of the currents. Let the direction of the current at A be downwards through the paper, that at B upwards ; if i is the strength of either current, the magnetic potential at a point P is, Art. 202, equal to 2i [< PAB ± 2'7rn] - 2i [ir-< PBA ± 21^1]. 332 MAGNETIC FORCE DUE TO CURRENTS. [CH. X This may be written 47n (n + m) - < APB x 2i ; hence along an equipotential line the angle APB is con- stant, hence the equipotential lines are the series of circles passing through AB. The lines of magnetic force are at right angles to the equipotential lines, they are therefore the series of circles having their centres along AB such that the tangents to them from 0, the middle point of AB, are of the constant length OA. The lines of magnetic force and the equipotential lines are represented in Fig. 97. Fig. 97. The direction of the magnetic force is easily found as follows. If PT is the direction of the magnetic force at P, then since PT is the normal to the circle round 206] MAGNETIC FORCE DUE TO CURRENTS. 833 APB, the angle BPT is equal to the complement of the angle PAB. The magnetic force i? at P is the resultant of the forces 2ilAP at right angles to ^P and ^ijBP at right angles to BP. Resolving these along PT, we have P = -f' cos ABP + -^ cos BAP AP BP 2iAB AP.BP' Thus the intensity of the magnetic force at P varies inversely as the product of the distances of P from A and B. At a point on the line bisecting AB at right angles AP = BP, and along this line, which may be called the axis of the current, the magnetic force is inversely pro- portional to the square of the. distance from A and B; the direction of the force is parallel to the axis. At a point whose distances from A and B are large compared with AB we may put AP = BP = OP, in this case the magnetic force varies inversely as OP-^, and the direction of the force makes with OP the same angle as OP makes with the line at right angles to AB. 206. Number of tubes of magnetic force due to the two currents which pass through a circuit con- sisting of two parallel wires. Let ^, P be the points where the axes of the two currents intersect a plane 334 MAGNETIC FORCfi DUE TO CURRENTS. [CH. X wires of the circuit cut the same plane. Then, Art. 204, the number of tubes of magnetic force due to A which AlG pass through GB per unit length = 1% log -j-j. . Similarly the number which pass through GB and are due to the current B is -2*^og-g^; hence the number through GB per unit length due to the current i at A and — i at B, is -.j, AG . BC\ We see from the symmetry of the expression that this is the number which would pass through the circuit AB due to currents H- i and — i at G and B respectively. When the circuits AB, GB are so situated that when the total number of tubes passing through GB due to the current in A, B is zero, the circuits AB, GB are said to be conjugate to each other. The condition for this is that loe: -i-FT—rrr; should vanish, or that ^ AB . BG AG _AB BG~ BB' another way of stating this result is that G and B must be two points on the same line of magnetic force due to the currents at A and B; this is equivalent to the condition that A and B should be points on a line of magnetic force due to equal and opposite currents at G and B. Since the lines of magnetic force due to the 206] MAGNETIC FORCE DUE TO CURRENTS. 335 currents A and B are a series of circles with their centres on AB it follows that if CD is conjugate to AB it will remain conjugate however CD is rotated round the point 0', 0' being the point where the line bisecting CD at right angles intersects AB. A case of considerable practical importance is when we have two equal circuits AB and CD, the current through A being in the same direction as that through G and that through B in the same direction as that through D, Let us consider the case when AB and CD are equal and parallel and so placed that the points A, B, D, C are at the corners of a rectangle. Then if i is the current flowing round each of the circuits ; fl the magnetic potential at a point P will, by Art 202, be given by the equation n = — 2i6 — 2i(j) 4- constant, where 6 and are the angles subtended respectively by AB and CD at P. The lines of magnetic force are the curves which cut these at right angles, along such a line is constant, where ?'i, 7\, r.^, r^ are the distances of a point on the line from A, B, C, D respectively. The lines of magnetic force are represented in Fig. 98. There are two points E, F where the magnetic force vanishes ; these points are on the line drawn through 0, the centre of the rectangle, parallel to the sides A B and CD\ we can easily prove that OE is equal to OA, 336 MAGNETIC FORCE DUE TO CURRENTS. [CH. X At a point P on the axis of the current, i.e. on the line through at right angles to AB, the magnetic force is parallel to the axis and is by Art. 205 equal to 2i.AB' 2^ . CD p if OP = X, AB equal to Fig. 98. 2a, AG = 2d, the magnetic force at P is 4m 4- 4m a^ -}- (a) -{■ df a' + id-xf This is, neglecting the fourth and higher powers of x, equal to 8m f Sd' - g^ J a^ + rf^l (a^ + rfO'*"r thus, if VScZ = a, the term in x- disappears and the lowest power of X which appears in the expression for the magnetic force is the fourth. Thus with this relation between the size of the coils and the distance between them the force near varies very slowly as we move along the axis. 207] MAGNETIC FORCE DUE TO CURRENTS. 337 The number of tubes of magnetic force which pass through one circuit when a current i flows round the other may, by using the result given in Art. 206, easily be proved to be equal to 4^ log BG AG' 207. Direct and return currents flowing^ uni- formly through two parallel and infinite planes. K 3 L N r ^ Fm. 99. Let the two parallel planes be at right angles to the plane of the paper and let this plane intersect them in the lines AB, CD. Let a current i flow upwards at right angles to the plane of the paper through each unit length oiAB and downwards through each unit length of CD. Let EF be the section of the plane parallel to AB and CD and midway between them. We shall prove that the magnetic force between the planes is uniform and parallel to EF, being thus parallel to the planes in which the currents are flowing and at right angles to the currents. We shall begin by proving that the magnetic force has no component at right angles to the planes in which the currents are flowing. This is evidently true by T. E. 22 338 MAGNETIC FORCE DUE TO CURRENTS. [CH. X symmetry at all points in the plane midway between AB and CD; we can prove it is true at all points in the following way. Take. a rectangular parallelepiped one of whose faces is in the plane whose section is EF, let another pair of faces be parallel to the plane of the paper and the third pair perpendicular to the line EF, The total normal magnetic induction over this closed surface vanishes. Since the currents are uniformly distributed in the infinite planes, the magnetic induction will be the same at all points in a plane parallel to those in which the currents are flowing. Hence the total magnetic induction over the pairs of faces of the parallelepiped which are at right angles to the parallel planes will vanish : for the induction at a point on one face will be equal to that at a corresponding point on the opposite face, and in the one case it will be along the inward normal, in the other along the outward. Hence since the total induction over the parallelepiped is zero the induction over one of the faces parallel to the planes must be equal and opposite to that over the opposite face. But one of these faces is in the plane EF where the magnetic induction normal to the face vanishes ; hence the total normal induction over the other face must vanish, and since the induction is the same at each point at the face the induction can have no component at right angles to this face, i.e. at right angles to the planes in which the currents are flowing. This proof applies to all parts of the field, whether between the planes or outside them. To prove that the force parallel to the currents vanishes, we take a rectangle PQRS with two sides PQ, RS parallel to the currents, the other sides PS, QR being at right angles to the planes of the currents. No current I 207] MAGNETIC FORCE DUE TO CURRENTS. 339 flows perpendicularly through this rectangle, hence (Art. 201) the work done when unit magnetic pole is taken round its circumference is zero. But since the magnetic force parallel to PS, RQ vanishes, the work done on unit pole, if ^ is the force along PQ, F' that along RS, is equal to {F'-F')PQ. Since this vanishes F=F', i.e. F is constant throughout the field, and since it vanishes at an infinite distance it must vanish throughout the field. We have now proved that throughout the field the components of the magnetic force in two directions at right angles to each other vanish, hence the magnetic force, where it exists, must be parallel to EF, Fig. 99. By drawing a rectangle in the space outside the planes with one pair of its sides parallel to EF we can prove that the force parallel to EF also vanishes outside the planes, so that in this region there is no magnetic force. To find the magnitude of the magnetic force H between the planes, take a rectangle such as LMNK, Fig. 99, cutting one of the planes, the sides of the rectangle being respectively parallel and perpendicular to EF. The quan- tity of current flowing through this rectangle is i x LM, since i flows through each unit of length of the plane; hence 47ri x LM is equal to the work done in taking unit magnetic pole round the rectangle. But this work is H X LM, since no work is done when the pole is moving along MN, NK and KL, hence we have HxLM=4>irixLM, r H = 4>7n. Thus the magnetic force is independent of the distance between the planes. 22—2 340 MAGNETIC FORCE DUE TO CURRENTS. [CH. X 208. Solenoid. We can apply exactly the same method to the very important case of an infinitely long right circular solenoid, i.e. an infinitely long right circular cylinder round which currents are flowing in planes perpendicular to the axis. Such a solenoid may be con- structed by winding a right circular cylinder uniformly with wire, the planes of the winding being at right angles to the axis of the cylinder, so that between any two planes at right angles to the axis and at unit distance apart there are the same number of turns of wire. We can show by the same method as in Art. 207, that inside the cylinder the radial magnetic force vanishes, and that the force parallel to the axis of the cylinder is uniform, that out- side the cylinder the magnetic force vanishes : and that if H is the magnetic force inside the cylinder parallel to the axis II=4f7r (current flowing between two planes separated by unit distance). If there are n turns of wire wound round each unit length of the cylinder and i is the current flowing through the wire, this equation is equivalent to H=4)Trm. The preceding result is true whatever be the shape of the cross section of the cylinder on which the wire is wound, provided the number of turns of wire between two parallel planes at unit distance apart perpendicular to the axis of the cylinder is uniform. Endless Solenoids. Near the ends of a straight solenoid the magnetic field is not uniform and ceases to be parallel to the axis of the cylinder and equal to 4f7rni. We can, however, avoid this irregularity if we wind the wire 208] MAGNETIC FORCE DUE TO CURRENTS. 341 on a ring instead of on a straight cylinder. Suppose the ring is generated by the revolution of a plane area about an axis in its own plane which does not cut it, and let the ring be wound with wire so that the windings are in planes through the axis of the ring and so that the number of windings between two planes which make an angle 6 with each other is equal to nO-l^ir ; n is thus the whole number of windings on the ring. Then we can prove as in Art. 207 that the magnetic force vanishes outside the solenoid, and that inside the solenoid the lines of magnetic force are circles having their centres on the axis of the solenoid and their planes at right angles to the axis. Let H be the magnetic force at a distance r from this axis ; the work done on unit pole when taken round a circle whose radius is r and whose centre is on the axis and plane perpen- dicular to it is 2'7rrH ; this by Art. 201 is equal to 47r times the current flowing through this circle, and is thus equal to 47^?^^, if i is the current flowing through one of the turns of wire. Hence ^irrH = 4>'jrni or H= . r Thus the force is inversely proportional to the distance from the axis. The preceding proof will apply if the solenoid is wound round a closed iron ring ; if however there is a gap in the iron it requires modification. Let Fig. 100 represent a section of the solenoid and suppose that ABDG is a gap in the iron, the faces of the iron being planes passing through the axis of the solenoid. Let this axis cut the plane of the paper in 0. 342 MAGNETIC FORCE DUE TO CURRENTS. [CH. X Let P be a point on the face of one of the gaps, B the magnetic induction in the iron at right angles to OP, Fig. 100. then since the normal magnetic induction is continuous B will also be the magnetic induction in the air. Hence if /jl is the magnetic permeability of the iron, the magnetic force in the iron is B/jn while that in the air is B. If OP = r, the work done in taking unit pole round a circle whose radius is r is — (27r-0)-\-Brd, where 6 is the angle subtended by the air gap at the axis of the solenoid. Hence by Art. 201 we have or rB B = (27r-0) /^ + e = 4!7nii 2)11x1 -J Hi + 2^(^-i) This formula shows the great effect produced by even a very small air gap in diminishing the magnetic induction. 209] MAGNETIC FORCE DUE TO CURRENTS. 343 Let us take the case of a sample of iron for which At-l = 1000, then if ^/27r = 1/100, i.e. if the air is only one per cent, of the whole circuit, the value of B is only one-eleventh of what it would be if the iron circuit were complete, while even though ^/27r were only equal to 1/1000 the magnetic induction would be reduced one-half by the presence of the gap. We can explain this by the tendency which the tubes of magnetic induction have to leave air and run through iron. If the magnetic force in the solenoid due to the current circulating round it is in the direction of the arrow, the face AB of the gap will be charged with positive magnetism, the face CD with negative. If this distribution of magnetism existed in air, tubes of mag- netic induction starting from AB and running through the air to CD would be pretty uniformly distributed in the field ; in this case they would only be in the solenoid for a short part of their course. But as soon as the solenoid is filled with soft iron these tubes forsake the air and run through the iron, and as they are in the opposite direction to the tubes due to the current they diminish the magnetic induction in the iron. 209. Ampere's Formula. We saw, Art. 136, that the magnetic force exerted by a magnetic shell of uniform strength is that which would be produced if each unit of length at a point P on the boundary of the shell exerted a magnetic force at Q equal to sin OjPQ^, where 6 is the angle between PQ and the tangent at P to the boundary of the shell : the direction of the magnetic force at Q is at right angles to both PQ and the tangent to the boundary at P. Since the magnetic force due to the shell is by 844 MAGNETIC FORCE DUE TO CURRENTS. [CH. X Ampere's rule the same as that due to a current flowing round the boundary of the shell, the intensity of the current being equal to the strength of the shell, it follows that the magnetic force due to a linear current may be calculated by supposing an element of current of length ds at P to exert at Q a magnetic force equal to ids sin djPQ^ where i is the strength of the current, and 6 the angle between PQ and the direction of the current at P: the direction of the magnetic force being at right angles both to PQ and to the direction of the current at P. The direction of the magnetic force is related to the direction of the current, like rotation to translation in a right-handed screw working in a fixed nut. 210. Magnetic force due to a circular current. The preceding rule will enable us to find the magnetic force along the axis of a circular current. Let the plane of the current be at right angles to the plane of the paper. Let the current intersect this plane A B Fig. 101. in the points A, B, Fig. 101, flowing upwards at A and downwards at B. Let be the centre of the circle round which the current is flowing, P a point on the axis of the circle. The force at P will by symmetry be along OP. If i is the intensity of the current, then the force at P due to an element ds of the current at A will be at right 210] MAGNETIC FORCE DUE TO CURRENTS. 345 angles to the current at A, i.e. it will be in the plane of the paper, it will also be at right angles to ^P : the magnitude of this force is ids/AP^, hence the component along OP is equal to ., OA By symmetry each unit length of the current will furnish the same contribution \o the magnetic force along the axis at P: hence the magnetic force due to the circuit is equal to Thus the force varies inversely as the cube of the distance from the circumference of the circle. At the Fm. 102. centre of the circle AP=OA, hence the magnetic force at the centre is equal to 27ri OA' 346 MAGNETIC FORCE DUE TO CURRENTS. [CH. X and thus if the current remains of the same intensity varies inversely as the radius of the circle. The lines of magnetic force round a circular current are shown in Fig. 102. 'The plane of the current is at right angles to the plane of the paper and the current passes through the points A and B. 211. A case of some practical importance is that of two equal circular circuits conveying equal currents and placed with their axes coincident. Let A, B; G, D be the points in which the currents, which are supposed to flow in planes at right angles to the plane of the paper, cut this plane, the currents flowing upwards at A and 0, downwards at B and D: let P be a point on the common axis of the two circuits. The magnetic force at P is, if i is the intensity of the current through either circuit, equal to where a is the radius of the circuits. If 2d is the distance between the planes of the circuits, and x = OP, where is the point on the axis midway between the planes of the currents, the magnetic force at P is 27rta- \ ^j. H 5 + terms in af^ and higher powers of x> . Thus if a = 2d, that is if the distance between the currents is equal to the radius of either circuit, the lowest power of x in the expression for the magnetic 212] MAGNETIC FORCE DUE TO CURRENTS. 347 force will be the fourth. Thus near where x is small the magnetic force will be exceedingly uniform. This disposition of the coils is adopted in Helmholtz's Galvanometer. 212. Mechanical Force acting on an electric current placed in a magnetic field. The mechanical forces exerted by currents on a mag- netic system are equal and opposite to the forces exerted by the magnetic system on the currents. Since the forces exerted by the currents on the magnets are the same as those exerted by Ampere's system of magnetic shells, it follows that the mechanical forces on the currents must be the same as those on the magnetic shells ; hence the determination of the mechanical forces on a system of currents can be effected by the principles investigated in Art. 135. Introducing the intensity of the current instead of strength of the magnetic shell we see from that Article that the force in any direction acting on a circuit conveying a current i is equal to i times the rate of increase of the number of unit tubes of magnetic induction passing through the circuit, when the circuit is displaced in the direction of the force. In many cases the deduction from this principle given on page 216, is useful, as it shows that the forces on the current are equivalent to a system of forces acting on each element of the circuit. If i is the strength of the current, ds the length of an element at P, B the magnetic induction at P, the angle between ds and P, then the force on the element is equal in magnitude to idsB sind, and its direction is at right angles both to ds and P. The relation between the direction of the mechanical force and the directions of the current and the magnetic induction is shown in 348 MAGNETIC FORCE DUE TO CURRENTS. [CH. X the accompanying figure, where the magnetic induction is supposed drawn upwards from the plane of the paper. Fig. 103. 213. Couple acting on a plane circuit placed in a uniform magnetic field. Let A be the area of the circuit, i the intensity of the current, (j) the angle between the normal to the plane of the circuit and the direction of the magnetic induction. The number of unit tubes of magnetic induction due to the uniform field passing through the circuit is lAB coscj), where B is the strength of the magnetic induction in the uniform field, and this does not change as the circuit is moved parallel to itself; there are therefore no translatory forces acting on the system. The number of tubes passing through the circuit changes however as the circuit is rotated, and there will therefore be a couple acting on the circuit; the moment of the couple tending to increase is by the last Article equal to the rate of increase with <^ of the number of unit tubes passing through the circuit, that is to -J- (lAB cos<^) = — iAB sincp. 214] MAGNETIC FORCE DUE TO CURRENTS. 349 The couple vanishes with , and hence the circuit tends to place itself with its normal along the direction of the magnetic induction, and in such a way that the direction of the lines of magnetic induction thread the circuit so that the direction of the magnetic induction through the circuit and the direction in which the^^r^:ent flows round it are related like translation and ro^tii5ii in a right- handed screw working in a fixed nut. 214. Force between two infinitely long straight parallel currents. Let the currents be at right angles to the plane of the paper, intersecting this plane in A and B, let the intensity of the currents be i, i' respec- tively, and let the currents come from below upwards through the paper. Then, by Art. 202, the magnetic force at B due to the current through A is equal to AB' and is at right angles io AB \ hence, by Art. 212, the mechanical force per unit length on the current at B is equal to 2u^ AB' and since it acts at right angles both to the current and to the magnetic force, it acts along AB. By the rule given in Art. 212, we see that if the currents are in the same direction the force between them is an attraction, if the currents are in opposite directions the force between them is a repulsion. Hence, we see that straight parallel currents attract or repel each other according as they are flowing in the same or opposite directions with a force which varies inversely as the distance between them. 350 MAGNETIC FORCE DUE TO CURRENTS. [CH. X 215. Mechanical force between two circuits^ each circuit consisting of a pair of infinitely long parallel straight conductors. Let the currents be all perpendicular to the plane of the paper and let the currents of the first and second pairs intersect the plane of the paper in ^, J5 and C, D respectively: we shall consider the case when the circuits are placed symmetri- cally and so that the line EF bisects both AB and CD at right angles. Let the current i flow upwards through Fio. 104. the paper at A, downwards at B, the current ^' upwards through the paper at (7, downwards at D. The force between the circuits will by symmetry be parallel to EF. Between the currents at A and C there is an attraction along CA equal per unit length to 2ii' AC' the component of this parallel to EF is 2u' AC' EF. Between the currents B and C there is a repulsion along BC equal per unit length to 2ii' 216] MAGNETIC FORCE DUE TO CURRENTS. 351 the component of this parallel to EF is Hence on each unit length of G there is a force parallel to FE and equal to there is an equal force acting in this direction on each unit length of D ; hence the total force per unit length on the circuit CD is an attraction parallel to EF equal to li' EF=a), AE=a, CF=h, this is equal to 1 \{a - by + x^ (a + by + x^\ this vanishes when oo = and when x is infinite. Hence there must be some intermediate value of oo when the attraction is a maximum. This value of x is easily found to be given by the equation a;2 = 1 {2 Ja'' + b^-a^¥-(a' + ¥)} : when a — b is very small this gives 00 = a — b, when b/a is very small a V3 216. Force between two coaxial circular cir- cuits. The solution of the general case requires the use of more analysis than is permissible in this work : there 352 MAGNETIC FORCE DUE TO CURRENTS. [CH. X are however two important cases which can be solved by elementary considerations. The first of these is when the radii of the circuits are nearly equal, and the circuits are so close together that the distance between their planes is a very small fraction of the radius of either circuit. In this case the force per unit length of each circuit is approximately the same as that between two infinitely long straight parallel circuits, the distance between the straight circuits being equal to the shortest distance between the circular ones. Thus if i, % are the currents through the circular circuits, whose radii are respectively a and 6, and x is the distance between the planes of the circuits, the attraction between the parallel circuits is at right angles to the planes of the circuits and is approximately equal to ^iraii'x {a - by + x^ ' This is a maximum when x = a — h; that is, when the distance between the planes of the circuits is equal to the difference of their radii. Another case which is easily solved is that of two co- axial circular circuits, the radius of one being small com- pared with that of the other. Let i be the intensity of the current flowing round the large circuit whose radius is a, i' the current round the small circuit whose radius is h ; let X be the distance between the planes of the circuits. Then since 6 is very small compared with a, the magnetic force due to the large circuit will be approximately uniform a over the second circuit and equal to 27ria^ / (a^ + x"^) , its value at the centre of that circuit. Thus the number of 217] MAGNETIC FORCE DUE TO CURRENTS. 353 unit tubes of magnetic induction due to the first circuit which pass through the second circuit is equal to 27rHa%^ Hence by Art. 210 the force on the second circuit in the direction in which x increases, i.e. the repulsion between the circuits, is equal to ^irHHa^h' ~ ^ — 1- Thus the attraction between the circuits is equal to {a? + x^f This is a maximum when x = a/2, so that the attraction between the circuits is greatest when the distance between their planes is half the radius of the larger circuit. In the more general case when the radii have any values, there is unless the radii are equal a position in which the attraction is a maximum. When we use the attraction between currents as a means of measuring their intensities, the currents ought to be placed in this position, for not only is the force to be measured greatest in this case, but it is also practically independent of any slight error in the proper adjustment of the distance between the coils. 217. Coefficients of Self and Mutual Induction. The coefficient of self induction of a circuit is defined to be the number of unit tubes of magnetic induction which pass through the circuit when it is traversed by unit current and when there are no other currents nor permanent magnets in its neighbourhood. T. E. 23 354 MAGNETIC FORCE DUE TO CURRENTS. [CH. X The coefficient of mutual induction of two circuits A and B is defined to be the number of unit tubes of magnetic induction which pass through B when unit current flows round A, and there are no other currents nor permanent magnets in the neighbourhood of the circuits. We see from Art. 137 that the coefficient of mutual induction is also equal to the number of unit tubes of induction which pass through A when unit current flows round B. If the circuits consist of several turns of wire, then in the preceding definitions we must take as the number of tubes of magnetic induction which pass through the circuit, the sum of the number of tubes of magnetic induction which pass through the different turns of the circuit. We see from the preceding definitions that if we have two circuits A and B, and if the currents i, j flow respectively through these circuits, then the number of tubes of magnetic induction which pass through the circuits A and B are respectively, Li + Mj, and Mi + Nj, where L and N are the coefficients of self-induction of the circuits A and B respectively, and M is the coefficient of mutual induction between the circuits. The results given in the preceding Articles enable us to calculate the coefficients of self-induction in some simple cases. In the case of the long straight solenoid discussed in Art. 208, when unit current flows through the wire the magnetic force in the solenoid is 47rw, where n is the number of turns per unit length ; hence if A is the area of the core of the solenoid, and if the core is filled with 217] MAGNETIC FORCE DUE TO CURRENTS. 355 air, the number of unit tubes of magnetic induction pass- ing through each turn of wire is equal to 4!7rnA, and since there are n turns per unit length, the coefficients of self- induction of a length I of the solenoid is equal to 4>7rnHA. If the core were filled with soft iron of permeability fi, then the number of unit tubes of magnetic induction which pass through each turn of wire is ^irn^A and the coefficient of self-induction of a length I is ^irnH/jbA. If the iron instead of completely filling the core only partially fills it, then if B is the area of the core occupied by the iron, the coefficient of self-induction of a length I is ^irnH [fiB + A-B]. Consider now the coefficient of mutual induction of two solenoids a and 0. The coefficient of mutual in- duction will vanish unless one of the solenoids is inside the other, for the magnetic force due to a current through a solenoid vanishes outside the solenoid. Hence when a current flows through a no lines of induction will pass through y8 unless ff is either inside a or completely surrounds it. Let ^ be inside a. Let B be the area of the solenoid yS, and let m be the number of turns of wire per unit length. Then if unit current flows through a, the magnetic force inside is 47r/i, where n is the number of turns per unit length. Hence if there is no iron inside the solenoids, the number of tubes of magnetic induction passing through each turn of fi is 4>7rnB, and since there are m turns per unit length, the coefficient of mutual induction of a length I of the two solenoids is ^urnmlB. We see, by Art. 216, that the coefficient of mutual induction between a large circle of radius a and a small 23—2 356 MAGNETIC FORCE DUE TO CURRENTS. [CH. X one of radius b, with their planes parallel and the line joining their centres at right angles to their planes is equal to (a' + x^f ' where oc is the distance between the planes. If we have two circuits a, 0, each consisting of two infinitely long parallel straight conductors, the current flowing up one of these and down the other, then by Art. 206, the coefficient of mutual induction between a and ^ is, per unit length, equal to ^ , AC .BD where A, B, C, D are respectively the points where the wires of the circuits a and 13 intersect a plane at right angles to their common direction. The current through the conductor intersecting this plane in A is in the same direction as that through the conductor passing through C. 218. We can express the energy in the magnetic field due to a system of currents very easily in terms of the currents and the coefficients of self and mutual in- duction of the circuits. We proved Art. 162 that the energy per unit length in a unit tube of induction at F is equal to R/Stt, where R is the magnetic force at P. The tube of induction is a closed curve, and the total amount of energy in this tube is equal to where ds is an element of length of the tube and XRds denotes the sum of all the products Rds for the tube. 218] MAGNETIC FORCE DUE TO CURRENTS. 357 But ^Rds is the work done on unit pole when it is taken round the closed curve formed by the tube of induction, and this by Art, 201 is equal to 47r times the sum of the currents encircled by the curve. Hence the energy in a tube of induction is equal to i (the sum of the currents encircled by the tube). Hence the whole energy in the magnetic field is equal to half the sum of the products obtained by multiplying the current in each circuit by the number of tubes of mag- netic induction passing through that circuit. Thus if we have two circuits A and B, and if i, j are the currents through A and B respectively, L, N the coefficients of self-induction of A and B, M the coefficient of mutual induction between these circuits, then the numbers of tubes of magnetic induction passing through A and B respectively are Li + Mj, and Mi + Nj. Hence the energy in the magnetic field around this circuit is \i{Li-\-Mj)^\j{Mi+Nj) = ^M+Mij-\-\Nj\ If we have only one circuit carrying a current i, .then if L is its coefficient of self-induction, the energy in the magnetic field is \Li\ Thus the coefficient of self-induction is equal to twice the energy in the magnetic field due to unit current. We may use this as the definition of coefficient of self- induction, and this definition has a wider application than 358 MAGNETIC FORCE DUE TO CURRENTS. [CH. X the previous one. The definition in Art. 217 is only applicable when the currents flow through very fine wires, the present one however is applicable when the current is distributed over a conductor with a finite cross section. Thus let us consider the case where we have a current flowing through an infinitely long cylinder whose radius is OA, the direction of flow being parallel to the axis of the cylinder, and where the return current flows down a thin tube, whose radius is OB coaxial with this cylinder. Fig. 105. Let i be the current which flows up through the cylinder and down through the tube, let us suppose that the current through the cylinder is uniformly distributed over its cross section. The magnetic force will vanish outside the tube, for since as much current flows up through the cylinder as down through the tube, the total current flowing through any curve enclosing them both vanishes, and therefore the work done in taking unit pole round a circle with centre and radius greater than that of the tube will vanish. Since the magnetic force due to the currents must by symmetry be tangential to this circle and have the same value at each point on its 218] MAGNETIC FORCE DUE TO CURRENTS. 359 circumference, it follows that the magnetic force vanishes outside the tube. We can prove as in Art. 202 that at a point P between the cylinder and the tube the magnetic force is equal to 2f where r = OP. At a point P inside the cylinder the magnetic force is 2ir "^' where a = OA, the radius of the cylinder. By Art. 162 the energy per unit volume is equal to fiH^/S-n; where H is the magnetic force ; hence if fi is the magnetic permeability of the cylinder, the magnetic energy between two planes at right angles to the axis of the cylinder and at unit distance apart is equal to JoA r^ ^ttJo a' Stt , OB i' ^"^0^ + 4^- Hence, since the coefficient of self-induction per unit length is twice the energy when the current is unity, it is equal to In this case the coefficient of self-induction will be very much greater when the cylinder is made of iron than when it is made of a non-magnetic metal like copper. For take the case when OB = e . OA, where e = 2718, the base of the Napierian logarithms ; then the self-induction for copper, for which fi is equal to unity, is equal to 2*5 per unit 360 MAGNETIC FORCE DUE TO CURRENTS. [CH. X length, but if the cylinder is made of a sample of iron whose magnetic permeability is 1000, the coefficient of self-induction per unit length is 502. Thus in this case the material of the conductor through which the current flows produces an enormous effect, much greater than it does in the case of the solenoids. The self-induction depends upon the way in which the current is distributed in the cylinder; thus if the current instead of spreading uniformly across the sectioxi of the cylinder were concentrated on the surface, the magnetic force inside the cylinder would vanish, while that in the space between the tube and the cylinder would be the same as before, hence the energy would now be so that the coefficient of self-induction would now be 2 log (OB/OA), thus it would be less than before and in- dependent of the material of which the cylinder is made. Measurement of Current and Resistance. Galvanometers. 219. The magnetic force produced by a current may be used to measure the intensity of the current. This is most frequently done by means of the tangent galvanometer, which consists of a circular coil of wire placed with its plane in the magnetic meridian. If the magnetic field is not wholly due to the earth, the plane of the coil must contain the resultant magnetic force. At the centre of the coil there is a magnet which can turn freely about 219] MAGNETIC FORCE DUE TO CURRENTS. 361 a vertical axis. When the magnet is in equilibrium its axis will lie along the horizontal component of the mag- netic force at the centre of the coil, thus when no current is flowing through the coil the axis of the magnet will be in the plane of the coil. A current flowing through the coil will produce a magnetic force at right angles to the plane of the coil, proportional to the intensity of the current. Let this magnetic force be equal to €ri where i is the intensity of the current flowing through the coil and G a quantity depending upon the dimensions of the coil. G is called the ' Galvanometer constant.' Let // be the horizontal component of the magnetic force at the centre of the coil. Then the resultant magnetic force at the centre of the coil has a component H in the plane of the coil and a component Gi at right angles to it, hence if 6 is the angle which the resultant magnetic force makes with the plane of the coil, tan^ = g" (1). When the magnet is in equilibrium its axis will lie along the direction of the resultant magnetic force, hence the passage of the current will deflect the magnet through an angle 6 given by equation (1). As the current is pro- portional to the tangent of the angle of deflection, this instrument is called the tangent Galvanometer. The smaller we can make H, the external magnetic force at the centre of the coil, the larger will be the angle through which a given current will deflect the magnet. By placing permanent magnets in suitable positions in the neighbourhood of the coil we can partly neutralize the earth's magnetic field at the centre of the coil : in this way 362 MAGNETIC FORCE DUE TO CURRENTS. [CH. X we can reduce H and increase the sensitiveness of the galvanometer. A magnet for this purpose is shown in Fig. 106, which represents an ordinary type of galvano- meter. Fig. 106. Another method of increasing the sensitiveness of the instrument is employed in the 'astatic galvanometer.' In this galvanometer (Fig. 107) we have two coils A and B in series, so arranged that the current circulates round them in opposite directions. Thus, if the magnetic force at the centre of the upper coil is upwards from the plane of the paper that at the centre of the lower coil will be downwards. Two magnets a, /9, mounted on a common axis, are placed at the centres of the coils A and B re- spectively, the axes of magnetization of these magnets point in opposite directions ; thus as the magnetic forces at the centres of the two coils due to the currents are also in opposite directions, the couples due to the currents acting on the two magnets will be in the same direction. 219] MAGNETIC FORCE DUE TO CURRENTS. 363 The couples arising from the external magnetic field will however be in opposite directions: if the external Fig. 107. magnetic field is uniform and the moments of the two magnets very nearly equal, the couple tending to restore the magnet to its position of equilibrium will be very small, and the galvanometer will be very sensitive. The larger we make G the greater will be the sensi- tiveness of the galvanometer. If the galvanometer consists of a single turn of wire wound into a circle of radius a, then (see Art. 210) G = ^irja. If there are n turns close together and arranged so that the distance between any two turns is a very small fraction of the radius of the turns, then G is approximately iTrnja. If the galvanometer consists of a coil of rectangular cross section, the sides of the rectangle being in and at right angles to the plane of the coil, and if 26 is tlie breadth of this rectangle (measured at right angles to the plane of the coil), 2a the depth in the plane of the coil, n the number of turns of wire passing through unit area, then taking as axis oi x 364 MAGNETIC FORCE DUE TO CURRENTS. [CH. X the line through the centre of the coil at right angles to its plane and as axis of 3/ a line through the centre at right angles to this, we have G = 27rn rb rc+a yi^^ J -bJ c-a (af + y'^dxdy f ' where c is the mean radius of the coil. A ^2^ n : '^'^ D E f\ Ki f I H Q Fig. 108. If 26, 2(f) are the angles subtended at the centre by AB, CD, Fig. 108, this reduces to 6 G = 4}7rnh log cot cot I In sensitive galvanometers the hole in the centre for the magnet is made as small as possible, so that the inner windings have very small radii ; when this is the case, we TT may put (f) = — , and then G = 4!7rnh log cot - 219] MAGNETIC FORCE DUE TO CURRENTS. 365 In this case when the area of the cross section of the coil is given, i.e. when 2¥ cot 6 is given, we can prove that G^ is a maximum when g log cot ^ = 2 cos 0, the solution of which is ^ = 16° 46': this makes the breadth bear to the depth the ratio of 1 to 1'61. The sensitiveness of modern galvanometers is very great, some of them will detect a current of 10~^^ amperes. It would take a current of this magnitude centuries to liberate 1 c.c. of hydrogen. Since while rcTTTTr = sin Q cos Q. (Bi/i) Thus for a given absolute increment of i, SO will be greatest when 6 is zero, and for a given relative increment, Bd, or the change in deflection, will be greatest when 6/ = 45°. In some cases it is important to have the magnetic field near the magnet as uniform as possible. This can be attained (see Art. 211) by using two equal coils placed parallel to one another and at right angles to the line joining their centres, the distance between the coils being equal to the radius of either. The magnet is then placed on the common axis of the two coils and midway between them. S66 MAGNETIC FORCE DUE TO CURRENTS. [CH. X 220. Sine Galvanometer. In this galvanometer, Fig. 109, the coil itself can move about a vertical axis, its Fig. 109. position being determined by means of a graduated hori- zontal circle. In using the instrument the coil is placed so that when no current goes through it the magnetic axis of the magnet at its centre is in the plane of the coil. When a current passes through the coil the magnet is deflected out of this plane, and the coil is now moved round until the axis of the magnet is again in the plane of the coil. When this is the case the components of the magnetic force at right angles to the plane of the coil due respectively to the current and to the external magnetic field must be equal and opposite. If H is the external magnetic force, (j) the angle through which the coil has been twisted when the axis of the magnet is again in the plane of the coil, the external force at right angles to the plane of the coil is H sin . If ^ is the current through the coil, G the magnetic force at its 221] MAGNETIC FORCE DUE TO CURRENTS. 867 centre when the wires of the coil are traversed by unit current, then the magnetic force at right angles to the coil due to the current is Gi : hence when this is in equi- librium with the component due to the external field, Hsm(f> = Gi, or '^—jT Sin . If the torsional couple vanishes when (f> is zero, the couple when the coil is twisted through an angle <^ will be proportional to ^ ; let it equal t<^, then when there is equilibrium, we have iBAn cos = t<^, T± BAn cos if (ft is small this equation becomes approximately • _ ''"^ ^~~BA^' 222. Ballistic Galvanometer. A galvanometer may be used to measure the total quantity of electricity passing 222] MAGNETIC FORCE DUE TO CURRENTS. 369 through its coil, if this passes so quickly that the magnet of the galvanometer has not time to appreciably change its position while the electricity is passing. Let us suppose that when no current is passing the axis of the magnet is in the plane of the coil, then if i is the current passing through the plane of the coil, G the galvanometer constant, i.e. the magnetic force at the centre of the coil when unit current passes through it, m the moment of the magnet, the couple on the magnet while the current is passing is Gim. If the current passes so quickly that the magnet has not time sensibly to depart from the magnetic meridian while the current is flowing, the earth's magnetic force will exert no couple on the magnet. Thus if K is the moment of inertia of the magnet, 6 the angle the axis of the magnet makes with the magnetic meridian, the equation of motion of the magnet during the flow of the current is thus if the magnet starts from rest the angular velocity after a time t is given by the equation dO c' K ^rr= Gm dt Jo idt If the total quantity of electricity which passes through the galvanometer is Q and o) the angular velocity com- municated to the magnet, we have therefore K(o = GmQ. This angular velocity makes the magnet swing out of the plane of the coil : if H is the external magnetic force T. E. ^ 24 370 MAGNETIC FORCE DUE TO CURRENTS. [CH. X at the centre of the coil, the equation of motion of the magnet is, if there is no retarding force, ir^ + miysin(9 = 0. Integrating this equation we get K {('^j' - A + 2mH (1 - cos ^) = 0. If ^ is the angular swing of the magnet, the angular velocity vanishes when ^ = ^, hence K7r times the current flowing through that circuit. Now the magnetic field inside the metal, and therefore the work done when unit pole passes round a closed circuit, is unaltered by the impulse, hence the current flowing through any such closed curve is also un- altered by the impulse ; hence, as there were no currents through it before the impulse acted, there will be none generated by the impulse. In other words, the currents generated in a mass of metal by an electric impulse are entirely on the surface of the metal, and the inside of the conductor is free from currents. 231. The currents will not remain on the surface, they will rapidly diffuse through the metal and die away. We can find the way the currents distribute themselves after the impulse stops by the use of the two fundamental principles of electro-dynamics, (1) that the work done by the magnetic forces when unit pole travels round a closed circuit is equal to 47r times the quantity of current flowing through the circuit, (2) that the total electro- motive force round any closed circuit is equal to the rate of diminution of the number of tubes of magnetic induc- tion passing through the circuit. Let u, V, w be the components parallel to the axes of OS, y, z of the electric current at any point, a,/S, 7 the com-, ponents of the magnetic force at the same point. The axes are chosen so that if x is drawn to the east, y to the north, z is upwards. Consider a small rectangular circuit ABGD, the sides AB, EG being parallel to the axes of z and y respectively. Let AB = 2h, BC= 2k. Let a, y8, 7 be the components of magnetic force at 0, the centre of the 394 ELECTROMAGNETIC INDUCTION. [CH. XI rectangle ; x,y,z the coordinates of ; let the coordinates of P, a point on AB,\>& x,y -\-k, z-\-^; the z component of the magnetic force at P will be approximately ^^^dz^^dy^' Let now a unit magnetic pole be taken round the rectangle A BCD, the direction of motion round A BCD being related to the positive direction of oo like rotation and translation in a right-handed screw. The work done on unit pole as it moves from A to B will be i::hi^*p)< dy which is equal to 2/17 + 2hk-^ ; the work done on the pole as it moves from C to D is -2hy+2hkp. We may show similarly that the work done on unit pole as it moves from B to G is equal to -2k^-2hk^f, and when it moves from D to A to 2kl3-2hk^. dz Adding these expressions we see that the work done on unit pole as it travels round the rectangle A BCD is equal (|-S)«'- The quantity of current passing through this rectangle is equal to 4iuhk, 2:31] ELECTROMAGNETIC INDUCTION. 395 hence since the work done on unit pole in going round the rectangle is equal to ^tt times the current passing through the rectangle, see Art. 201, we have 47r X ^uhk = ( 3^ — "^ ) ^^^^ ^^-=|-£ (^)- By taking rectangles whose sides were parallel to the axes of X and z, and of x, y we should get in a similar way ^-4:-£ (^)- *-4f-| (^)- If X, F, Z are the components of the electric intensity at 0, we can prove by a similar process that the work done on unit charge of electricity in going round the rectangle ABGD is equal to dy dz J If a, 6, c are the components of magnetic induction at 0, the number of tubes of magnetic induction passing through the rectangle is a x ^hk ; hence the rate of diminu- tion of the number of unit tubes is equal to at But by Faraday's law (Art. 226) the work done on unit charge in going round the circuit is equal to the rate of diminution in the number of tubes of magnetic induction passing through the circuit, hence dt \dy dz J 396 ELECTROMAGNETIC INDUCTION. [CH. XI or similarly da di db dt dc di dZ dy dX dz dY dx dY dz dZ dx d_X_ dy (4). Let us consider the case when the variable part of magnetization is induced, so that da _ da db _ d^ dc _ dy di~^di' dt~^dt' di~^di' where /jl is the magnetic permeability. If a is the specific resistance of the metal in which the currents are flowing, and if the currents are entirely conduction currents, au = X, o-v = F, aw = Z. We have by equation (1) du _ d dc d db dt dy dt dz dt ' hence by putting F= S^ is infinite ; in this case L' = L. When the secondary circuit is completed through electric lamps &c., S is in practice small compared with Np, so that L' — L — M'^jN. Thus the completion of the circuit causes a great diminution in the value of the apparent self-induction of the primary circuit. The work done per unit time in the transformer is equal to the mean value of E cos 'pt . X, it is thus equal to 1 E^ cos a 2 {Z V + ^ ^1 E'R ~ 2 L'y -f- R' • When the secondary circuit is broken S is infinite and therefore L' = L, R = R, and the work done on the trans- former per unit time or the power spent on it is equal to 1 E'E 2 LY + ^' * When the circuit is completed and S is small compared 408 ELECTROMAGNETIC INDUCTION. [CH. XI with Np, L' = L- M'lJSr, R' = R-\- M'S/N% and then the power spent is equal to This is very much greater than the power spent when the secondary circuit is not completed ; this must evidently be the case, as when the secondary circuit is completed lamps are raised to incandescence, the energy required for this must be supplied to the transformer. The power spent when the secondary circuit is not completed is wasted as far as useful effect is concerned, and is spent in heating the transformer. The greater the coefficient of self-induction of the primary, the smaller is the current sent through the primary by a given electromotive force, and the smaller the amount of power wasted when the secondary circuit is broken. When the secondary circuit is closed the self-induction of the primary is diminished from L to L'; since there is less effective self-induction in the primary, the current through it, and consequently the power given to it, is greatly increased. We see from the expression just given that the power absorbed by the transformer is greatest when n^-^{^-n)p' that is, when When there is no magnetic leakage, i.e. when LN=M\ 236] ELECTROMAGNETIC INDUCTION. 409 the power absorbed continually increases as the resist- ance in the secondary diminishes ; when however LN is not equal to if' the power absorbed does not necessarily increase as S diminishes, it may on the contrary reach a maximum value for a particular value of S, and any diminution of 8 before this value will be accompanied by a decrease in the energy absorbed by the transformer. The greater the frequency of the electromotive force, the larger will be the resistance of the secondary when the absorption of power by the transformer is greatest. When the frequency is very great, such as, for instance, when a Leyden jar is discharged (see page 422), the critical value of the resistance in the secondary may be exceedingly large. In this case the difference between the maximum absorption of power and that corresponding to /Sf = may be very great. Thus when S = 0, the power absorbed is equal to 1 E'R 2 L'Y + i^' ' or approximately for very high frequencies 1 E^ 2 X'y ' while the maximum power absorbed is 1 E^ 4 L'p' which exceeds that when >Sf = in the proportion of L'p to 2E. The currents x, y in the primary and secondary are represented by the equations x = A cos {pt — a), y = B cos {jjt — /3). 410 ELECTROMAGNETIC INDUCTION. [CH. XI Thus the ratio of the maximum value of the current in the primary to that in the secondary is BjA: by equation (5), we have B ^ Mp or, when Np is large compared with S, B_M A" N' ^ — a = TT. If the primary and secondary coils cover the same length of the core, and are wound on a core of great permeability, then M/N is equal to m/n, where m is the number of turns in the primary, and n the number in the secondary. If we have a lamp whose resistance is s in the secondary the potential difference between its electrodes is sy, i.e. sB cos {pt — /3). The maximum value of this expression is sB ; substi- tuting the value of B, we find that this value is equal to M J. This is greatest when L'=0, in which case it is equal to M ^ and this, if S is small compared with Np^ is equal to ■ 237] ELECTROMAGNETIC INDUCTION. 411 If R is small compared with SM^/N^ this is ap- proximately Thus if for example M/N= 20, the maximum current through the secondary is 20 times that through the primary; while the electromotive force between the terminals of the lamp is approximately Now s is always smaller than S, as S is the resistance of the whole secondary circuit, while s is the resistance of only a part of it : the electromotive force between the terminals of any lamp is thus in this case always less than 1/20 of the electromotive force between the terminals of the secondary. In getting this value we have assumed the conditions to be those most favourable to the production of a high electromotive force in the secondary; if there is any magnetic leakage, i.e. if L' is not zero, then at high frequencies the electromotive force in the secondary would be very much less than the value just found, in fact where there is any magnetic leakage, the ratio of the electromotive force in the secondary to that in the primary is indefinitely small when the frequency is infinite. 237. Distribution of rapidly alternating currents. When the frequency of the electromotive force is so great that in the equations of the type dx dii L -r + M -y- + ,..Rx = external electromotive force, at at the term Rx depending on the resistance is small com- pared with the terms Ldx/dt, Mdyjdt depending on 412 ELECTROMAGNETIC INDUCTION. [CH. XI induction, which if the electromotive force is supposed to vary as cos^^, will be the case when Lp, Mp are large compared with R ; the equations determining the currents take the form -^(Lx-\-My + ...) = external electromotive force, " dt ' where N is the number of tubes of induction due to the external system passing through the circuit whose co- efficient of self-induction is L. We see from this that Lx + My + . . . + N = constant, and since x, y . . . N all vary harmonically, the constant must be zero. Now Lx -f- My + ... is the number of tubes of magnetic induction which pass through the circuit we are considering, and are due to the currents flowing in this and the neighbouring circuits, while N is the number of tubes passing through the same circuit due to the ex- ternal system. Hence the preceding equation expresses that the total number of tubes passing through the circuit is zero. The same result is true for any circuit. Now consider the case of the currents induced in a mass of metal by a rapidly alternating electromotive force. The number of tubes of magnetic induction which pass through any circuit which can be drawn in the metal is zero, and hence the magnetic induction must vanish through- out the mass of the metal. The magnetic force will con- sequently also vanish throughout the same region. But since the magnetic force vanishes, the work done when unit pole is taken round any closed curve in the region must also vanish, and therefore by Art. 201 the current flowing 288] ELECTROMAGNETIC INDUCTION. 413 through any closed curve in the region must also vanish ; this implies that the current vanishes throughout the mass of metal, or in other words, that the currents generated by infinitely rapidly alternating forces are confined to the surface of the metal, and do not penetrate into its interior. We showed in Art. 230 that the currents generated by an electrical impulse started from the surface of the con- ductors and then gradually diffused inwards. We may approximate to the condition of a rapidly alternating force by supposing a series of positive and negative impulses to follow one another in rapid succession. The currents started by a positive impulse have thus only time to diffuse a very short distance from the surface before the subsequent negative impulse starts opposite currents from the surface ; the effect of these currents at some distance from the surface is to tend to counteract the original currents, and thus the intensity of the current falls off rapidly as the distance from the surface of the conductor increases. 238. The amount of concentration of the current de- pends on the frequency of the electromotive force and of the conductivity of the conductor. If the frequency is in- finite and the conductivity finite, or the frequency finite and the conductivity infinite, then the current is confined to an indefinitely thin skin near the surface of the conductor. If, however, both the frequency and the conductivity are finite, then the thickness of the skin occupied by the current is finite also, while the magnitude of the current diminishes rapidly as we recede from the surface. Any increase in the frequency or in the conductivity increases the concentration of the current. 414 ELECTROMAGNETIC INDUCTION. [CH. XI The case is analogous to that of a conductor of heat, the temperature of whose surface is made to vary har- monically, the fluctuations of temperature corresponding to the alterations in the surface temperature diminish in intensity as we recede from the surface, and finally cease to be appreciable. The fluctuations, however, with a long period are appreciable at a greater depth than those with a short one. We may for example suppose the temperature of the surface of the earth to be subject to two variations, one following the seasons and having a yearly period, the other depending on the time of day and having a daily period. These fluctuations become less and less apparent as the depth of the place of observation below the surface of the earth increases, and finally they become too small to be measured. The annual variations can, however, be detected at depths at which the diurnal variations are quite inappreciable. This concentration of the current near the surface of the conductor, which is sometimes called ' the throttling of the current,' increases the resistance of the conductor to the passage of the current. When, for example, a rapidly alternating current is flowing along a wire, the current will flow near to the outside of the wire, and if the frequency is very great the inner part of the wire will be free from current ; thus since the centre of the wire is free from current, the current is practically flowing through a tube instead of a solid wire. The area of the cross section of the wire, which is effective in carrying this rapidly alternating current, is thus smaller than the effective area when the current is continuous, as in this case the current distributes itself uniformly over the whole of the cross section of the wire. As the effective area for 239] ELECTROMAGNETIC INDUCTION. 415 the rapidly alternating currents is less than that for con- tinuous currents, the resistance, measured by the heat pro- duced in unit time when the total current is unity, is greater for the alternating currents than for continuous currents. 239. Distribution of an alternating current in a Conductor. The equations given in Art. 231 enable us to find how an alternating current distributes itself in a conductor. We shall consider the case which presents the least analytical difficulties, but which will serve to illustrate the laws of the phenomenon we are discussing. This case is that of an infinite mass of a conductor bounded by a plane face. Take the axis of x at right angles to this face, and the origin of coordinates in the face; let the currents be everywhere parallel to the axis of z, and the same at all points in any plane parallel to the face of the conductor. Then if //, is the magnetic permeability and a the specific resistance of the conductor, w the current at the point x, y, z at the time t parallel to the axis of z, we have by the equations of Art. 231, . dw [d^w d^w d^w\ or, since w is independent of y and z . dw d^w *'^'*d?='^d^ (!)• We shall suppose that the currents are periodic, making p/27r complete alternations per second. We may put, writing i for J^l, W = €^^* ft), where &> is a function of x, but not of t Substituting this value of w in equation (1) we get 416 ELECTROMAGNETIC INDUCTION. [CH. XI or if 'n? = ^Trfiipja, The solution of this is where A and B are constants. Now „ = |l^l%-i _|2ir/^>^ (1+i). 1 We shall suppose that the conductor stretches from x=0 to a? = 00 and that the cause which induces the currents lies on the side of the conductor for which x is negative. It is evident that in this case the magni- tude of the current cannot increase indefinitely as we recede from the face nearest to the inducing system ; in other words, w cannot be infinite when x is infinite : this condition requires that B should vanish ; in this case we have and therefore Thus li w = A cos pt when x = 0, w==A6 ^ " ^ coslp^ - ( -j a at a distance x from the surface. 239] ELECTROMAGNETIC INDUCTION. 417 This result shows that the maximum value of the cuiTent at a distance x from the face is proportional to ,-^^f- Thus the magnitude of the current diminishes in geometrical progression as the distance from the face increases in arithmetical progression. In the case of a copper conductor exposed to an electro- motive force making 100 alternations per second, /a = 1, o- = 1600, ^ = 27rxl00; hence {27r/xp/o-}^ = 7r/2, so that the maximum current is proportional to e ^ . Thus at 1 cm. from the surface the maximum current would only be "208 times that at the surface, at a distance of 2 cm. only '043, and at a distance of 4 centimetres less than 1/500 part of the value at the surface. If the electromotive force makes a million alternations per second [^ir^pjay = SOtt ; the maximum current is thus proportional to e"^^, and at the depth of one millimetre is less than one six-millionth part of its surface value. The concentration of the current in the case of iron is even more remarkable. Consider a sample of iron for which /a = 1000, a = 10000, exposed to an electro- motive force making 100 alternations per second, so that p = 27r X 100. In this case {27r/Ap/o-)^= 20 approximately, and thus the maximum current at a depth of one milli- metre is only "13 times the surface value, while at 5 millimetres it is less than one twenty-thousandth part of its surface value. If the electromotive force makes a million alternations per second, then for this specimen of iron j27r/x/}/(r}* T. E. 27 418 ELECTROMAGNETIC INDUCTION. [CH. XI is approximately 2000, and the maximum current at the distance of one-tenth of a millimetre from the surface is about one five-hundred-millionth part of its surface value. We see from the preceding expressions for the current that the distance required to diminish the maximum cur- rent to a given fraction of its surface value is directly proportional to the square root of the specific resistance, and inversely proportional to the square root of the number of alternations per second. 240. Magnetic Force in the Conductor. The currents in the conductor are all parallel to the axis of z, and are independent of the coordinates y, z. Now the equations of Art. 231 may be written in the form da _ fdw dv\ dh _ (du dw\ 'Tt'^^Kdy'dzj' 'Jt'^^Kdi"!^)' dc _ fdv ^ du\ dt \dx dy) ' where a, h, c are the components of the magnetic induc- tion, u, V, w those of the currents. In the case we are considering u=v = 0, and w is independent of y and z ; hence a = c = 0, and the magnetic induction is parallel to the axis of y. Thus the currents in the plate are accompanied by a magnetic force parallel to the surface of the plate and at right angles to the direction of the current. From the above equations we have db _ dw dt^ dx^ Hence 6 = - A e"*^ cos 241] ELECTROMAGNETIC INDUCTION. 419 and by Art. 239 w = Ae-"^^ cos {pt — mx\ where m = {^ir^pjo-y. (^t-mx-'^ = Ae-""^ cos lpt-mx--\ . Thus the magnetic force in the conductor diminishes as we recede from the surface according to the same law as the current. 241. Mechanical Force acting on the Con- ductor. When a current flows in a magnetic field a mechanical force acts on the conductor carrying the cur- rent (see page 347). The direction of the force is at right angles to the current and also to the magnetic induction, and the magnitude of the force per unit length of the conductor is equal to the product of the current and the magnetic induction at right angles to it. In the case we are considering the magnetic induction and the current are at right angles. If tv is the intensity of the current, the current flowing through the area dxdy is wdxdy\ hence the force on the volume dxdydz parallel to x, and in the positive direction of x, is equal to — wh dx dy dz. The total force parallel to x acting on the conductor is Inwhdxdy dz, but since h and lu are both independent of y and z, the force acting on the conductor per unit area of its face is / whdx. 27—2 420 ELECTROMAGNETIC INDUCTION. [CH. XI Now if a, P, y are the components of the magnetic force ~~ dx dy^ hence, since h — /xy3, we see that the force on the con- ductor parallel to x is 47r j dx' where /3o is the value of /5 when ^ = 0, i.e. at the surface of the conductor, and yS^ is the value of ^ when x= oo , But it follows from the expression for b given in the last article that y9^ = ; hence the force on the conductor parallel to ^v per unit area of its face is equal to Sir ' This is always positive, and hence the conductor tends to move along the positive direction of ^ ; in other words, the conductor is repelled from the system which induces the currents in the conductor. These repulsions have been shown in a very striking way in experiments made by Professor Elihu Thomson and also by Dr Fleming. In these experiments a plate placed above an electro- magnet round which a rapidly alternating current was circulating, was thrown up into the air, the repulsion between the plate and the magnet arising from the cause we have just investigated. The expression ^^ is the repulsion at any instant, OTT but since ySo is proportional to cos (pt + e) the mean value 242] ELECTROMAGNETIC INDUCTION. 421 of P^ is H'^12 if H is the maximum value of y8o. Hence the mean value of the repulsion is equal to IGtt* 242. The screening off of Electromagnetic In- duction. We have seen in Art. 240 that the magnetic force diminishes rapidly as we recede from the surface of the conductor, and becomes inappreciable at a finite distance, say d, from the surface. At a point P whose distance from the surface is greater than d we may neglect both the current and the magnetic force. Thus the electro- magnetic action of the currents in the sheet of the con- ductor whose thickness is rf just counterbalances at P the electromagnetic action of the original inducing system situated on the other side of the face of the conductor. Hence the slab of thickness d may be regarded as screening off from P the electromagnetic effect of the original system. In the investigation in Art. 240 we sup- posed that the conductor was infinitely thick, but since the currents are practically confined to the slab whose thick uess is d, it is evident that the screening is done by this layer and that no appreciable advantage is gained by increasing the thickness of the slab beyond d. The thickness d of the slab required to screen off the magnetic force depends upon the frequency of the alternations and on the magnetic permeability and specific resistance of the conductor. By Arts. 239 and 240 the current and magnetic force at a distance x from the surface are proportional to g-maj^ where m =■ [27rfjbplcr]^ ; hence for a thickness d to reduce the magnetic force to an inappreciable fraction of its surface value md must be considerable. If we regard 422 ELECTROMAGNETIC INDUCTION. [CH. XI the system as screened off when the magnetic effect is reduced to a definite fraction of its undisturbed value, then d the thickness of the screen is inversely propor- tional to m. The greater the frequency the thinner the screen. Thus from the examples given in Art. 239 we see that if the system makes a million oscillations a second, a screen of copper less than a millimetre thick will be perfectly efficient, while a screen of iron a very small fraction of a millimetre in thickness will stop prac- tically all induction. If the system only makes 100 alter- nations a second, the screen if of copper must be several centimetres and if of iron several millimetres thick. 243. Discharge of a Leyden Jar. One of the most interesting applications of the laws of induction of currents is to the case of a Leyden jar, the two coatings of which are connected by a conducting circuit possessing self-induction. Let us consider a jar whose inside A is connected to the outside B by a circuit whose resistance is R and whose coefficient of self-induction is L. Let i be the current flowing through the circuit from A to B. Then by the laws of the induction of currents di . L-T.-\-Ri— electromotive force tending to increase i = F-.-n (1). If V^ and Vs al-e respectively the potentials of A and By Q the charge on the inside of the jar, and G the capacity of the jar, then or v,-r,) = ^. 243] ELECTROMAGNETIC INDUCTION. 423 The alteration in the charge is due to the current flowing through the conductor, and i is the rate at which the charge is diminishing, so that dQ ' " dt ' Substituting this value of i in equation (1), we get ^W + ^di-^G-^ (2>- The form of the solution of this equation will depend upon whether the roots of the quadratic equation are real or imaginary. Let us first take the case when they are imaginary, i.e. when ^<4^. In this case the solution of (2) takes the form e=^.-S%os[(^-g)S+«} (3), where A and a are arbitrary constants. We see from this expression that Q is alternately positive and negative and vanishes at times following one another at the interval 1(1 R']^ ^/|za-4z4 • The charge Q is thus represented by a harmonic function whose amplitude decreases in geometrical progression as the time increases in arithmetical progression. 424 ELECTROMAGNETIC INDUCTION. [CH. XI The discharge of the jar is oscillatory, so that if for example, to begin with, the inside of the jar is charged positively, the outside negatively ; then on connecting by the circuit the inside and the outside of the jar, the posi- tive charge on the inside diminishes ; when however it has all disappeared there is a current in the circuit, and the inertia of this current keeps it going, so that positive electricity still continues to flow from the inside of the jar ; this loss of positive electricity causes the inside to become charged with negative electricity, while the outside gets positively charged. Thus the jar which had originally positive on the inside, negative on the outside, has now negative on the inside, positive on the outside. The poten- tial difference developed in the jar by these charges tends to stop the current and finally succeeds in doing so. When this happens the charges on the inside and outside would be equal and opposite to the original charges if the re- sistance of the circuit were negligible ; if the resistance is finite the new charges will be of opposite sign to the old ones, but smaller. The current now begins to flow in the opposite direction, and goes on flowing until the inside is again charged positively, the outside negatively ; if there were no resistance the charges on the inside and outside would regain their original values, so that the state of the system would be the same as when the dis- charge began ; if the resistance is finite the charges are smaller than the original ones. The system goes on then as before until the charges become too small to be ap- preciable. The charges in the jar and the currents in the wire are thus periodic, the charges surging backwards and forwards between the coatings of the jar. The oscillatory character of the discharge was sus- 243] ELECTROMAGNETIC INDUCTION. 425 pected by Henry from observations on the magnetization of needles placed inside a coil in the discharging circuit. The preceding theory was given by Lord Kelvin in 1853. The oscillations were detected by Feddersen in 1857. The method he used consisted in putting an air break in the wire circuit joining the inside to the outside of the jar. This air break is luminous when a current passes through it, and shines out brightly when the current passing through it is great, and is dark when the current vanishes. Hence if we observe the image of this air space formed by reflection at a rotating mirror, it will if the discharge is oscillatory be drawn out into a band with dark and bright spaces, the interval between two dark spaces depending on the speed of the mirror and the frequency of the electrical vibrations. Feddersen observed that the appearance of the image of the air break formed by a rotating mirror was of this character. He showed moreover that the oscillatory character of the discharge was destroyed by putting a large resistance in the circuit, for he found that in this case the image of the air space was a broad band of light gradually fading away in intensity instead of a series of bright and dark bands. When the discharge is oscillatory the frequency of the discharges is often exceedingly large, a frequency of a million complete oscillations a second being by no means a high value for such cases. We see by the expression (3) that when R = 0, the time of vibration is 27r JLG ; thus this time is increased when the self-induction or the capacity is increased. By inserting coils with very great self-induction in the circuit, Dr Oliver Lodge has produced such slow electrical vibrations that the sounds generated by the successive discharges form a musical note. I R\_ 1 V 4i2 oj 426 ELECTROMAGNETIC INDUCTION. [CH. XI la the preceding investigation we have supposed that B? was less than 4Z/(7; if however R is greater than this value, the solution of equation (2) changes its character, and we have now where — Xi, — Xq are the roots of the quadratic equation XX2 + i2x+i = 0. Hence ^^^S+aAS'^^ X-^ ""^ 2Z V 4Z2 GL If we take <= when the circuit is closed, then dQ,ldt vanishes when ^ = and we get, if Qo is the value of Q when ^ = 0, A- The solution of this equation is <.= . '^-'"0'^-") ^ (2). E COS (pt — a) da E cos (pt- a) . and thus -v- = ' -[ (•>), 244] ELECTROMAONETIC INDUCTION. 429 {'-cy where tan a = ^ . Comparing these equations with those of Art. 233 we see that the circuit behaves as if the jar were done away with and the self-induction changed from L to L — 1/Cp\ We also see from (3) that if Cp^ is greater than 1/2Z, the current produced by the electromotive force in the circuit broken by the jar (whose resistance is infinite) is actually greater than the current which would flow if the jar were replaced by a conductor of infinite con- ductivity. If Cp^ =1/L the apparent self-induction of the circuit is zero, and the circuit behaves like an inductionless closed circuit of resistance E. Thus by cutting the circuit and connecting the ends to a condenser of suitable capacity we can increase enormously the current passing through the circuit. We can perhaps see the reason for this more clearly if we consider the behaviour of the mechanical system, which we have used to illustrate the oscillatory discharge of a Leyden jar, viz. the rectilinear motion of a mass attached to a spring and resisted by a frictional force proportional to the velocity. Suppose that X, an external force, acts on this system ; then at any instant X must be in equilibrium with (1) the resultant of the rate of diminution of the momentum of the mass, (2) the force due to the compression or extension of the spring, (3) the resistance. If the frequency of X is very great, then for a given momentum (1) will be very large, so that unless (1) is counterbalanced by (2) a finite force of very great frequency will produce an exceedingly small momentum. Suppose however the frequency of the ex- ternal force is the same as that of the free vibrations of 430 ELECTROMAGNETIC INDUCTION. [CH. XI the system when the friction is zero, then when the mass vibrates with this frequency, (1) and (2) will balance each other, so that all the external force has to do is to balance the resistance ; the system will therefore behave like one without either mass or stiffness resisted by a frictional force. 245. A circuit containing^ a condenser is parallel with one possessing self-induction. Let ABC, AEG, Fig. 118, be two circuits. Let L be the coefficient of self-induction of ABC, R the resistance of this circuit, C the capacity of the condenser in AEG, r the resistance of wires leading from A and G to the plates. Fig. 118. Then if i is the current through ABG, x the charge on the plate nearest to -4, we have, neglecting the self-induction of the circuit AEG, -r di r,. dx X since each of these quantities is equal to the electromotive force between A and G. If i = cos pt, then X — ^ ^ \ sm (jpt + a). {i+^i' where a = tan~^ ~ + tan~^ — 7^ . B, rpG 245] ELECTROMAGNETIC INDUCTION. 431 Hence J= jM±E cos (pt + a). -\-r Thus the maximum current along AEC is to that along ABC as ^L'^ + R^ is to a/ j^^.^ + r\ or, if we can neglect the resistances of the wires to the condenser, as ^Ly + R^ : 1 1 Op. We see that for very high frequencies practically all the current will go along the condenser circuit. Thus when the frequency is very high a piece of a circuit with a little electrostatic capacity will be as efficacious in robbing neighbouring circuits of current as if the places where the electricity accumulates were short-circuited by a conductor. 246. Lenz^s Law. When a circuit is moved in a magnetic field in such a way that a change takes place in the number of tubes of magnetic induction passing through the circuit, a current is induced in the circuit; the circuit conveying this current being in a magnetic field will be acted upon by a mechanical force. Lenz's Law states that the direction of this mechanical force is such that the force tends to stop the motion which gave rise to the current. This result follows at once from the laws of the induction of currents. For suppose Fig. 119 Fig. 119. 432 ELECTROMAGNETIC INDUCTION. [CH. XI represents a circuit which, as it moves from right to left, encloses a larger number of tubes of induction passing through it from left to tight. The current induced will tend to keep the number of tubes of induction unaltered, so that since the number of tubes of magnetic induction due to the external magnetic field which pass through the circuit from left to right increases as the circuit moves tow^ards the left, the tubes due to the induced current will pass through the circuit from right to left. Thus the magnetic shell equivalent to the induced current has the positive side on the left, the negative on the right. Since the number of tubes of induction due to the external field which pass through this shell in the negative direction, i.e. which enter at the positive and leave at the negative side, increases as the shell is moved to the right, the force acting on the shell is, by Art. 212, from left to right, which is opposite to the direction of motion of the circuit. There is a simple relation between the mechanical and electromotive forces acting on the circuit. Let P be the electromotive force, X the mechanical force parallel to the axis of x, i the current flowing round the circuit, u the velocity with which the circuit is moving parallel to oc, N the number of unit tubes of magnetic induction passing through the circuit. Then ^" dt' and if the induced current is due to the motion of the circuit dN dN hence dt dx .u\ P = - u dN dx 246] ELECTROMAGNETIC INDUCTION. 433 Again, by Art. 212, we have „ .dN SO that Xii = — Pi. If we wish merely to find the direction of the current induced in a circuit moving in a magnetic field, Lenz's law is in many cases the most convenient method to use. An example of this law is afforded by the coil revolving in a magnetic field (Art. 234) ; the action of the magnetic field on the currents induced in the coil produces a couple which tends to stop the rotation of the coil. The magnets of galvanometers are sometimes surrounded by a copper box, the motion of the magnet induces currents in the copper, and the action of these currents on the magnets by Lenz's law tends to stop the magnet, and thus brings it to rest more quickly than if the copper box were absent. The quickness with which the oscillations of the moving coil in the Desprez D'Arsonval Galvanometer (Art. 221) subside is another example of the same efifect ; when the coil moves in the magnetic field currents are induced in it, and the action of the magnetic field on these currents stops the coil. Again, if a magnet is suspended over a copper disc, and the disc is rotated, the movement of the disc in the magnetic field induces currents in the disc ; the action of the magnet on these currents tends to stop the disc, and there is thus a couple acting on the disc in the direction opposite to its rotation. There must, however, be an equal and opposite couple acting on the magnet, i.e. there must be a couple on the magnet in the direction of rotation of the disc ; this couple, if the magnet is free to move, will set it rotating in the T. E. 28 434 ELECTROMAGNETIC INDUCTION, [CH. XI direction of rotation of the disc, so that the magnet and the disc will rotate in the same direction. This is a well-known experiment ; the disc with the magnet freely suspended above it is known as Arago's disc. Another striking experiment illustrating Lenz's law is to rotate a metal disc between the poles of an electro-magnet, the plane of the disc being at right angles to the lines of magnetic force ; it is found that the work required to turn the disc when the magnet is 'on' is much greater than when it is 'off.' The extra work is accounted for by the heat produced by the currents induced in the disc. 247. Methods of determining the coefQcients of self and mutual induction of coils. When the coils are circles, or solenoids, the coefficients of induction can be calculated. When, however, the coils are not of these simple shapes the calculation of the coefficients would be difficult or impossible ; they may, however, be determined by experiment by means of the following methods. 248. Determination of the coefficient of self- induction of a coil. Place the coil in BD, one of the c Fig. 120. arms of a Wheatstone's Bridge, and balance the bridge for steady currents, insert in CD a ballistic galvanometer, and place a key in the battery circuit. When this key 248] ELECTROMAGNETIC INDUCTION. 435 is pressed down so as to complete the circuit, although there will be no current through the galvanometer when the currents get steady, yet a transient current will flow through the galvanometer, in consequence of the electro- motive forces which exist in BD arising from the self- induction of the coil. This current though only transient is very intense while it lasts and causes a finite quantity of electricity to pass through the galvanometer, producing a finite kick. We can calculate this quantity as follows : an electromotive force E in BD will produce a current through the galvanometer proportional to E, let this cur- rent be IcE. In consequence of the self-induction of the coil there will be an electromotive force in BD equal to where L is the coefficient of self-induction of the coil and i the current passing through the coil. This electromotive force will produce a current q through the galvanometer where q is given by the equation If Q is the total quantity of electricity which passes through the galvanometer d -'/: the integration extending from before the circuit is com- pleted until after the currents have become steady. The right-hand side of this equation is equal to 28—2 436 ELECTROMAGNETIC INDUCTION. [CH. XI where i^ is the value of i when the currents are steady. By the theory of the ballistic galvanometer, given in Art. 222, we see that if 6 is the kick of the galvanometer where T is the time of swing of the galvanometer needle, G the galvanometer constant, and H the horizontal com- ponent of the earth's magnetic force. Hence we have kLiQ = sin ^6 . — ^ (1). Let us now destroy the balance of the Wheatstone's Bridge by inserting a small additional resistance r in BD, this will send a current p through the galvanometer. To calculate p we notice that the new resistance has approximately the current {^ running through it, and the effect of its introduction is the same as if an electromotive force rio were introduced into DB, this as we have seen produces a current krio through the galvanometer ; hence p = kriQ. This current will produce a permanent deflection <^ of the galvanometer, and by Art. 219 ^ = tan <^ ^ , TT or krio = tan <^ ^ . .(2). Hence from equations (1) and (2), we get sin i|9 T tan<^ TT 249] ELECTROMAGNETIC INDUCTION. 437 249. Determination of the coefficient of mutual induction of a pair of coils. Let A and 5, Fig. 121, represent the pair of coils of which A is placed in series with a galvanometer, and B in series with a battery ; this second circuit being provided with a key for breaking or closing the circuit. Fig. 121. Let R be the resistance of the circuit containing A, Suppose that originally the circuit containing B is broken and that the key is then pressed down, and that after the current becomes steady the current i flows through this circuit. Then before the key was pressed down no tubes of magnetic induction pass through the coil A, while when the current % flows through B the number of such unit tubes is Mi, where M is the coefficient of mutual induction between A and B. Thus the circuit containing A has received an electrical impulse equal to Miy so that Q, the quantity of electricity flowing through the galvanometer will be Mi\R, and if Q is the kick of the galvanometer, we have using the same notation as before. We can eliminate 438 ELECTROMAGNETIC INDUCTION. [CH. XI a good many of the quantities by a method somewhat similar to that used in the last case. Cut the circuit con- taining the coil A and connect its ends to two points on the circuit B separated by a small resistance >S ; then if R is very large compared with 8 this will not alter appreciably the current flowing round B\ on this supposition the current flowing round the galvanometer circuit will be *> R^8 and if <^ is the corresponding deflection of the galvano- meter ^-^i = tan^.f (2). Hence from equations (1) and (2), we get UB sini^r ilf= R-h S tan 7r times the current flowing through the circuit : the convention made on both the electrostatic and magnetic systems is that J9 is a quantity of no dimensions and always equal to unity. We shall for the present leave the dimensions oi p undecided. The dimensional equation connecting the electric and magnetic quantities is therefore p X H X L =i, when H is magnetic force, L a length and { a current. Taking this relation and starting with the electric charge we can get by the equations given in Art. 252 the dimensions of all the electrical and magnetic quantities in terms of M, L, T, p,K: or starting with the magnetic pole we can get them in terms of M, L, T, p, /i. The results for some of the most important electrical quantities are given in the following table. I Quantity. Symbol. Dimensions in Dimensions in terms terms of K and p. of /t and p. Charge e Km^DT-' PfL-^M^D Electric intensity F K-m^L-^T-' p-'fiiM^DT-^ Potential difference V K-m^UT-' p-'fim^DT-- Current i K^M^DT-^^ pti-^M^DT-^ Resistance R K-'L-'T p-'fiLT-' Electric polariza- tion D KhM^L-*T-' PfjL-^M^L-i Capacity C KL p^fi-'L-'T- Specific inductive capacity K K p'fi-'L-'T' Strength of Mag- netic pole m pK-^M^D ^hM^DT-' T. E. 29 450 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII Quantity. SymboL Dimensions in Dimensions in terms terms of K and p. of ix and p. Magnetic force H p-'K^M^DT"^ fi-^M^L-^T-' Magnetic induc- tion B pK-^M^L-l fi^M^L-^T-' Magnetic per- meability fjL p'K-'L-'T' fi We see from this table that the dimensions of K, fi, and p must on all systems of measurement be connected by the relation ^^=5r2 = (velocityy. On Maxwell's theory of the electric field j^/V/xiT is equal to the velocity with which electric disturbances travel through a medium whose magnetic permeability is fi and specific inductive capacity K. On the electrostatic system of units K is of no dimen- sions, as the specific inductive capacity of air is taken as unity whatever may be the unit of mass, length and time. Also on this system p is by hypothesis of no dimensions, being always equal to unity. Hence the dimensions of the electrical quantities on this system of units are got by omitting p and K in the third column of the table. On the electromagnetic system of units /jl is of no dimensions, the magnetic permeability of air being taken as unity whatever the units of mass, length and time ; p is also of no dimensions on this system. Hence the dimen- sions of the electrical quantities on this system of units are got by omitting yu. and p from the fourth column in the table. Another system of units could be got by taking fM and 254] DIMENSIONS OF ELECTRICAL QUANTITIES. 451 K as of no dimensions and p a velocity. If this velocity were taken equal to the ratio of the electromagnetic unit charge to the electrostatic unit, then the unit of electric charge on this system would be the ordinary electrostatic unit of that quantity while the unit magnetic pole would be the unit as defined on the electromagnetic system. This system would thus have the advantage that the electric quantities would be as defined in the electrostatic system, while the magnetic quantities would be as defined in the magnetic system, and we should not have to introduce any new definitions : whereas if we use the electrostatic system we have to define all the magnetic quantities afresh, and if we use the electromagnetic system we have to re-define all the electrical ones\ This system is however never used in practice ; the electromagnetic system or one founded upon it is uni- versally used in Electrical Engineering, and the electro- static system is used for special classes of investigations. 254. The units of resistance, of electromotive force, of capacity on the electromagnetic system are either too large or too small to be practically convenient : hence new 1 It should be noticed that it is only when the electromagnetic system of units is used that ' magnetic induction ' has the meaning assigned to it in Art. 152. If we use any other system of units in which we start with electrical quantities, the ' magnetic induction through unit area ' appears as the quantity whose rate of variation is equal to p times the electromotive force round the boundary of the area. The magnetic induction defined in this way is always proportional to the magnetic induction as defined in Art. 152. The two are however only identical on the electromagnetic system of units. With the definition of Art. 152 the magnetic induction is of the same dimensions as magnetic force since they are both the mechanical force on a unit pole when placed in cavities of different shapes. 29—2 452 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII units which are definite multiples or submultiples of the electromagnetic units are employed. These units and their relation to the electromagnetic system of units (when the units of length, mass and time are the centimetre, gramme and second) are given in the following Table. The unit of resistance is called the Ohm and is equal to 10^ electromagnetic units. The unit of electromotive force is called the Volt and is equal to W electromagnetic vinits. The unit of current is called the Ampere and is equal to 10~^ electromagnetic units. The unit of charge is called the Coulomb and is equal to 10"^ electromagnetic units. The unit of capacity is called the Farad and is equal to 10~* electromagnetic units. The Microfarad is equal to 10~^^ electromagnetic units. The Ampere is the current produced by a Volt through an Ohm. We shall now proceed to explain the methods by which the various electrical quantities can be measured in terms of these units : when the quantity is so measured it is said to be determined in absolute measure. 255. Determination of a Resistance in Absolute measure. The method given in Art. 223 enables us to compare two resistances, and thus to find the ratio of any resistance to that of an arbitrary standard such as the resistance of a column of mercury of given length and cross section when at a given temperature. In order to make use of the electromagnetic system of units we must find the number of electromagnetic units in our standard resistance, or what amounts to the same thing we must 255] DIMENSIONS OF ELECTRICAL QUANTITIES. 453 be able to specify a conductor whose resistance is the electromagnetic unit of resistance. The first method we shall describe, that of the re- volving coil, was suggested by Lord Kelvin and carried out by a committee of the British Association, who were the first to measure a resistance in absolute measure. The method was also one of those used by Lord Rayleigh and Mrs Sidgwick in their determination of the Ohm. When a coil of wire spins about a vertical axis in the earth's magnetic field, currents are generated in the coil; these currents produce a magnetic force at the centre of the coil. If a magnet is placed at the centre of the coil, this magnetic force gives rise to a couple on the magnet tending to twist the magnet in the direction in which the coil is rotating. The resistance of the coil may be deduced from the deflection of the magnet as follows. Let H be the horizontal component of the earth's magnetic force, A the area enclosed by one turn of the coil, n the number of turns, 6 the angle the plane of the coil makes with the magnetic meridian, let the coil revolve with uniform velocity w, so that we may put The number of tubes of magnetic induction passing through the coil is equal to iiAH^inO, and the rate of diminution of this is — nAHco cos (at. Hence, if L is the coefficient of self-induction of the coil, R its resistance, and i the current flowing through the coil, the current being taken as positive when the lines of 454 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII magnetic force due to the current and those due to the earth pass through the circuit in the same direction, we have di L ^,+Ri = — nAHco cos cot at Hence, as in Art. 233, we have nAHay ,„ . r • ,. i = - p., a x 2 1^ ^^^ ^^ + ^^ ^^" ^^1- Nowif unit current through the coil produces a magnetic force at the centre, the current i through the coil will produce a magnetic force Gi cos (ot at right angles to the magnetic meridian, and a force Gi sin wt along the magnetic meridian, since 6 = wt Hence the magnetic force due to the currents in the coil has a component nAEGcoR nAHGco ,^ . ^ . . „ ^. - 2W^^m - 2(Wr-^I>) t^ "^^ 2^^ + ^" ^^" 2^^'' at right angles to the magnetic meridian ; and a component nAHGLro' nAHGco .^ . « . r o .i along the magnetic meridian. Now suppose we have a magnet at the centre of the coil, and let the moment of inertia of this magnet be so great that the time of swing is very large compared with the time of revolution of the coil. The magnetic force acting on the magnet due to the current induced in the coil consists, as we see, of two parts, one constant, the other periodic, the frequency being twice that of the revolution of the coil. By making the moment of inertia of the magnet great enough we may make the effect of the periodic terms as small as we please ; we shall suppose that the magnet is heavy enough to allow us to neglect 255] DIMENSIONS OF ELECTRICAL QUANTITIES. 455 the effect of the periodic terms; when this is done the magnetic force at the centre becomes equal to nAHGcoR 2 (R^ + o)'L') at right angles to the magnetic meridian, and to nAHGLco'' 2(R' + (D''L') along it. Hence if (f> is the angle the magnet at the centre makes with the magnetic meridian 1 nAHGfoR ^ , 2 R' + co^L^ ^^^^= I nAHG L^ ' 2 R^ + co'^L' 1 nAGcoR , ^ 2R'+(d'L'- or tan (/> = 1 nAGLco' 2 R^ + (o^D This equation enables us to find R, SiS A, G, L can be calculated from the dimensions of the rotating coil. When Leo is small compared with R the equation reduces to the simple form , 1 nA Go) When the coil consists of a single ring of wire of radius a, n = l, A= 7ra\ G = 27r/a ; hence tan0 = — ^ . Thus by this method we compare R, which, by Art. 243, is of the dimensions of a velocity, with the velocity of a point on the spinning coil. 456 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII The preceding investigation is only approximate as we have neglected the magnetic field due to the magnet placed at the centre of the ring. 256. Iiorenz's Method. This was also one of the methods used by Lord Rayleigh and Mrs Sidgwick in their determination of the Ohm. It depends upon the principle that if a conducting disc spins in a magnetic field which is symmetrical about the axis of rotation, and if a circuit is formed by a wire, one end of which is con- nected to the axis of rotation while the other end presses against the rim of the disc, an electromotive force pro- portional to the angular velocity will act round the circuit. We can determine this electromotive force by finding the couple acting on the disc when a current flows round this circuit. Let / be the current flowing through the wire. When this current enters the disc it will spread out ; let q be the radial current crossing unit length of the circumference of a circle of radius r at the point defined by 6. Let rdrdd be an element of the area of the disc. The radial current flowing through this area is equal to q rd6. Hence by Art. 210, if H is the magnetic force normal to the disc at this area the tangential mechanical force acting on the area is equal to Hq rdr dO. The moment of this force about the axis of the disc is equal to Hqr'drdd; hence the couple acting on the disc is equal to IP Hqr^drdd, the integration being extended over the area of the disc. 256] DIMENSIONS OF ELECTRICAL QUANTITIES. 457 Since the current flowing across a circle drawn on the disc with its centre at the centre of the disc must equal /, the current flowing into the disc, we have [qrde I. Since the magnetic field is symmetrical about the axis of rotation, II is independent of 6, hence the couple acting on the disc is equal to JHr dr. If N is the number of tubes of magnetic induction passing through the disc -/ H^irr dr. and thus the couple acting on the disc is equal to Now suppose in this circuit there is a battery w^hose electromotive force is E, then in the time 8t the work done by the battery is EIBt ; this work is spent in heating the circuit and in driving the disc. The angle turned through by the disc in this time is wBt, if w is the angular velocity of the disc ; hence the mechanical work done is equal to ^ INcoht ZTT By Joule's law the mechanical equivalent of the heat pro- duced in the circuit is equal to 458 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII where R is the resistance of the circuit. Hence we have by the Conservation of Energy Em = RP8t-\-^INcoEt, LIT E-l-Nt^ /= ^ ; hence there is a counter electromotive force in the circuit equal to This case iUustrates the remark made in Art. 224, as from Ampere's law of the mechanical force acting on currents in a magnetic field we have deduced, by the aid of the principle of the conservation of energy, the expres- sion for the electromotive force due to induction, and have thus proved by dynamical principles that the induction of currents is a consequence of the mechanical force exerted by a magnet on a circuit conveying a current. In Lord Rayleigh's experiments the disc was placed between two coils through which a current passed and the axes of the disc and of the two coils were coincident. The magnetic field acting on the disc may be considered as approximately that due to the current through the coils, as this field is very much more intense than that due to the earth. Hence if i is the current through the coils, M the coefficient of mutual induction between the coils and a circuit coinciding with the rim of the disc, Hence the electromotive force due to the rotation of the disc is Miw ~2^* 257] DIMENSIONS OF ELECTRICAL QUANTITIES. 459 The experiment was arranged as in the diagram ; a galvanometer was placed in the circuit connecting the Fig. 123. centre of the disc and the rim and this circuit was con- nected to two points P, Q in the circuit in series with the coils, and the resistance between P and Q was adjusted until no current passed through the galvanometer. If R is the resistance between P and Q and if a current i flows through PQ the E.M.F. between P and Q will be Ri, but since there is no current through the galvanometer this balances the electromotive force due to the rotation of the disc ; hence Mi(o May Since M can be calculated from the dimensions of the coil and the disc, this formula gives us R in absolute measure. 257. The method given in Art. 249 for determining a coefficient of mutual induction in terms of a resistance may be used to determine a resistance in absolute measure. If we use a pair of coils whose coefficient of mutual induction Ri = or 460 DIMENSIONS OF ELECTRICAL QUANTITIES. [cH. XII can be determined by calculation, then equation (2) of Art. 249 will give the absolute measure of a resistance. This method has been employed by Mr Glazebrook. The result of a large number of experiments made by the preceding methods is that the Ohm is the resistance at 0° C. of a column of mercury 106*3 cm. long and 1 sq. millimetre in cross section. For a comparison of the relative advantages of the preceding methods the student is referred to a paper by Lord Rayleigh in the Philosophical Magazine for November, 1882. 258. Absolute Measurement of a Current. A current may be determined by measuring the attraction between two coils placed in series with each other and with their planes parallel and at right angles to the line joining their centres. If i is the current through the coils, M the coefficient of mutual induction between the coils, x the distance between their centres, the attraction between the coils is equal to dM .^ dx By attaching one of the coils to the scale-pan of a balance and keeping the other fixed we can measure this force, and hence if we calculate dMjdx from the dimensions of the coils we can determine i in absolute measure. The unit current is very conveniently specified by the amount of silver deposited from a solution of silver nitrate through which this current has been flowing for a given time. Lord Rayleigh found that the Ampere is the current which flowing uniformly for one second would cause the deposition of '001118 grammes of silver. 260] DIMENSIONS OF ELECTRICAL QUANTITIES. 461 259. The unit electromotive force is that acting on a conductor of unit resistance when conveying unit current. A practical standard of electromotive force is the Clark cell (Art. 183), whose electromotive force at T Centigrade is equal to 1-434 {1 - -00077 {t - 15)} volts. 260. Ratio of Electrostatic and Electromag- netic Units. We saw in Art. 252 that the ratio of the measur3 of any electrical quantity on the electrostatic system of measurement to the measure of the same quantity on the electromagnetic system is always some power of a certain quantity which we denoted by "-y," and which is the ratio of the electromagnetic unit of electric charge to the electrostatic unit. The measurement of the same electrical quantity on the two systems of units will enable us to find " v." The quantity which has most frequently been measured with this object is the capacity of a condenser. The electro- static measure of the capacity can be calculated from the dimensions of the condenser; thus the electrostatic measure of the capacity of a sphere is equal to its radius ; the ca- pacity of two concentric spheres of radii a and 6 is abl(b — a); the capacity of two coaxial cylinders of length I radii a and h is i^/log b/a. Thus if we choose a condenser of suitable shape the electrostatic measure can be calculated from its dimensions. The electromagnetic measure can be determined by the following method due to Maxwell. One of the arms AC of a Wheatstone's Bridge is cut at P and Q, one plate of the condenser is connected to P, the other to a vibrating piece 462 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII R which oscillates backwards and forwards between P and Q ; when R comes into contact with Q the condenser gets charged, when into contact with P it gets discharged. The current through the galvanometer may be divided into two parts. There is first a steady current which flows through AD when no electricity is flowing into the con- denser, this we shall denote by y. Besides this there is at times a transient current which flows while the condenser is being charged. We shall suppose that each time the condenser is being charged a quantity of electricity equal to F flows through DA in the opposite direction to y. Then if the condenser is charged n times a second the amount which Fig. 124. flows through the galvanometer owing to the charging of the condenser is nY. If the time of swing of the galvanometer needle is very long compared with 1/n of a second this will produce the same effect on the galvanometer as a steady current whose intensity is nY flowing from D to A. Thus if nY = y, the current due to the repeated charging of the condenser will just balance the steady current and there will be no deflection of the galvanometer. We now proceed to find F. This is evidently equal to the quantity of electricity which would flow from A to D 260] DIMENSIONS OF ELECTRICAL QUANTITIES. 463 if there were no electromotive force in the wire BG and the plates of the condenser with the gi-eatest charge they acquire in the experiment were connected to P and Q re- spectively. Let ^ be the current from the condenser along PA during the discharge, Y the current along AD, W the current along BD. Let the resistances of AB, BG, GD, DB, DA, be c, a, 7, yS, a respectively. Let the coefficients of self-induction of these circuits be L^, L^, L^, L^, L^ re- spectively. Then from the circuit ABD, we have , dJ'Y J. d'{Y-Z) J. d'W , dY ^'W^^'~~dt^ ^'^dF^^'lU {dY dZ\ ^dW ^ Integrating from just before discharging until after the condenser is completely discharged, and remembering that both initially and finally Y, Z, W vanish, we have aY+c(Y-Z)-^W = (1), where Y, Z, W are the quantities of electricity which have passed during the discharge through AD, PA, and 5Z) re- spectively. Similarly from the circuit DBG, we have (^ + y+a)W-\-(y-{-a)Y-aZ = (2). We find from equations (1) and (2) Now Z is the maximum charge in the condenser; hence if G is capacity of the condenser and A and C 464 DIMENSIONS OF ELECTRICAL QUANTITIES. [CH. XII the potentials of A and C respectively when the charge is a maximum, i.e. when no current is flowing into the con- denser, If 2/ is the current flowing through AD when no current is flowing in the condenser, and D denotes the potential of D A - D = ay. Hence by equation (3) But when there is no deflection of the galvanometer nY=:y; hence n(7 ~ /9 (7 + a) + (a + c) (^ + 7 + a) 1°^ "^ '^ "^ /3 ^"^ "^ "^^1 • If we know the resistances and n we can deduce from this equation the value of C in electromagnetic measure. In practice the resistance of the battery a is very small compared wdth the other resistances, hence putting a = 0, we find approximately \i + - --^ 1 C7 _ ( 7 (n + c + /3) nC /3 J _ (c< + c + /3)(/S+7) 260] DIMENSIONS OF ELECTRICAL QUANTITIES. 465 By this method we find the electromagnetic measure of the capacity of a condenser ; the electrostatic measure can be found from its dimensions. Now by Art. 252 electrostatic measure of a condenser t;^ = electromagnetic measure of the same condenser Experiments made by this method show that v = S X 10^^ cm./sec. very nearly. 30 CHAPTER XIII. Dielectric Currents and the Electromagnetic Theory of Light. 261. The Motion of Faraday Tubes. Dielectric Currents. In Chapter xi. we considered the relation between the currents in the primary and secondary circuits when an alternating current passes through the primary circuit, we did not however discuss the phenomena occurring in the dielectric between the circuits. As we regard the dielectric as the seat of the energy due to the distribution of the currents, the study of the effects in the dielectric is of primary importance. We owe to Maxwell a theory, now in its main features universally accepted, by which we are able to completely determine the electrical con- ditions, not merely in the conductors but also in every part of the field. We shall also see that Maxwell's views lead to a comprehensive theory of optical as well as of electrical phenomena, and enable us by means of elec- trical principles to explain the fundamental laws of Optics, Before specifying in detail the principles of Maxwells theory, we shall endeavour to show by the consideration of some simple cases that in considering the relation between the work done in taking unit magnetic pole round a CH. XIIL 261] DIELECTRIC CURRENTS. 467 closed circuit and the current flowing through that circuit (see Art. 201), we must include under the term current effects other than the passage of electricity through con- ducting media, if we are to retain the conception that the dielectric is the seat of the energy in electric and magnetic phenomena. Let us consider the case of a long, straight, cylindrical conductor carrying an alternating electric current. In the dielectric around this wire there is a magnetic field, and according to the views enunciated in Art. 162, there is in a unit volume of the dielectric at a place where the magnetic force is H an amount of energy equal to /juH'^IHtt. As the alternating current changes in intensity, the energy in the surrounding field changes, and this change in the energy must be due to the motion of energy from one part of the field to another, the energy moving radially towards or away from the wire conveying the current. If the dielectric medium possesses inertia, and if its properties in any way resemble those of any kind of matter with which we are acquainted, the energy cannot travel from one place to another with an infinite velocity. As the alternating current changes, the energy in the field will change also ; when the current is passing through its zero value, it is evident that the magnetic energy cannot now vanish throughout the field, for we assume that the enei-gy travels at a finite rate, and it is only a finite time since the current was finite. If the magnetic energy did vanish it would imply that the energy could travel over a distance, however great, in a finite time. If, however, the magnetic energy does not vanish simul- taneously all over the field, there must be places where 30—2 468 DIELECTRIC CURRENTS. [CH. XIII the magnetic force does not vanish. But the current through the conductor vanishes and there are no magnetic substances in the field. Hence we conclude that unless we assume that the energy in the magnetic field can travel from one place to another with an infinite velocity, we must admit that in a variable field magnetic forces can arise apart from magnets or electric currents through conductors. 262. Let us now see if we can find any clue as to what produces the magnetic field under these circumstances. Let us consider the following simple case. Let A, B (Fig. 125) O Fig. 125. be two vertical metal plates forming a parallel plate condenser and let the upper ends of these plates be con- nected by a wire of high resistance. Suppose that initially the plate A is charged with a uniform distribution of positive electricity while B is charged with an equal distribution of negative electricity. If the plates are dis- connected, horizontal Faraday tubes at rest will stretch from one plate to the other. When the plates are connected by the wire the horizontal Faraday tubes will move vertically upwards towards the wire. Let v be the velocity of these tubes, and a- the surface density of the I 262] DIELECTRIC CURRENTS. 469 electricity on the plates, then the upward current passing across unit length in the plate A and the downward current in B are equal to va. By Art. 207 these currents will produce a uniform magnetic field between the plates, the magnetic force being at right angles to the plane of the paper and its magnitude equal to ^irva. If iV is the number of Faraday tubes passing through unit area of a plane in the dielectric parallel to the plates of the condenser N =(t. Thus the magnetic force between the planes is equal to ^irNv. The condition of things be- tween the plates is such that we have the Faraday tubes moving at right angles to themselves, and that we have also a magnetic force at right angles both to the Faraday tubes and to the direction in which they are moving ; while the intensity of this force is equal to 47r times the product of the number of tubes passing through unit area and the velocity of these tubes. Let us now see what are the consequences of gene- ralizing this result, and of supposing that the relation between the magnetic force and the Faraday tubes which exists in this simple case is generally applicable to all magnetic fields. Suppose then that whenever we have movements of the Faraday tubes we have magnetic force and conversely, and that the relation between the magnetic force and the Faraday tubes is that the magnetic force is equal to 47r times the product of the 'polarization* (Art. 69) and the velocity of the Faraday tubes at right angles to the direction of polarization. The direction of the magnetic force being at right angles to both the direction of polarization and the direction in which the Faraday tubes are moving. 470 DIELECTRIC CURRENTS. [CH. XIII We shall begin by considering what on this view is the physical meaning of H' x 00' where 00' is a line so short that the magnetic force may be regarded as constant along its length, and //' is the component of the magnetic force along 00'. Let OA (Fig. 126) represent in magnitude and direction the velocity of the Faraday tubes, and OP the polarization ; then if OB represents the magnetic force, OB will be at right angles to OA and OP and equal to 47r. 0^. OP sin <^, where is the angle POA. The component H' of the magnetic force along 00' will be 47r . OA . OP sin <^ cos 0, where 6 is the angle BOO'. Thus we have H' X 00' = 47r .OA.OP. 00' sin 0COS 6 = 247rA (1), where A is the volume of the tetrahedron three of whose sides are OA, OP, 00'. 262] DIELECTRIC CURRENTS. 471 Let us now find the number of Faraday tubes which cross 00' in unit time. To do this, draw 00 and O'D equal and parallel to AO, OA being the velocity of the Faraday tubes. Then the number of tubes which cross 00' in unit time is the number of tubes passing through the area OCDO\ The area of the parallelogram OGDO' is equal to 0.4x00' sin ^00'. The number of tubes passing through it is therefore OP X sin <9' xOAx OO'siiiAOO' (2), where 6' is the angle between OP and the plane of the parallelogram OCDO' ; this is the same as the angle between OP and the plane AOO'. But 6A = OP X sin 6' x OA x 00' sin ^100', where A as before is the volume of the tetrahedron POO' A. Hence from (1) and (2) we see that H' X 00' = 47r (number of Faraday tubes crossing 00' in unit time). Thus JH'ds where the integral is taken round a closed curve is equal to 47r times the number of tubes which pass inwards across the curve in unit time. In Art. 201 JH'ds was taken as equal to 47r times the currents flowing through the space enclosed by the curve, and the only currents discussed in that article were currents flowing through conductors : we shall now con- sider what interpretation we must attach to the new expression we have just found for JH'ds. 472 DIELECTRIC CURRENTS. [CH. XIII In the first place, any tube which in unit time passes inwards across one part of the curve and outwards across another part will not contribute anything to the total number of tubes passing across the closed curve, for its contribution when it passes inwards is equal and opposite to its contribution when it passes outwards. Hence all the tubes we need consider are those which only cross the curve once, which pass inwards across the curve and do not leave it within unit time. These tubes may be divided into two classes, (1) those which remain within the curve, (2) those which manage to disappear without again crossing the boundary. The first set will increase the total polarization over any closed surface bounded by the curve, and the number of those which cross the boundary in unit time is equal to the rate of increase in this total polarization. The existence of the second class of tubes tithe 1 %eire i C33Z tube Fig. 127. depends upon the passage of conductors, or of moving charged bodies through the area bounded by the curve. 262] DIELECTRIC CURRENTS. 473 Thus suppose we have a metal wire passing through the circuit, then the tubes which cross the boundary may run into this wire and be annulled, the disappearance of each unit tube corresponding to the passage of unit electricity along the wire ; or the tube may have one end on the wire and cross the circuit, its end running along the wire ; the passage of such a tube across the boundary means the passage of a unit of electricity along the wire, or one end of the tube may be on a charged body which moves through the circuit. Thus the number of tubes of class (2) which cross the circuit in unit time is equal to the number of units of electricity which pass in that time along conductors or charged bodies passing through the circuit, i.e. it is equal to the sum of the conduction and convection currents flowing throug the circuit. Hence the work done when unit pole is taken round a closed circuit is equal to 47r times the sum of the conduction and convection currents flowing through that circuit and the rate of increase of the total polarization through the circuit. From this we see that a change in the polariza- tion through the circuit produces the same magnetic effect as a conduction current whose intensity is equal to the rate of increase of the polarization. We shall call the rate of increase in the polarization the dielectric current. The recognition of the magnetic effects due to these dielectric currents is the fundamental feature of Maxwell's Theory of the Electric Field. We have given a method of regarding the magnetic field which leads us to ex- pect the magnetic effects of dielectric currents. It must be remembered, however. Maxwell's theory consists in the expression of this result and is not limited to any particular method of explaining it. 474 DIELECTRIC CURRENTS. [CH. XIII 263. Propagation of Electromagnetic Disturb- ances. We shall now proceed to show that Maxwell's theory leads to the conclusion that an electric disturbance is propagated through air with the velocity of light. We can employ the equations we deduced in Art. 231, if we regard u, v, w the components of the current, as the components of the sum of the dielectric, convection, and conduction currents. If X, F, Z are the components of the electric intensity, and IT its specific inductive capacity, then the X, y, z components of the polarization are respectively ^x, fy. fz, 47r 47r 47r the components of the dielectric currents are therefore K dX KdY KdZ 47r dt * 4>7r dt ' 4!7r dt ' If (T is the specific resistance of the medium, the components of the conduction current are X Y Z cr ' a ' a* Hence u, v, w the components of the total effective current are given by the equations K dX X 47r at cr ^KdY_^Y 4!7r dt a- ' K dZ Z 47r at or Hence substituting these values of u, v, w in the equa- tions of Art. 231 we get, using the notation of that Article, 263] DIELECTRIC CURRENTS. 475 the following equations as the expression of Maxwell's Theory, (47r dt a-) dy dz ' (K dY Y) da d^ ^"^ \^7r dt '^ cr] ~ dz dx' ^^(K dZ ^Z)dp da (47r dt a-} dx dy' da dZ dY dt ~ dy dz * dh dX dZ dt" dz dx' dc dY dX dt "" dx dy ' Let us now consider the case of a dielectric for which o- is infinite, so that all the currents are dielectric currents; putting a infinite in the preceding equations, and a = fia, h = /x/3, c = fjuy, we get j^dX^dy _d§\ dt dy dz v- ^ _da _^drf ^ It'di^dxC (1)> dZ dp da dt dx da ^^-dt=^ dZ dy dp ^dX ^ dt ~ dz dy^dY ^ dt ~Tx dy^ dY\ dz dZ dx ^ dX .(2). 476 DIELECTRIC CURRENTS. [CH. XIII Differentiating the first equation in (1) with respect to t, we get j.d?X _ d drj d djS df dy dt dz dt Substituting the values of dy/dt, dff/dt, and noticing that by (1) dX dY dZ dx dy dz is independent of the time, we get ^d'X d'X d'X d'X ^^^f=^^'^df-^l^ (^>- We may by a similar process get equations of the same form for F, Z, a, h, c. To interpret these equations let us take the simple case when the quantities are independent of the coordi- nates X, y. Equation (3) then takes the form If we put i=z dp dz^ t JlJ^K' and change the variables from z and ^ to f and t], we get d^dnfj The solution of which is ^'i-hX-'-M '"'' where F and / denote any arbitrary functions. 263] DIELECTRIC CURRENTS. 477 Since F(z — t/JfjbK) remains constant as long as z -tjj fiK is constant, we see that if a point travels along the axis of z in the positive direction with the velocity l/J/juK, the value of F(z — tlJ/j.K) will be constant at this point. Hence the first term in equation (5) represents a value of X travelling in the positive direction of the axis of z with the velocity I/J/jlK. Similarly the second term in (5) represents a value of X travelling in the negative direction along the axis of z with the velocity l/Jjj,K. For example, suppose that w^hen ^ = 0, X is zero except between ^ = + e, z= —e where it is equal to unity, and suppose further that dX/dt is everywhere zero when ^ = 0. Then equation (5) shows that after a time t X = ^ between z = .=^ -- e, and z = — 1=^^ + e 2 J^,K JtiK ' and between ^ = -. — e, and z=— 7= + e, and is zero everywhere else. Thus the quantity repre- sented by X travels through the dielectric with the velocity IjJ^K. It is shown in treatises on Differential Equations that equation (3), the general form of the equation (4), represents a disturbance travelling with the velocity l/J^K. Thus Maxwell's theory leads to the result that electric and magnetic effects are propagated through the dielectric with the velocity IjJ^K. Let us see what this velocity is when the dielectric is air. Using the electromagnetic system of units we have 478 DIELECTRIC CURRENTS. [CH. XIII for air a = 1, K=— where v is the ratio of the electro- magnetic unit of electricity to the electrostatic unit (Art. 253). Hence on Maxwell's theory electric and magnetic effects are propagated through air with the velocity "v." Now experiments made by the method described in Art. 260 lead to the result that within the errors of experiment v is equal to the velocity of light through air. Hence we conclude that electromagnetic effects are propagated through air with the velocity of light. This result led Maxwell to the view that since light travels with the same velocity as an electromagnetic disturbance, it is itself an electromagnetic phenomenon; a wave of light being a wave of electric and magnetic disturbances. 264. Plane Electromagnetic Waves. Let us con- sider more in detail the theory of a plane electric wave. If/, g, h are the components of the electric polarization in such a wave, l^ m, n the direction cosines of the normal to the wave front, and \ the wave length, then we may put 27r /=/o cos — {Ix 4- my -{■ nz — Vt), A. g = go cos — (Ix 4- niy + 7iz — Vt), A, h = hQ cos —-(lx-\- my + nz —Vt) ; A, where V is the velocity of propagation of the wave, and fot 9o> ho quantities independent of x, y, z or t Since df^dg^dh^^ dx dy dz ' we have Z/i + mg^ + nho — 0, and therefore If + mg + nh = 0. 264] DIELECTRIC CURRENTS. 479 Thus the electric polarization is perpendicular to the direction of propagation of the wave. By equation (2), Art. 263, we have da^dZ_dY dt~ dy dz ' and Z=^h, Y^^g. Hence da 47r 27r. . . 27r ., , , tt^. 17^ ~u;T~ l^'^o ~ '^9o] sm -z- {ooc -\- my + nz - Vt), a = -17Y7 (^^0 — '^^^o) cos — (lo) + my -\-nz — Vt) ; or since a = ^irViiiig — mh); similarly l3 = 4i7rV{lh-nf), y = 4<7rV(7nf-lg). Hence la 4- m/3 + ny = 0, so that the magnetic force is at right angles to the direction of propagation of the wave, and since fa + g^ + hy = {h the magnetic force is perpendicular also to the electric polarization. Since ja2 + ^2 ^. ^2ji = 47r F IP + ^= + h']\ the resultant magnetic force is 47rF times the resultant electric polarization. 480 DIELECTRIC CURRENTS. [CH. XIII Hence in a plane electric wave, and therefore on Maxwell's theory in a plane wave of light, there is in the front of the wave an electric polarization, and at right angles to this, and also in the wave front, there is a magnetic force bearing a constant ratio to the polariza- tion. We shall see in Art. 267 that in a plane polarized light wave the electric polarization is at right angles to, and the magnetic force in, the plane of polarization. In strong sunlight the maximum electric intensity is about 10 volts per centimetre, and the maximum magnetic force about one-fifth of the horizontal magnetic force due to the earth in England. 265. Propagation by the Motion of Faraday- Tubes. The results obtained by the preceding analysis follow very simply from the view that the magnetic force is due to the motion of the Faraday tubes. The electro- A B Fig. 128. motive force round a circuit moving in a magnetic field is equal to the rate of diminution of the number of tubes I 265] DIELECTRIC CURRENTS. 481 of magnetic induction passing through the circuit. Thus let P, Q/Fig. 128) be two adjacent points on a circuit, P', Q' the position of these points after the lapse of a time Bt. Then the diminution in the time Bt of the number of tubes of magnetic induction passiug through the circuit of which PQ forms a part may, as in Art. 135, be shown to be equal to the sum of the number of tubes which pass through the sum of the areas PP'Q'Q. The number passing through PP'Q'Q is equal to PQ X PP' X 5 sin sin ^, where B is the magnetic induction, the angle it makes with the plane PP'Q'Q, and 6 the angle between PP' and PQ. If V is the velocity with which the circuit is moving PF = VBt. Thus the rate of diminution in the number of tubes passing through the circuit is SPQ.Fi^ sin sin l9. Hence we may regard the electromotive force round the circuit as equivalent to an electric intensity at each point P of the circuit whose component along PQ is equal to VB sin <^ sin 6. As the component of this intensity parallel to B and V vanishes, the resultant intensity is at right angles to B and Y and equal to i^Tsin^lr, where -v/r is the angle between B and V. In this case the circuit was supposed to move, the tubes of induction being at rest, we shall assume that the same expression holds when the circuit is at rest and the tubes of mag- netic induction move with the velocity V across an element of the circuit at rest. Let us now introduce the view that the magnetic force is due to the motion of the Faraday tubes. Let OA (Fig. 129) T. E. 31 482 DIELECTRIC CURRENTS. [CH. XIII represent the velocity of the Faraday tubes, OP the electric polarization, and OB the magnetic induction, which in a Fig. 129. non-crystalline medium is parallel to the magnetic force and therefore (see page 470) at right angles to OP and OA. By what we have just proved the electric intensity is at right angles to OB and OA, and therefore along 00. Now in a non-crystalline medium the electric intensity is parallel to the electric polarization; hence OP and OG must coincide in direction ; hence the Faraday tubes move at right angles to their length. Again, if E is the electric intensity, by what we have just proved E=BV (1). But if H is the magnetic force, /x the magnetic permea- bility, and by Art. 262 H=4^itVP (2), where P is the electric polarization. 266] DIELECTRIC CURRENTS. 483 Hence by (1) and (2) ^= 47r/i F^P. If K is the specific inductive capacity of the dielectric 47r hence we have V'^= l/fiK. The tubes therefore move with the velocity 1/\//jlK at right angles to their length. 266. Evidence for MaxwelPs Theory. We shall now consider the evidence furnished by experiment as to the truth of Maxwell's theory. We have already seen that Maxwell's theory agrees with facts as far as the velocity of propagation through air is concerned. We now consider the case of other dielectrics. The velocity of light through a non-magnetic dielectric whose specific inductive capacity is K is on Maxwell's theory equal to \jj K. Hence velocity of light in this dielectric velocity of light in air ^. specific inductive capacity of air specific inductive capacity of dielectric * But by the theory of light this is also equal to 1 31—2 484 DIELECTRIC CURRENTS. [CH. XIII where n is the refractive index of the dielectric. Hence on Maxwell's theory n^= electrostatic measure of the specific inductive capacity. In comparing the values of ri- and K we have to re- member that the electrical conditions under which these quantities are on Maxwell's theory equal to one another, are those which hold in a wave of light where the electric intensity is reversed millions of millions of times per second. We have at present no means of directly measur- ing K under these conditions. To make a fair comparison between ii? and K we ought to take the value of K determined for electrical oscilla- tions of the same frequency as those of the vibrations of the light for which n is measured. As we catmot find K for vibrations as rapid as those of the visible rays, the other alternative is to use the value of n for waves of very great wave length ; we shall call this value n^. The process by which n^ is obtained is not however very satisfactory. Cauchy has given the formula connecting n with the wave length X, which holds accurately within the limits of the visible spectrum, unless the refract- ing substance is one which shows the phenomenon known as 'anomalous dispersion.' To find ^^ we apply this em- pirical formula to determine the refractive index for waves millions of times the length of those used to determine the constants A, B, G which occur in the formula. For these reasons we should expect to find cases in which K is not equal to n^, but though these cases are numerous there are many others in which K is approximately equal to n\^. A list of these is given in the following table : 267] DIELECTRIC CURRENTS. 485 Name of Substance K nl Paraffin 2-29 2022 Petroleum spirit 1-92 1-922 Petroleum oil 207 2075 Ozokerite 213 2-086 Benzene 2-38 2-2614* Carbon bisulphide 2-67 2-678* As examples where the relation does not hold, we have Glass (extra dense flint) 10-1 2-924* Calcite (along axis) 7-5 2197* Quartz (along optic axis) 4-55 2-41* Distilled water 76 1-779* Maxwells Theory of Light has been developed to a considerable extent and the consequences are found to agree well with experiment. In fact the electromagnetic is the only theory of light yet advanced in which the difficulties of reconciling theory with experiment do not seem insuperable. 267. Hertz's Experiments. The experiments made by Hertz on the properties of electric waves, on their reflection, refraction, and polarization furnish perhaps the most striking evidence in support of Maxwell's theory, as it follows from these experiments that the properties of these electric waves are entirely analogous to those of light waves. We regret that we have only space for an exceedingly brief account of a few of Hertz's * These a Hum light. 486 DIELECTRIC CURRENTS. [CH. XIII beautiful experiments; for a fuller description of these and other experiments on electric waves with their bearings on Maxwell's theory, we refer the reader to Hertz's own account in 'Electrical Waves' and to Recent Researches on Electricity and Magnetism by J. J. Thomson. We saw in Art. 243 that when a condenser is dis- charged by connecting its coatings by a conductor, elec- trical oscillations are produced, the period of which is approximately 2irjLG where G is the capacity of the condenser, and L the coefficient of self-induction of the circuit connecting its plates. This vibrating electrical system will, on Maxwell's theory, be the origin of electrical waves, which travel through the dielectric with the ve- locity V and whose wave length is VlirY JLG. By using condensers of small capacity whose plates were connected by very short conductors Hertz was able to get electrical waves less than a metre long. This vibrating electrical system is called a vibrator. Hertz used several forms of vibrators ; the one used in the experiment we are about to describe consists of two equal brass cylinders placed so that their axes are coincident. The two cylinders are connected to the two poles of an induction coil. When this is in action sparks pass between the cylinders. The cylinders corre- spond to the plates of the condenser, and the air be- tween the cylinders (whose electric strength breaks down when the spark passes) to the conductor connecting the plates. The length of each of these cylinders is about 12 cm., and their diameters about 3 cm. ; their sparking ends are well polished. 267] DIELECTRIC CURRENTS. 487 To detect the presence of the electrical waves, Hertz used a very nearly closed metallic circuit, such as a piece of wire, bent into a circle, the ends of the wire being exceedingly close together. When the electric waves strike against this detector very minute sparks pass between the terminals ; these sparks serve to detect the presence of the waves. Recently Professor Lodge has introduced a still more sensitive detector. It is founded on the fact discovered by Branly that the electrical resistance of a number of metal turnings, placed so as to be loosely in contact with each other, is greatly affected by the impact of electric waves, and that all that is necessary to detect these waves is to take a glass tube, fill it loosely with iron turnings, and place the tube in series with a battery and a galvanometer. When the waves fall on the tube the resistance, and therefore the deflection of the galvano- meter, is altered. The analogy between the electrical waves and light waves is very strikingly shown by Hertz's experiments with parabolic mirrors. If the vibrator is placed in the focal line of a parabolic cylinder, and if the Faraday tubes emitted by it are parallel to this focal line ; then if the laws of reflection of these electric waves are the same as for light waves, the waves emitted by the vibrator will, after reflection from the cylinder, emerge as a parallel beam ; and will therefore not diminish in intensity as they recede from the mirror. When such a beam falls on another para- bolic cylinder, the axis of whose cross section coincides with the axis of the beam, it will be brought to a focus on the focal line of the second mirror. The parabolic mirrors used by Hertz were made of 488 DIELECTRIC CURRENTS. [CH. XIII sheet zinc, and their focal length was about 12o cm. The vibrator was placed so that the axes of the cylin- ders coincided with the focal line of one of the mirrors. The detector, which was placed in the focal line of an equal parabolic mirror, consisted of two pieces of wire ; each of these wires had a straight piece about 50 cm. long, and was then bent at right angles so as to pass through the back of the mirror, the length of the bent piece being about 15 cm. The ends of the two pieces coming through the mirror were bent so as to be ex- ceedingly near to ' each other. The sparks passing between these ends were observed from behind the mirror. The mirrors are represented in Fig. 130. .J ^1 Fig. 130. Reflection of ElectHc Waves. To show the reflection of these waves the mirrors were placed side by side so that their openings looked in the same direction and their axes converged at a point distant about 8 metres from the min'ors. No sparks passed between the points of the detector when the vibrator was in action. If 267] DIELECTRIC CURRENTS. 489 however a metal plate about 2 metres square was placed at the intersection of the axes of the mirrors, and at right angles to the line which bisects the angle between the axes, sparks appeared at the detector. These sparks however disappeared if the metal plate was turned through a small angle. This experiment shows that the electric waves are reflected and that, approximately at any rate, the angle of incidence is equal to the angle of reflection. Refraction of Electric Waves. To show the refraction of these waves Hertz used a large prism made of pitch. This was about 1"5 metres high, and it had a refracting angle of 30° and a slant side of 1'2 metres. When the electric waves from the mirror containing the vibrator passed through this prism, the sparks in the detector were not excited when the axes of the two mirrors were parallel, but sparks were produced when the axis of the mirror containing the detector made a suitable angle with that containing the vibrator. When the system was adjusted for minimum deviation, the sparks were most vigorous in the detector when the angle between the axes of the mirrors was equal to 22°. This would make the refractive index of pitch for these electrical waves equal to r69. Electric Analogy to a plate of Tourmaline. If a properly cut tourmaline plate is placed in the path of a plane polarized beam of light incident at right angles on the plate, the amount of light transmitted through the tourmaline plate depends upon its azimuth. For one particular azimuth all the light will be stopped. 490 DIELECTKIC CURRENTS. [CH. XllI while for an azimuth at right angles to this the maximum amount of light will be transmitted. If a screen be made by winding metal wire round a large rectangular framework so that the turns of the wire are parallel to one pair of sides of the frame, and if this screen be interposed between the mirrors when they are facing each other with their axes coincident, then it will stop the sparks in the detector when the turns of the wire are parallel to the focal lines of the mirrors, and thus to the Faraday tubes proceeding from the vibrator: the sparks will however recommence if the framework is turned through a right angle so that the wires are perpen- dicular to the focal lines of the mirror. If this framework is substituted for the metal plate in the experiment on the reflection of waves, the sparks will appear in the detector when the wires are parallel to the focal lines of the cylinders and will disappear when they are at right angles to them. Thus this framework reflects but does not transmit Faraday tubes parallel to the wires, while it transmits but does not reflect Faraday tubes at right angles to them. It thus behaves towards the transmitted electrical waves as a plate of tourmaline does towards light waves. By using a framework wound with exceedingly fine wires placed very close together Du Bois and Rubens have recently succeeded in polarizing in this way radiant heat whose wave length though greater than that of the rays of the visible spectrum is exceedingly small compared with that of electric waves. Angle of Polarization. When light polarized in a plane at right angles to the plane of incidence falls upon a plate of refracting substance, 20 7] DIELECTKIC CURRENTS. 491 and the normal to the wave front makes with the normal to the refracting surface an angle tan~^ //,, where fi is the refractive index, all the light is refracted and none re- flected. When light is polarized in the plane of incidence some of the light is always reflected. Trouton has obtained a similar effect with electric waves. From a wall 3 feet thick reflections were ob- tained when the Faraday tubes proceeding from the vibrator were perpendicular to the plane of incidence, while there was no reflection when the vibrator was turned through a right angle so that the Faraday tubes were in the plane of incidence. This proves that on the electromagnetic theory of light we must suppose that the Faraday tubes are at right angles to the plane of polariza- tion. A very convenient arrangement for studying the properties of electric waves is described in a paper by Professor Bose in the Philosophical Magazine for January 1897. CHAPTER XIV. Thermoelectric Currents. 268. Seebeck discovered in 1821 that if in a closed circuit of two metals the two junctions of the metals are at different temperatures an electric current will flow round the circuit. If, for example, the ends of an iron and of a copper wire are soldered together and one of the junc- tions is heated a current of electricity will flow round the circuit ; the direction of the current is such that the current flows from the copper to the iron across the hot junction, provided the mean temperature of the junctions is not greater than about 600° Centigrade. The current flowing through the thermoelectric circuit represents a certain amount of energy, it heats the circuit and may be made to do mechanical work. The question at once arises, What is the source of this energy ? A discovery made by Peltier in 1834 gives a clue to the answer to this question. Peltier found that when a cur- rent flows across the junction of two metals it gives rise to an absorption or liberation of heat. If it flows across the junction in one direction heat is absorbed, while if it flows in the opposite direction heat is liberated. If the current flows in the same direction as the current at the CH. XIV. 269] THERMOELECTRIC CURRENTS. 493 hot junction in a thermoelectric circuit of the two metals heat is absorbed; if it flows in the same direction as the current at the cold junction of the circuit heat is liberated. Thus, for example, heat is absorbed when a current flows across an iron-copper junction from the copper to the iron. The heat liberated or absorbed is proportional to the quantity of electricity which crosses the junction. The amount of heat liberated or absorbed when unit charge of electricity crosses the junction is called the Peltier Effect at the temperature of the junction. Now suppose we place an iron-copper circuit with one junction in a hot chamber and the other junction in a cold chamber, a thermoelectric current will be produced flowing from the copper to the iron in the hot chamber, and from the iron to the copper in the cold chamber. Now by Peltier's discovery this current will give rise to an absorption of heat in the hot chamber and a libera- tion of heat in the cold one. Heat will be thus taken from the hot chamber and given out in the cold. In this respect the thermoelectric couple behaves like an ordinary heat-engine. 269. The experiments made on thermoelectric currents are all consistent with the view that the energy of these currents is entirely derived from thermal energy, the cur- rent through the circuit causing the absorption of heat at places of high temperature and its liberation at places of lower temperature. We have no evidence that any energy is derived from any change in the molecular state '494 THERMOELECTRIC CURRENTS. [CH. XIV of the metals caused by the passage of the current or from anything of the nature of chemical combination going on at the junction of the two metals. Many most important results have been arrived at by treating the thermoelectric circuit as a perfectly re- versible thermal engine and applying to it the theorems which are proved in the Theory of Thermodynamics to apply to all such engines. The validity of this application may be considered as established by the agreement be- tween the facts and the result of this theory. There are however thermal processes occurring in the thermoelectric circuit which are not reversible, i.e. which are not reversed when the direction of the current flowing through the circuit is reversed. There is the conduction of heat along the metals due to the difference of temperatures of the junctions, and there is the heating effect of the current flowing through the metal which, by Joule's law, is pro- portional to the square of the current and is not reversed with the current. Inasmuch as the ordinary conduction of heat is independent of the quantity of electricity passing round the circuit, and the heat produced in accordance with Joule's law is not directly proportional to this quantity, it is probable that in estimating the connection between the electromotive force of the circuit, which is the work done when unit of electricity passes round the circuit, and the thermal effects which occur in it, we may leave out of account the conduction effect and the Joule effect and treat the circuit as a reversible engine. If this is the case, then, as Lord Kelvin has shown, the Peltier effect cannot be the only reversible thermal effect in the circuit. For let us assume for a moment that the Peltier effect is the only reversible thermal effect in the 269] THERMOELECTRIC CURRENTS. 495 circuit. Let Pj be the Peltier effect at the cold junction whose absolute temperature is T^, so that Pi is the mechanical equivalent of the heat liberated when unit of electricity crosses the cold junction ; let Pa be the Peltier effect at the hot junction whose absolute temperature is Ta, so that Pa is the mechanical equivalent of the heat absorbed when unit of electricity crosses the hot junction. Then since the circuit is a reversible heat-engine, we have (see Maxwell's Theory of Heat) _ work done when unit of electricity goes round the circuit But the work done when unit of electricity goes round the circuit is equal to E, the electromotive force in the circuit, and hence E={T,-T,).?^. Thus on the supposition that the only reversible thermal effects are the Peltier effects at the junctions, the electromotive force round a circuit whose cold junc- tion is kept at a constant temperature should be pro- portional to the difference between the temperatures of the hot and cold junctions. Gumming, however, showed that there were circuits where, when the temperature of the hot junction is raised, the electromotive force diminishes instead of increasing, until when the hot junc- tion is hot enough the electromotive force is reversed and the current flows round the circuit in the reverse direc- tion. This reasoning led Lord Kelvin to suspect that besides the Peltier effects at the junction there were 496 THERMOELECTRIC CURRENTS. [CH. XIV reversible thermal effects produced when a current flows along an unequally heated conductor, and by a laborious series of experiments he succeeded in establishing the existence of these effects. He found that when a current of electricity flows along a copper wire whose tempera- ture varies from point to point, heat is liberated at any point P when the current at P flows in the direction of the flow of heat at P, i.e. when the current is flowing from hot places to cold, while heat is absorbed at P w^hen the current flows through it in the opposite direc- tion. In iron, on the other hand, heat is absorbed at P when the current flows in the direction of the flow of heat at P, while heat is liberated when the current flows in the opposite direction. Thus when a current flows along an unequally heated copper wire it tends to diminish the differences of temperature, while when it flows along an iron wire it tends to increase those differences. This effect produced by a current flowing along an unequally heated conductor is called the Thomson effect. Specific Heat of Electricity, 270. The laws of the Thomson effect can be con- veniently expressed in terms of a quantity introduced by Lord Kelvin and called by him the * specific heat of the electricity in the metal.' If a is this ' specific heat of electricity,' A and B two points in a wire, the temperatures of J. and B being respectively t^ and t^, and the difference between t^ and t^ being supposed small, then a is defined by the relation, a (ti — t^) = heat developed in AB when unit of electricity passes through AB from A to B. » 270] THERMOELECTRIC CURRENTS. 497 The study of the thermoelectric properties of con- ductors is very much facilitated by the use of the thermo- electric diagrams introduced by Professor Tait. Before proceeding to describe them we shall enunciate two results of experiments made on thermoelectric circuits which are the foundation of the theory of these circuits. The first of these is, that if E^ is the electromotive force round a circuit when the temperature of the cold junction is 4 and that of the hot junction t^^E^. the electro- motive force round the same circuit when the temperature of the cold junction is t^, and that of the hot junction t^, then E^ + E^ will be the electromotive force round the circuit when the temperature of the cold junction is ^qj and that of the hot junction t^. It follows from this result that E, the electromotive force round a circuit whose junctions are at the temperatures t^ and ^i, is equal to J t where Qdt is the electromotive force round the circuit when the temperature of the cold junction is ^ — ^dt, and the temperature of the hot junction is t-^^dt The quantity Q is called the thermoelectric power of the circuit at the temperature t. The second result relates to the electromotive force round circuits made of different pairs of metals whose junctions are kept at assigned temperatures. It may be stated as follows: If E^c is the electromotive force round a circuit formed of the metals A, G, E^c that round a circuit formed of the metals J5, 0, then E^ic — Ebc is the electromotive force acting round the circuit formed of the T. E. 32 498 THERMOELECTRIC CURRENTS. [CH. XIV metals A and B ; all these circuits being supposed to work between the same limits of temperature. 271. Thermoelectric Diagrams. The Thermo- electric line for any metal (A) is a curve such that the ordinate represents the thermoelectric power of a circuit of that metal and some standard metal (usually lead) at a temperature represented by the abscissa. The ordinate is taken positive when for a small difference of temperature the current flows from load to the metal A across the hot junction. It follows from Art. 270, that if the curves a and ^ represent the thermoelectric lines for two metals A and B, then the thermoelectric power of a circuit made of the metals A and B at an absolute temperature repre- sented by ON will be represented by RS, and the electromotive force round a circuit formed of the two R ' l_a 1 ^^;^ S G — y3 ^ F N M Fig. 131. metals A and B when the temperature of the cold junction is represented by OL, that of the hot junction by OM, will be represented by the area EFOH. Let us now consider a circuit of the two metals A and B with the junctions at the absolute temperatures OXi, 0X2, where OL^ and OL^ are nearly equal. Then 271] THERMOELECTRIC CURRENTS. 499 the electromotive force round the circuit (i.e. the work done when unit of electrical charge passes round the circuit) is represented by the area EFGH. Consider now the thermal effects in the circuit. We have Peltier eifects Fig. 132. at the j unctions ; suppose that the mechanical equivalent of the heat absorbed at the hot junction when unit of electricity crosses it is represented by the area Pi, let the mechanical equivalent of the heat liberated at the cold junction be represented by the area P^. There are also the Thomson effects in the unequally heated metals; suppose that the mechanical equivalent of the heat liberated when unit of electricity flows through the metal A from the hot to the cold junction is represented by the area iTj, and that the mechanical equivalent of the heat liberated when unit of electricity flows through B from the hot to the cold junction is represented by 500 THERMOELECTRIC CURRENTS. [CH. XIV the area K^. Then by the First Law of Thermodynamics, we have sivesi EFGH = P,-P,-hK,-K^ (1). The Second Law of Thermodynamics may be expressed in the form that if -H" be the amount of heat absorbed in any reversible engine at the absolute temperature t, then z In our circuit the two junctions are at nearly the same temperature, and we may suppose that the temperature at which the absorption of heat corresponding to the Thomson effect takes place is the mean of the tempera- tures of the junctions, i.e. ^(OXi + OL^). Hence by the Second Law of Thermodynamics, we have - -^ - ^ 4- - ^"^Z^^^ (^\ ''~ OL, OL,\{Oh-\-OQ ^ ^• Hence from (1) and (2) we get area EFGH = \ |^^ 4- ^1 {OL, - 0L,\ or since OL^ is very nearly equal to OL^ and therefore Pj is very nearly equal to Pg, this gives approximately area EFGH = ^(0L,- OL,). But when OL^ is very nearly equal to OL^, the area EFGH^GH{OL,-OL,), so that P, = GH,OL„ 271] THERMOELECTRIC CURRENTS. 501 SO that Pi is represented by the area GHUV. Now Pj is the Peltier effect at the temperature represented by OXj, hence we see that at any temperature Peltier effect = (thermoelectric power) (absolute temperature), or P=Q^, when t is the absolute temperature. By the definition of Art. 270 we see that if cti is the specific heat of electricity for the metal ^1, 0-3 that for B^, then But by (1) area EFGH =P,-P, + K, - K, , and Pi = area GHVU, P, = area FEST. Hence K^ -K, = area SEHV - area TFG U = (tan ^1 — tan 6^ OL^ x L^L^, where 6^, 62 are the angles which the tangents at E and F to the thermoelectric lines for A and B make with the axis along which temperature is measured. Hence cTi — a2= (tan ^1 — tan ^2) OL^ (3). When the temperature interval L^L^ is finite the areas UGHV and FESL will still represent the Peltier effects at the junction, the area TFGU the heat absorbed when unit of electricity flows along the metal B from a place where the temperature is OXg to one where it is OXj. 502 THERMOELECTRIC CURRENTS. [CH. XIV The preceding results are independent of any assump- tion as to the shape of the thermoelectric lines. The results of the experiments made by Professor Tait and others show that over a considerable range of tempera- tures these lines are straight for most metals and alloys, while Le Roux has shown that the 'specific heat of electricity ' for lead is excessively small. Let us assume that it is zero and suppose that the diagram represents the thermoelectric lines of metals with respect to lead : then since these lines are straight, 6 is constant for any metal and o-g vanishes when it refers to lead, the value of o- the ' specific heat of electricity ' in the metal is by (3) given by the equation 2)(^-ro), 271] THERMOELECTRIC CURRENTS. 503 where 1\ is the neutral temperature for the two metals and is given by the equation r„ = ti tan ^1 — 4 tan 0, f J ^ tan ^1 — tan $2 The electromotive force round a circuit formed of these metals, the temperatures of the hot and cold junc- tions being t^, t^ respectively, is equal to '"^^ Qdt = (tan e, - tan 6^) {T, - T,) {\ {T, + T,) - T,). This vanishes when the mean of the temperatures of the junctions is equal to the neutral temperature. If the temperature of one junction is kept constant the electromotive force has a maximum or minimum value when the other junction is at the neutral temperature. In Fig. 133 the thermoelectric lines for a number of metals are given. The figure is taken from a paper by Noll, Wiedemann s Annalen, vol. 53, p. 874. The abscissae represent temperatures each division being 50° C, the ordinates represent the E.M.F. for a temperature difference of 1° C. each division representing 2 '5 microvolts. To find the E.M.F. round a circuit whose junctions are at ti and ^2 degrees we multiply the ordinate for K^i + ^a) degrees by (t^ — ti). X, \ v<^ ^ ^ y- < ""•••^ -^^ >< ^ **^^ •%., '*«»^ >< ^ t "Af "Cu^ ^ Cu PS ^ ?.) 1 Bra?? -■■*^^v"' __CUJ6 ) Rrass^ _^ ^ .,^><:::>,^^^^-<. Al **-*^^ /^ ^^ >^, 1 '"^-^ f^ ^^ ^2:^ 4:^ \ ^'^ ^ =:^ \ \ \ ^^^^C^ ^ \ ^s ^ N. \ \ \ V %., X \ \ \ \ <^\ V ^ \ ^*N N \ i^ \ 250 200 150 100 50 50 Fig. 133. 100 150 200 250 INDEX. (The numbers refer to the pages.) Absolute measurement of a resistance 452, 456 of a current 460 Alternating currents, distribution of 411, 415 Ampere's law 321, 343 Angle, Solid 210 Anode 278 Axis of a magnet 190 Ballistic galvanometer 368 Boundary conditions for two dielectrics 121 for two magnetisable sub- stances 254 for two conductors carrying currents 317 Bunsen's cell 296 Capacity of a condenser 84 of a sphere 84 of two concentric spheres 85, 134 of two parallel plates 90, 127 of two coaxial cylinders 93, 134 specific inductive 113 Capacities, comparison of two 105 Cathode 278 Cavendish experiment 33 Charge of electricity 8 Charge, unit 12 Circular currents magnetic force due to 344 force between two 351 T. E. Clark's Cell 296, 461 Coefficients of capacity 43 of induction 43 of self-induction 353, 434, 440 of mutual induction 353, 437, 438 Condensers 84 in parallel 108 in cascade 109 parallel plate 90 Condenser in an alternating current circuit 428, 430 Conductors 9 Conjugate conductors 306 Coulomb's Law 36, 120 Couples between two magnets 197 on a current in a magnetic field 348 Currents electric 276 strength of 278 magnetic force due to 320, 326 distribution of steady 302, 311 distribution of alternating 411, 415 dielectric 473 Cylinder electric intensity due to 21 capacity of 93, 134 Daniell's Cell 294 Declination, magnetic 226 33 506 INDEX. Dielectric currents 473 Dielectric plane and an electrified point 162 — sphere in an electric field 165 Dimensions of electrical quantities 446 Dip 228 Discharge of Leyden jar 422 Dissipation function 310 Distribution of steady currents 302, 311 alternating currents 411, 415 currents in a conductor 393 currents due to an impulse 891 Diurnal variation 234 Doublet, electric field due to 151 Duperrey's lines 231 Dynamical system illustrating induction 384 Electrification by friction 1 positive and negative 2 by induction 4 Electric intensity 13 Electric potential 25 Electric screens 51 Electric images 138 Electric currents 276 Electroscope 5 Electrolysis 278, 280 Electrolyte, e. m. f. required to liberate ions of 298 Electromagnetic induction 374 Faraday's law of 380 Neumann's law of 380 screening 421 wave, plane 478 Electrometers 97 quadrant 98 Electromotive force of a cell 289 Ellipsoids in magnetic field 267 Energy in the electric field 37, 70, 120 of a shell in a magnetic field 213 in the magnetic field 263 due to a system of currents 357 Equipotential surface 26 Faraday's laws of Electrolysis 280 laws of electromagnetic induc- tion 380 tubes 67, 468, 480 tubes, tension in 73 tubes, pressure perpendicular to 75 Force lines of 60 tubes of 65 on an uncharged conductor 82 on an electrified system 53 between electrified bodies 12 on charged conductor 56 between bodies in a dielectric 127 on a dielectric 135 between magnets 185 due to a magnet 193, 195 between two small magnets 201 on a shell in a magnetic field 214 on a current in a magnetic field 347 Galvanometer tangent 360 sine 366 ballistic 368 Desprez D'Arsonval 367 resistance of 373 Gauss's proof of law of force be- tween poles 204 Gauss's theorem 14 Grove's Cell 296 Heat produced by a current 286, 308 Hertz's experiments 485 Hysteresis 250 Impulse, distribution of currents induced by 391 Impedance 400 Induction magnetic 240 electromagnetic 374 total normal electric 13 Insulators 9 INDEX. ;o7 Intensity electric 13 of magnetization 191 Inversion 168 Ions 279 Isoclinic lines 229 Isogonic lines 229 Jar, Leyden 107 Joule's Law 287 Kirchhoff's Laws 302 Law of force between electrified bodies 12, 31 Lenz's Law 431 Leyden Jar 107 in parallel 108 in cascade 109 discharge of 422 Light, Maxwell's Theory of 478 et seq. Lines of force 25, 60 refraction of 124 Lorenz's Method 456 Magnets 184 Magnet pole of 189 axis of 190 moment of 190, 208 potential due to 191 resolution of 192 force due to 193, 195 couple on 197 Magnetic force 188 disturbances 236 potential 188 shell 209 shell, force due to 218 shell, force acting on 214 shielding 259 induction 240 induction, tubes of 242 permeability 245 retentiveness 250 susceptibility 245 declination 226 dip 228 field, energy in 263 Magnetic field due to current 320 due to two straight currents 331 due to circular current 344 Magnets, action between two small 197 Magnetization, intensity of 191 Magnetized sphere, field due to 219 Maxwell's Theory 473, 478 Model illustrating magnetic in- duction 384 Moment of magnet, determination of 208 Mutual induction, coefficient of 353 determination of 437 comparison of 438 Neumann's Law of electromagnetic induction 380 Neutral temperature 502 Ohm's Law 282 Ohm, determination of 452, 456 Parallel plate condenser 90 Peltier Effect 493 Periodic electromotive force 398 Permeability, magnetic 245 affected by temperature 249 Plane uniformly electrified 22 Plane and electrified point 140 Planes parallel, separated by dielectric 127 two parallel and electrified point 174 magnetic force due to currents in parallel 337 Polarization in a dielectric 118 of a battery 297 Pole, unit 187 Poles of a magnet 189 Potential electric 25 of charged sphere 27 of a magnet 191 Propagation of electromagnetic dis- turbance 474 108 INDEX. Ratio of units 461 Refraction of lines of force 124 Resistance electric 282 of conductors in series 283 of conductors in parallel 284 specific 286 measurement of 371 absolute 452, 456 Resolution of a magnet 192 Retentiveness, magnetic 250 Rotating circuit 401 Saturation, magnetic 248 Screening electric 51 electromagnetic 4^1 magnetic 259 Secondary circuit, effect of on apparent self-induction and re- sistance 388 Self induction coefficient of 353 coefficient of, of a solenoid 355 coefficient of, of two parallel circuits 356 determination of 434 comparison of 440 Shell, magnetic 209 Sine galvanometer 366 Specific inductive capacity 113 determination of 136 Specific resistance 286 Specific Heat of Electricity 496 Sphere electric intensity due to 20 potential due to 27 capacity of 84, 85 and an electrified point 142 in a uniform electric field 150 inversion of 169 Sphere magnetic field due to 219 in a uniform magnetic field 258 Spheres intersecting at right angles 155 in contact 176 Solenoid 340 Solid angle 210 Surface density 36 Susceptibility 245 Tangent galvanometer 360 Temperature, effect of, on magnetic permeability 249 Terrestrial magnetism 225 Thermoelectric currents 492 diagrams 498 Thomson Effect 496 Transformers 404 Tubes of electric force 65 Faraday 67, 468, 480 Faraday, tension in 73 Faraday, pressure perpendicu- lar to 75 of magnetic induction 242 Units electrostatic system 442 electromagnetic system 444 Variation in magnetic elements 233 diurnal 234 Voltaic cell 288 Wave Electromagnetic 478 Wheatstone's Bridge 303, 372 Work done when unit pole is taken round a circuit 323 CAMBRIDGE : PRINTED BY J. & C. F. CLAY. AT THE UNIVERSITY PRESS. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW RENEWED BOOKS ARE SUBJECT TO IMMEDIATE RECALL LIBRARY, UNIVERSITY OF CALIFORNIA, DAVIS Book Slip-10m-l,'63(D506884)458 2922^k Call Number: QC$18 Thomson, J.J. Elements of the 1897 "^^ Thoy'^OY) QC5/G T4- 292254