GIFT OF The Distribution of Current and the Variation of Resistance in Linear Conductors of Square and Rectangular Cross-Section when Carrying Alternating Currents of High Frequency BY HIRAM WHEELER EDWARDS A DISSERTATION IN PARTIAL SATISFACTION OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY SUBMITTED TO THE FACULTY OF THE COLLEGE OF NATURAL SCIENCES UNIVERSITY OF CALIFORNIA MAY i, 1911 The Distribution of Current and the Variation of Resistance in Linear Conductors of Square and Rectangular Cross-Section when Carrying Alternating Currents of High Frequency BY HIRAM WHEELER EDWARDS \\ A DISSERTATION IN PARTIAL SATISFACTION OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY SUBMITTED TO THE FACULTY OF THE COLLEGE OF NATURAL SCIENCES UNIVERSITY OF CALIFORNIA MAY i, 1911 [Reprinted from the PHYSICAL REVIEW, Vol. XXXIII. , No. 3; September ,;ri.j; ** j THE DISTRIBUTION OF CURRENT AND THE VARIATION OF RESISTANCE IN LINEAR CONDUCTORS OF SQUARE AND RECTANGULAR CROSS-SECTION WHEN CARRYING ALTERNATING CURRENTS OF HIGH FREQUENCY. BY HIRAM WHEELER EDWARDS. I. INTRODUCTION. HHE present paper undertakes the investigation of the virtual resist- * ance, inductance and current distribution of long, straight conduc- tors of square and rectangular cross-section, when carrying alternating cur- rents of high frequency. Experimental observations are given, which, within certain limits, corroborate the calculations for virtual resistance, but no attempt was made to measure the current intensity and inductance. Approximate formulas are derived by means of which the virtual resist- ance, the inductance and the current intensity may be calculated for particular cases. . v, The problem of determining the virtual resistance and inductance of long, straight, isolated conductors, when carrying alternating current is not a new one. In the oscillating current circuit of wireless telegraph, the increase of resistance due to the high frequencies of alternation is a very impotrant factor. The Tesla experiments illustrating the "skin- effect" show that in a number of cases the virtual resistance is many times the resistance offered to direct current. In the alternating-cur- rent railroads, the use of the rails for the return circuit is impracticable because of the increased resistance. 1 For long, straight, cylindrical wires, Maxwell 2 has developed an expres- sion for the virtual resistance and inductance. Rayleigh 3 has modified Maxwell's results by showing that the permeability of the material of the conductor, if greater than unity, causes a further increase in the resist- ance. Kelvin 4 solved the problem in a different manner, obtaining results similar, in final form, to those of Maxwell. He has given simpli- fied forms of expression for virtual resistance and inductance for the ^Standard Handbook, for El. Eng., Sec. 2, No. 122; Sec. n, No. 33. 2 Maxwell, E. & M., Vol. 2, third ed., p. 320. See also Gray, E. &. M. f Vol. 2, Pt. I, p. 325 ; and Thomson, Recent Researches in E. &. M., Chap. 4. 3 Rayleigh, Phil. Mag., Vol. 21, 1886, p. 381. < Kelvin, Math. & Phys. Papers, Vol. 3, p. 462. 244550 '1,5'.. HIRAM WHEELER EDWARDS. [VOL. XXXIII. particular cases of low and high frequency. In a recent paper, Nicholson 1 has published another solution of this problem. He considers first two parallel conductors carrying equal currents in opposite directions and then arrives at the case of the single wire by neglecting the effect of the second upon it. In a paper on the " Effective Resistance and Inductance of a Concentric Main," Russell 2 has given formulas for the virtual resistance and inductance of a concentric main with a solid inner con- ductor. The expression for the virtual resistance is conveniently arranged so that the resistance of either the core or the sheath may be calculated for any particular case. He shows how the formulas become simplified for the cases of direct current and alternating currents of low and high frequency. A consideration of the current density in the inner and outer conductors of the main is another feature of this paper. Mordey 3 has employed Kelvin's formulas in the numerical computation of the increase of resistance of a cylindrical conductor, when carrying alternating currents of commercial frequency, 80 to 133 complete cycles per second. Thus with a frequency of 80, the resistance of a copper conductor, 2.5 cm. in diameter, increases 17.5 per cent. The same per- centage increase is computated for a conductor of 2.24 cm. at 100 cycles per second, and also for a conductor of 1.036 cm. at 133 cycles per second. Fleming 4 has measured the ratio of the virtual to the direct current resistance of cylindrical wires using alternating currents of frequency somewhat less than half a million per second. For example, with a fre- quency of 440,000 using a copper wire, no. 14 standard wire gauge, he finds the ratio to be 5.46 by Russell's formula while his measured value is 5.90. He considers this agreement between the values, which differ by about eight per cent., as satisfactory. Values are given below for cylindrical wires which differ by less than one per cent., and for square and rectangular sectioned wires about four per cent. The experimental work on cylindrical wires, cited below, was performed with slightly damped oscillating currents, to determine whether the damping was great enough to make any appreciable difference between the results observed from damped currents and calculated by a formula derived upon the assumption that the currents were undamped. Fleming used undamped oscillations in obtaining his values. 2. EXPERIMENTAL METHOD. (a) Production of Currents of High Frequency. The two most common methods of producing alternating currents of high frequency are the 1 Nicholson, Phil. Mag., Vol. 17, 1909, p. 255. 2 Russell, Phil. Mag., Vol. 17, 1909, p. 524. 3 Mordey, Electrician, May 31, 1889, p. 94. 4 Fleming, Electrician, December 17, 1909, p. 38 i. 0.3-] THE DISTRIBUTION OF CURRENT. 186 i.e. singing-arc method and by the discharge of a condenser through an inductive resistance. Both methods were tried but greater success was obtained by the latter, so it was adopted in the experimental work herein recorded. The oscillations of current from a condenser are more or less damped, but the damping was reduced to such a magnitude, in the circuits here described, that the results did not appreciably differ from those which would have been obtained from using undamped currents. The scheme of connections is given diagramatically in Fig. I. T.C. is a large six-inch induction coil . I ts primary is connected to a source of direct-current supply. I is a Cunningham mercury-jet interrupter. It has a wide range of frequency of interruption and when in good working order can be depended upon to give a con- Fig. 1. stant root-mean-square, cur- rent in the secondary of the induction coil. C is a condenser the plates of which are separated by sheets of glass which can resist differences of potential of 30,000 volts or more. This condenser, when charged by the induction coil to the point of breaking down the spark-gap S, dis- charges through the variable inductance L. Possibly the greatest difficulty in maintaining the discharge of a con- denser with sufficient constancy of current, to enable one to use it for precise measurements, lies in the spark-gap used. A long series of ex- periments showed conclusively that the electrodes must be clean and bright, and separated by an unvarying distance. Zinc electrodes, which are suitable for some purposes, are soon covered by an oxide which causes the width of the gap to vary. Fine jets of mercury were finally selected as the electrodes. The jets were perpendicular to each other in a hori- zontal plane and at the point of crossing were about a millimeter apart and half a millimeter in diameter. Their surfaces remained always clean and bright and the distances between them seemed to be unvarying. With these arrangements oscillating currents were obtained, adjustable in frequency from twenty thousand to ten millions or more cycles per second, and sufficiently uniform for precise measurements. (b) Measurement of Intensity of Current. In measuring the root- mean-square intensity of high frequency currents, it was found that several precautions were necessary. An instrument was first made which consisted of two parallel bars, each seven centimeters long, one centimeter HIRAM WHEELER EDWARDS. [VOL. XXXIII. wide and three millimeters thick. The bars were connected by seven No. 40 B. & S. gauge high resistance wires, placed one centimeter apart. A nickel-iron thermo-couple was soldered to the middle point of each wire and was properly calibrated by direct current. The alternating current leads were connected to the ends of the bars from the same side. It was found that with a direct current flowing through the instrument each resistance wire conducted one seventh of the total current. An alternating current of frequency about 300,000 was sent through the instrument and readings were taken from the thermo-couples. It was found that the ratio of currents in the two extreme wires was about seven to nine, with the wire nearest the leads carrying the most current. This unequal distribution shows the necessity for caution in the constuction of an ammeter that is to be used for alternating currents of high frequency. To eliminate the danger of unequal current distribution an ammeter was made as is shown in Fig. 2. In place of the two parallel bars in the instrument described above, there were substituted two triangular blocks of copper, PI and PI, with their par- allel edges about three cen- timeters apart. The bases of the triangles measured 4 cm. and the altitudes 10 cm. The blocks were connected by fifteen no. 40 resistance wires, spaced about 2 mm. apart. On the center wire was soldered a thermo-couple T.C., which is connected to a sensitive D'Arsonval galvanometer G. Resistance wires as small as no. 40 were used so that one could be sure that the alternating current resistance, for frequencies up to 1,000,000, was not noticeably different from the direct-current resistance, and hence the instrument could be calibrated by direct current. For currents ranging between one and two amperes, fifteen resistance wires gives about the best sensitiveness. This form of ammeter may be used for a very wide range of current strength by selecting a proper number of resistance wires. In using the latter instrument to obtain data for this paper each reading of alternating current was calibrated by direct current within a few minutes after the alternating currents had passed through it. This eliminates the danger of incorrect readings caused by a change in room temperature. A wooden box protected the instrument from any air- drafts. Fig. 2. No. 3.] THE DISTRIBUTION OF CURRENT. ] 88 (c) Measurement of Frequency of Alternation. Fleming has devised an instrument for measuring the frequency of alternating currents which he calls a photographic spark counter. 1 Since it has been modified in some particulars it will be described here. There is first an enclosing box about 25 cm. square and 40 cm. high. See Fig. 3, which is a diagram showing the plan of the essential parts. A vertical shaft near the middle of one side has mounted on it a four-sided mirror, also a fly wheel to insure uniformity of rotation. A collimator tube projects from one side of the box in line with the mirror. The spark-gap is placed directly in front of the collimator tube. The light from the spark passes through a lens to the mirror and is there reflected at an angle of about ninety degrees to a slit n the side of the box. A suitable plateholder is held in front of the -F^ ^F--- slit and is so constructed that the photo- graphic plate may be raised or lowered by a string passing through the top of the holder. Ordinarily the plate was about twenty centi- meters from the mirror. For the higher fre- Fig. 3. quencies it was found necessary to extend the box so that the plate was about fifty centimeters from the mirror. The mirror was rotated by a motor driven by storage cells. The speed of rotation of the mirror could be made as much as one hundred revo- lutions per second, but was ordinarily rotated at a speed from sixty to eighty revolutions per second. A speed counter attached directly to the upper end of the mirror shaft, with the aid of a stop watch, gave a measure of the angular velocity of the mirror. The plate was raised or lowered by hand since it was not necessary to know the velocity with which it moved. The projection of the train of oscillations on the plate could easily be made less than the width of the plate by properly modifying the speed of the motor. The sigma brand of Lumiere plates was used with excellent results. The average distance between images of sparks was measured by means of calipers and standard scale. From the average distance between images, together with the dimensions of the apparatus involved, the frequency of the sparks was calculated. The accuracy of these measurements is probably within two or three per cent. In early trials with the apparatus, it was found that the upper limit of frequency of sparks easily counted was about one million per second. At frequencies higher than this, the successive images of individual sparks were fused together into bands, thirty to fifty or sixty in number, each band representing a complete oscillatory discharge of the condenser, and 1 Loc. cit. 189 HIRAM WHEELER EDWARDS. [VOL. XXXIII. the number of bands representing the number of discharges occurring in the well-known multiple manner. With lower frequencies the bands were drawn out into trains of oscillatory sparks. There were usually from twelve to eighteen complete oscillations in each train visible on the plate. Since the intensity of luminosity of the spark becomes gradually weaker, it is impossible to say how many oscillations there are in each train, for the sensitiveness of the plate limits the number of those that can be counted. The character of the photographs has been shown by a number of previous writers, among whom Trowbridge 1 has done extensive work. (d) The Differential Electric Thermometer. To compare the virtual resistance with the direct current resistance, a differential electric ther- mometer was used. This instrument was devised and used by Fleming. 2 As shown in Fig. I it consists of two glass tubes, T\ and T were plotted as shown in Fig. 4. The curve is drawn from the calculated values and the small circles indicate the experimental observations. Fig. 5 shows in a similar manner the nearness with which the experi- 100.000. 200,000. Fig. 5. 300,000. A/. No. 3.] THE DISTRIBUTION OF CURRENT. I 9 2 mental values check the calculated ones for the square-sectioned wire of which 2a = 0.070 cm. Replacing the square wires by wires of rectangular section, another series of observations was obtained, the results of which are shown in Fig. 6 and Fig. 7. The calculated values of R'/R for the wire of which i 00,000. 200,000. Fig. 6. 300,000. H. 2a = 0.039 cm - an d 2b = 0.0665 cm -> were used in drawing the curve of Fig. 6, the small circles indicating, as above, the experimental readings. 100,000. 2001000. Fig. 7. 300,000. N Similarly Fig. 7 was drawn for the wire of which 20, = 0.0485 cm. and 2b = 0.127 cm - With a wire as large as this last one, the formula does not give values which agree satisfactorily with the experimental values if the frequency is much greater than 150,000. Formulas (26) and (30) were used here in calculating the values for R'/R (see below). 3. MATHEMATICAL DEVELOPMENT. The conductor, rectangular in cross-section, is so long that end effects may be neglected. Other conductors are so far removed that their influence need not be considered. Referring to Fig. 8, a reference system is chosen, with the origin at the center of any cross-section, the X and Y axes parallel to the sides of the section, and the Z axis coinciding with the axis of the wire. Since the conductor is long, it may be assumed that no component of current flows parallel to the XY plane. The fol- lowing equation then holds: = ~to* + W ' (l) 193 HIRAM WHEELER EDWARDS, [VOL. XXXIII. ^b 2 d where, in agreement with Max- well's notation, w is the z compo- nent of current intensity, H is the z component of vector potential of magnetic induction and ^ is the permeability of the material of the conductor. The impressed electromotive in- tensity may be supposed equal at all points of the cross-section of the wire, assuming that the specific resistance is uniform. Let the electromotive force per unit length be Ee^ nt . Considering only instantaneous relations, the total electromotive in- tensity, inductive and non-inductive, in the wire is P = E - dH dt (2) Outside the wire the corresponding electromotive intensity is dH P=-.T- (3) If the specific resistance of the wire is p and the specific inductive capacity of the medium is K (4) Considering the inductive action only, dt' and eliminating w between equations (i) and (4) (5) (6) In the wire the current may be regarded as due to the conductivity alone, and if then 19 TT n9TT - o. (7) In the surrounding medium i/p may be neglected and putting h = n/V, No. 3-1 THE DISTRIBUTION OF CURRENT. 1 94 where F, the velocity of propagation of the disturbance, is \\V Kp, The general solution of (7) is H = A^+B^+CF+Dr* + Fe"^+Mr*^+Ne"^+Qr*^*, (9) where k' = fc/1/2, and if h' = h/l/2 the solution of (8) is _j_ jyjh'x^y _|_ Q' e -ih'^=-y ^ ( l ) By symmetry relations the value of H must remain unchanged if + x is written for x and + y for y in the equations (9) and (10). This shows at once that A = B, A' = B', C = D, C' = D' t F = M = N = Q and F' = M' = N' = Q' . Hence inside the wire H=2A cosh kx+2C cosh ky+2F (cosh k'x+y+ cosh k'xy), (n) and outside #=2^4' cosh ^wc+2C'cosh ihy+2F' (coshih'x+y + cosh ih'x y). (12) If the wire has a square cross-section instead of a rectangular section then x can be interchanged with y in (n) and (12) without affecting H. This necessitates the additional simplification that A = C and A' = C'. To evaluate the constants of (n) and (12), the continuity of the total electromotive intensity and magnetic field intensity, at the boundary, is used. If a and |8 are the x and y components of magnetic field intensity, then since the curl of the vector potential is the magnetic field intensity, at the point x = b, y = o, by the continuity of /3 the following relation is obtained from (n) and (12): kA sinh kb + 2k' F sinh k'b = ihA' sinh ihb + 2ih'F' sinh ih'b. (13) Similarly by the continuity of a at x = o, y = a, kC sinh ka + 2k' F sinh k'a = ihC' sinh iha + 2ih'F f sinh ih'a. (14) Also at the point x = b, y = a, for both a and ]S, sinh kb + k'F (sinh k'b + a + sinh jfe'& - a) = UL' sinh $* + ih'F'(sinh ih'b + a + sinh tfc'6 - a), kC sinh j^a + k'F (sinh k'b + a - sinh fc'6 - a) = tAC' sinh + ih'F'(&nhih'l+a - sinh ih'b -a). 195 HIRAM WHEELER EDWARDS. [VOL. XXXIII. Also by the continuity of the total electromotive intensity at x = b y = o and at x = o, y = a using equations (2) and (3), two more relations may be obtained: E 2inAcoshkb 4inFcoshk'b = 2inA'coshihb 2inF f coshih f b, (17) E 2inCcoshka4.inFcoshk'a = 2inC'coshiha 2inF'coshih'a. (18) From equations (13) to (18) the values of the constants may be found. The total current passing through any cross-section may be found by taking the line integral of the magnetic field intensity around the bound- ary. If 7 is the total current 7 = 4 I Pdy - 4 I adx. (19) Jo Jo Inserting the values of a and /3 in this equation and integrating gives the desired electromotive force equation, from which, expressions for the virtual resistance and inductance may be calculated. The determina- tions of the constants by equations (13) to (18) is so complicated as to be unmanageable. It is possible, however, to obtain approximate formu- las with mean value for the constants by assuming that A = C = F in equation (n) and neglecting the effect of the medium. The assumption that A = C means that the vector potential along the x axis is the same as along the y axis for equal values of the argument. This is true for a square sectioned wire but when considering a rectangular sectioned con- ductor it is only approximately true, the degree of approximation depend- ing upon the difference between the two dimensions. The justification for putting A F is based upon the substantiation of the results as calculated for particular cases from experimental observations, The values of R'/R for four particular cases, calculated by a formula based upon this assumption are given in Figs. 58. Since the observed values agree up to frequencies of 150,000, w r ith these calculated values, it must be that up to this limit F is nearly equal to A . For the two rectangular sectioned wires this limit is reached at frequencies of 150,000 and then the formulas fail, then the differences between the observed and calcu- lated values increases as the frequency increases. For the larger square sectioned wire the values agree within five per cent, up to a frequency of 150,000, and then the differences increase to about ten per cent, at a frequency of 265,000, and at 312,000 the values are again in close agree- ment. The differences for the smaller square sectioned wire are not greater than three per cent, for the range tested, namely up to a fre- quency of 257,000. No. 3-1 THE DISTRIBUTION OF CURRENT. 196 To obtain these approximate expressions equation (7) will be used, but put into more convenient form. Introducing the non-inductive part of the electromotive intensity, which may be called d\I//dz, into equation (4) and neglecting the effect of the medium, gives dt dH pW= -dz--V' (20) Eliminating w between this equation and equation (i) d*H d*H i idf\ VT + irir - k 2 1 H - - - ] = o. (21) dx 2 dy 2 \ n dzJ i d\L> Since- is independent of x or y, (21) may be written Yl uZ (22) The general solution of this equation is similar to the solution of (7) and has eight undetermined coefficients. This number may be reduced to three by the symmetry relations as is shown above. Assuming now that these three constants are all equal leads to the following solution: H = - , +2A (cosh kx+cosh ky+cosh k'x+y+caah k'x-y). (23) fl oZ The value of the single undetermined coefficient may be found by integrating the expression for current intensity over the surface of the cross-section. If 7 is the total current, 7 = 4 I I wdydx. Jo Jo Using the value of w as determined by equations (i) and (23), with the aid of the above integral, the value of A may be shown to be 2(ka sinh kb-}-kb sinh ka-\-2 cosh k'b+a 2 cosh k'ba) If at the point x = b, y = o, H is put equal to some constant, say L, times the current and if / be the length of the conductor considered, d\l/ multiplying equation (23) by tin and putting / = E, the electro- motive force, gives E = Lliny cosh kb+2 cosh k'b+i ka sinh kb+kb sinh ka+2 cosh k'b+a2 cosh k'ba (25/ 197 HIRAM WHEELER EDWARDS. [VOL. XXXIII. If the hyperbolic functions are expanded in series of powers of k and the numerator divided by the denominator, the following expression is obtained : \ (26) 45P Z _ 7rVJV 4 (-i72fr 8 + io7ofr 6 a 2 +28o76V-34o6 2 a 6 - 14175'P 4 l"*" J where R = Ip/^ab (the resistance of a length / for direct current) and N = n/2-ir, the number of alternations per second. Writing R' for the virtual resistance, gives R' _ _ ~R = 45P 2 (27) If the wire has a square cross-section the expression for the virtual resistance may be found by putting b = a in (26) R' , , R - 45P^" I4I75 -P 4 Formulas (26) and (27) may be used in calculating R'/R for particu- lar cases where the dimensions and frequency are of a magnitude which will give a value of 0.5 or less to the second term of the series. If the frequency and dimensions are large then the series is not rapidly con- vergent. The calculation of more terms of the series is troublesome. To obtain a complete expression the fraction in (25) may be expressed in exponential functions and then separated into real and imaginary parts. The real coefficient of 7 is then the desired expression for R'. The device used in making the necessary transformation is embodied in the following relations: / I27TU72 = V2i \ * n where 5=J 2 -^. ( 2 8) Hence each hyperbolic function may be expressed in somewhat the following manner: cosh kb = %(e kb + e~ 7cb ) = cos bS cosh bS + * sin bS sinh bS. No. 3.] THE DISTRIBUTION OF CURRENT. . 198 Making these substitutions in (25) and then rationalizing the denomina- tor of the fraction, gives Olid + iA) + i(iCi - A l D l ) . E = Lhnj + Hmrmr - 2 nrir~ ~ 2 9) where ^4i = cos 65 cosh bS + 2 cos cosh _- + i , 1/2 1/2 1 = sin bS sinh 65+2 sin sinh , 1/2 1/2 Ci = 05 cos 65 sinh 65 aS sin 65 cosh 65 + 65 cos a5 sinh aS 65 sin aS cosh a5+2 cos = S cosh -52 cos = 5 sinh = 5, 1/2 1/2 1/2 F2 Di = aS cos 65 sinh 65 + a5 sin 65 cosh 65 + 65 cos aS sinh a5 + 65 sin a5 cosh a5+ 2 sin =5 sinh -5 2 sin - 5 sinh :S. 1/2 1/2 1/2 1/2 Hence : _ , . ~R = ~T~ ~W+Df~ For the wire of square cross-section this expression becomes simplified and R' _ 4>jrna 2 A 2 D< 2 - 2 C 2 , , j? ~ * r-2 j_ n 2 ' ^ ' J< P ^ -r -^2^ where the change in subscripts indicate the changes in the constants which are obtained by putting 6 = a. It is shown above that these expressions can be used to calculate the virtual resistance within certain limits. Before leaving the mathematical side of the investigation an expres- sion for finding the intensity of current at any point of the cross-section of the conductor is to be developed. For present purposes a study of the distribution of current in a wire of square cross-section is just as instructive as that in a wire of rectangular cross-section. Using equation (i) for current intensity and (23) for vector potential the following expression, for the wire of square section, is obtained: J 99 HIRAM WHEELER EDWARDS. [VOL. XXXIII. The exponential functions of this expression may be separated into real and imaginay parts by a method of procedure similar to the method used above. us- -r - 3 ' (33 where AS = I sin Sx sinh Sx-{- sin ^S-y sinh .Sy+sin S sinh 5 1/2 V 2 + sin S sinh - V2 1/2 x ~\~ y x ~\~ y BS = cos Sx cosh Sx + cos .Sv cosh S'y + cos -=- S cosh ^ S 1/2 1/2 + cos S cosh 5, V2 V2 Cs = cos 1/2 a5 cosh 1/2 a.S+^-S' cosh aS sinh aS aS sin a,S cosh a5 2, Dz = sin 1/2 a^ 1 sinh 1/2 #5 + aS cos aS sinh a5 + aS sin a5 cosh a5. The equation (33) expressing current intensity at any point in the cross-section of the wire is like the general electromotive force equation in that it is a complex quantity. The form must be complex because part of the current is caused by the inductive action and the other part by the non-inductive action. If equation (33) were multiplied by p, the specific resistance, the resulting equation would be an electromotive force equation, which is in general complex. To obtain values of w at points in the cross-section of a particular wire under given conditions, the real and imaginary parts must be calculated separately, and then added by vector methods, or analyti- cally by finding the square root of the sum of their squares. 4. VARIATION OF CURRENT INTENSITY. Another purpose of this paper is to show the variation of current intensity or density of flow, over the sectional area of a long, straight conductor of square cross-section, carrying current of high frequency. The mathematical development leads to a computation of current intensity at any point of the section. Since no experimental method of measuring this quantity has thus far been devised, the equations derived from theo- retical considerations will be relied upon entirely. The justification for this course lies in the satisfactory agreement between experimental and theoretical results, already described in another part of this paper. It appears that the experimental difficulties in the measurement of current No. 3.] THE DISTRIBUTION OF CURRENT. 2OO intensity would be considerable. If the wire be separated into filaments, like the fine, parallel wires of a flexible cable, the conditions which cause the uneven distribution of current have been modified to such an extent that the measurement of current in each filament would be of little value in solving the problems proposed. Equation (33) will be used in the calculations which follow. A wire or bar, two centimeters square, has been selected for illustration. To simplify the calculation let the frequency be 4OO/7T 2 . This makes 5" of equation (33) unity. Since the wire is large in cross-section, even with low frequency the variation in current intensity will be similar to the variation in a smaller wire with higher frequency. The similarity may be seen from an inspection of formula (32). It will be noticed that wherever V N appears in the equation as a factor, a dimensional length Xj yora appears also as a factor. N appears elsewhere only in K 2 . If V Nd (where d is a dimension) is put equal to some constant, then I/ AT" and d may be varied individually, so long as their product remains constant, and the resulting value for w will be proportional to the frequency. Consider the square section as a special case of the rectangular section of Fig. 8 in which b equals a. Fig. 9 shows graphically the variation of current intensity along the three principal lines of the particular square selected. Curve A represents the variation from the origin along the line y = o to x = =*= a. Curve B shows the variation from the middle of any side on either axis along the edge to the vertex. This curve represents the symmetrical variation of current along eight different lines of the square. In curve C the variation of current intensity along a diagonal from the origin to any one of the four vertices is shown. Another graphical representation of the density of current flow over the section of the same square wire, at the same frequency of alternation, is made in Fig. 10. Here contour lines are drawn for one quadrant of the section, the difference between consecutive lines being 0.005 C.G.S. unit of current. The line nearest the origin indicates an intensity of 0.215 C.G.S. unit. The current is fairly uniform over the central region, but proceeding from the origin along a diagonal to a vertex the lines are drawn more closely together. It might be supposed at first sight that for the case of very high frequency the current would be localized entirely at the corners. That such is not the case may be seen by inspection of formula (33). Putting x and y both equal to zero, the numerator still contains N, which, although small in comparison with the denominator, gives w a value different from zero. 2OI HIRAM WHEELER EDWARDS. [VOL. XXXIII. In any electromagnetic phenomenon the law of the conservation of energy is valid. In circuits such as described above the energy is mani- Fig. 9. Fig. 10. fested in two ways, namely, in the production of heat and in the produc- tion of the magnetic field. There would be a minimum of heat produced if the current were distributed uniformly over the cross-section. The energy of the magnetic field would be a minimum if the current were entirely on the surface of the wire. Since both of these conditions can- not be satisfied simultaneously, a compromise is established which makes the total energy a minimum. This compromise is shown for a particu- lar case in Fig. n. To show the variation in current intensity for different frequencies, also to show that the variation in a small wire with high frequency is similar to the variation in a large wire with low frequency, the values of w for a wire having 2a = 0.070 cm., with a frequency of 101,320, have been calculated along the line corresponding to curve A of Fig. 9 and are shown in Fig. n. The increase in intensity from the smallest to the greatest value in the smaller wire is greater than in the larger wire, but the characteristics of the two curves are the same. A value of the frequency could be found that would make the increase the same in both wires. The effect of increasing the frequency of alternation of cur- rent in the same wire of square section is illustrated by the two curves of Fig. n. Both curves are for the line y = o from the origin to x = a. The length of one side of the section is 20, = 0.070 cm. Curve A shows the variation for a frequency of 101,320, and curve B for a frequency of No. 3.] THE DISTRIBUTION OF CURRENT. 1,000,000. The wire is supposed to carry the same total current in both cases, ten amperes. The effect of increasing the frequency is shown in a crowding of the current toward the boundaries of the wire. SUMMARY. 1. Alternating currents have been produced by the discharge of a condenser through an inductive resistance, with sufficient uniformity of current to render possible their use in precise measurements. 2. An ammeter is described which is suitable for the measurement of the intensity of alternating currents. 3. The frequency of alternating currents has been measured by a photographic spark counter for any frequency up to three quarters of a million per second. 4. By checking Maxwell's formula for the virtual resistance of cylin- drical wires, it is shown that the damping of the oscillating currents used is small enough to neglect. 5. The ratio of the virtual to the direct current resistance for copper wires of square and rectangular cross-section was measured by the use of a differential thermometer, and these observed values were checked by approximate formulas within certain limits. For the two square cross-sectioned wires tested this limit was for frequencies above one hundred and fifty thousand and for the two rectangular cross-sectioned wires the limit was about one hundred and fifty thousand. 6. Formulas are developed by certain assumptions in the general theory, which give approximate methods of calculating the ratio of the virtual to the direct current resistance. The validity of these assump- tions is based upon experimental results. 7. In the development of the expression from which the ratio of resistances is calculated, there occurs an expression for current intensity at any point of the cross-section of the conductor. This expression is used to calculate, for some particular cases, the distribution of current intensity within the cross-section. UNIVERSITY OF CALIFORNIA, May i, 1911. THE PHYSICAL REVIEW A JOURNAL OF EXPERIMENTAL AN