THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
DAVIS
GIFT OF
PROFESSOR H.B. WALKER
PHYSICAL
MEASUREMENTS
DUFF and EWELL
BLAKISTON'S SCIENCE SERIES
PHYSICAL
MEASUREMENTS
BY
A. WILMER DUFF
PROFESSOR OF PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE
AND
ARTHUR W. EWELL
PROFESSOR OF PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE
SECOND EDITION, REVISED AND ENLARGED
WITH 78 ILLUSTRATIONS
PHILADELPHIA
P. BLAKISTON'S SON & CO
1012 WALNUT STREET
1910
LIBRARY
(JNfVFRSITY nr rmtr/\nutn
COPYRIGHT, 1910, BY P. BLAKISTON'S SON & Co.
Printed by
The Maple Press
York, Pa.
PREFACE.
Our intention in writing this book was not to give an
account of physical laboratory methods in general, but to
describe a limited number of carefully chosen exercises such
as we have found in our experience to be suitable for the
laboratory work of students who have had a fair course
in General College Physics.
The descriptions of the exercises will usually fit apparatus
and conditions of considerable diversity, but many practical
details have been included where experience has shown that
they are necessary. Other instructors who may adopt the
book will probably find some of the exercises unsuited to
their classes, but the list is sufficiently extensive to afford
a considerable variety of selection.
The descriptions of apparatus are intended to be read by
the student with the apparatus before him. Hence elaborate
illustrations have been thought unnecessary. For an ex-
tended account of certain special topics, such as the theory
of the balance and the construction of galvanometers,
references to other works have been given.
Usually several text-books and special treatises have
been referred to at the beginning of the account of an experi-
ment. It is assumed that each student will have one of the
text-books and that some of the special works will be found in
the reference room of the laboratory. While the reference
is generally to the latest edition (at the present date, 1910),
those who have different editions will have no difficulty in
finding the passages referred to. Each instructor who uses
v
VI PREFACE
the book will exercise his discretion as to what preliminary
reading will be required and will issue the necessary instruc-
tions to his class.
We are indebted to Dr. Albert W. Hull for assistance in
reading the page proof. Many of the tables have been taken
from Ewell's Physical Chemistry.
CONTENTS.
PAGE
GENERAL INTRODUCTION i
i. Purpose of Course. 2. General Directions. 3. Reports.
4. Errors. 5. Errors of Observation. 6. Possible Error
of a Calculated Result. 7. General Method for the Possible
Error of a Result. 8. Some General Notes on Errors. 9.
Probable Error of a Mean. 10. Limits to Calculations, n.
Notation of Large and Small quantities. 12. Plotting of
Curves.
MECHANICS 13
13. The Use of a Vernier. 14. Vernier Caliper. 15. Microm-
eter Caliper. 16. Micrometer Microscope. 17. Com-
parator. 1 8. Spherometer. 19. Dividing Engine. 20.
Cathetometer. 21. Barometer. 22. The Balance. 23. Ad-
justment of Telescope and Scale. 24. Time Determination.
I. To Make and Calibrate a Scale.
II. Errors of Weights.
III. Volume, Mass, and Density of a Regular Solid.
IV. Mohr-Westphal Specific Gravity Balance.
V. Density by the Volumenometer.
VI. Density of Gases.
VII. Acceleration of Gravity by Pendulum.
VIII. Coefficient of Friction.
IX. Hooke's Law and Young's Modulus.
X. Rigidity (or Shear Modulus).
XL Viscosity.
XII. Surface Tension.
HEAT 63
25. Radiation Correction in Calorimetry. 26. The Beck-
mann Thermometer.
XIII. Thermometer Testing.
XIV. Temperature Coefficient of Expansion.
vii
VI11 CONTENTS.
PAGE
XV. Coefficient of Apparent Expansion of a Liquid.
XVI. Coefficient of Increase of Pressure of Air.
XVII. Pressure of Saturated Water Vapor.
XVIII. Hygrometry.
XIX. Specific Heat by Method of Mixture.
XX. Ratio of Specific Heats of Gases.
XXI. Latent Heat of Fusion.
XXII. Latent Heat of Vaporization.
XXIII. Latent Heat of Vaporization. Continuous-flow
method.
XXIV. Thermal Conductivity.
XXV. The Mechanical Equivalent of Heat.
XXVI. The Melting-point of an Alloy.
XXVII. Heat Value of a Solid.
XXVIII. Heat Value of a Gas or Liquid.
XXIX. Pyrometry.
SOUND 119
XXX. The Velocity of Sound.
XXXI. The Velocity of Sound by Kundt's Method.
LIGHT 124
27. Monochromatic Light. 28. Rule of Signs for Spherical
Mirrors and Lenses.
XXXII. Photometry.
XXXIII. Spectrometer Measurements.
XXXIV. Radius of Curvature.
XXXV. Focal Length of a Lens.
XXXVI. Lens Combinations.
XXXVII. Magnifying Power of a Telescope.
XXXVIII. Resolving Power of Optical Instruments.
XXXIX. Wave-length of Light by Diffraction Grating.
XL. Interferometer.
XLI. Rotation of Plane of Polarization.
ELECTRICITY AND MAGNETISM 153
29. Resistance Boxes. 30. Forms of Wheatstone's Bridge.
31. Galvanometers. 32. Correction for Damping of a Bal-
listic Galvanometer. 33. Galvanometer Shunts. 34.
Standard Cells. 35. Device for Getting a Small E. M. F.
36. Double Commutator. 37. Relations between Electrical
Units .
CONTENTS.
IX
XLII. Horizontal Component of the Earth's Magnetic
Field.
XLIII. Magnetic Inclination or Dip.
XLIV. Measurement of Resistance by Wheatstone's
Bridge.
XLV. Galvanometer Resistance by Shunt Method.
XLVI. Galvanometer Resistance by Thomson's Method
XLVII. Measurement of High Resistance (i).
XLVIII. Measurement of High Resistance (2).
XLIX. Measurement of Low Resistance (i).
L. Measurement of Low Resistance (2).
LI. Measurement of Low Resistance (3).
LI I. Comparison of Resistances by the Carey-Foster
Method.
LIII. Battery Resistance by Mance's Method.
LIV. Temperature Coefficient of Resistance.
LV. Specific Resistance of an Electrolyte.
LVI. Comparison of E. M. F.'s by High Resistance
Method.
LVII. Comparison of E. M. F.'s and Measurement of
Battery Resistance by Condenser Method.
LVIII. Calibration of Voltmeter.
LIX. Calibration of Ammeter.
LX. Comparison of Capacities of Condensers.
LXI. Absolute Determination of Capacity.
LXII. Coefficients of Self-induction and of Mutual Induc-
tion.
LXIII. Strength of a Magnetic Field by a Bismuth Spiral.
LXIV. Study of a Ballistic Galvanometer.
LXV. Magnetic Permeability.
LXVI. Magnetic Hysteresis,
j VXTTT / (a) Mechanical Equivalent of Heat.
\ (b) Horizontal Intensity of Earth's Magnetism.
LXVIII. Thermoelectric Currents.
LXIX. Elementary Study of Resistance, Self-induction,
and Capacity.
LXX. Self-induction, Mutual Induction, and Capacity,
Alternating Currents.
LXXI. Dielectric Constant of Liquids.
LXXII. Electric Waves on Wires.
TABLES
I. Four-Place Logarithms.
2 35
CONTENTS.
II. Trigmometrical Functions.
III. Reduction to Infinitely Small Arc.
IV. Barometer Corrections.
V. Density and Specific Volume of Water.
VI. Density of Gases.
VII. Density, Specific Heat, and Coefficient of Expansion of
Metals.
VIII. Density, Specific Heat, and Coefficient of Expansion of
Miscellaneous Substances.
XL Elastic Moduli.
X. Surface Tension.
XI. Coefficient of Viscosity.
XII. Specific Heats of Gases.
XIII. Vapor Pressure of Water.
XIV. Boiling-point of Water.
XV. Wet and Dry Bulb Hygrometer.
XVI. Vapor Pressure of Mercury.
XVII. Melting-points of Metals.
XVIII. Wave-lengths of Light.
XIX. Refractive Indices.
XX. Specific Rotatory Power.
XXL Photometric Table.
XXII. Specific Resistance and Temperature Coefficient of
Metals.
XXIII. Specific Resistance and Temperature Coefficient of
Solutions.
XXIV. Dielectric Constants.
INDEXT 255
PHYSICAL MEASUREMENTS.
INTRODUCTION.
1. Purpose of Course.
Intelligent work requires a clear perception of the end
in view. It is important, therefore, to remember that the
purpose of a course in Laboratory Physics is not only the
attainment, by personal experimentation, of a more definite
knowledge of the facts and principles of physics and an
acquaintance with the use of measuring instruments and
methods, but also the acquisition of a scientific habit of
accuracy and carefulness in observing and examining phe-
nomena and drawing conclusions therefrom.
2. General Directions.
Much time in the laboratory will be wasted unless some
preparation be made before coming to the laboratory. The
purpose and general method of the measurement to be made
should be examined with the aid of the text-book and some
of the references preceding the directions. This may
usually be done in a few minutes at home, whereas it might
require an hour or more in a laboratory where a number
of people are moving around.
The readings made in the laboratory should always be
recorded in a firmly bound book reserved for this purpose
only, and never on loose slips of paper or in a book that may
become dog-eared and untidy. When, for convenience or
of necessity, two work together at an experiment, each
should keep his own notes of the measurements made, and,
i
2 INTRODUCTION.
whenever possible, each should make a separate set of read-
ings for himself, and these should be as independent as
possible.
No operation should be performed or measurement
made unless the purpose and meaning of it are understood ;
otherwise it may be made imperfectly or some essential
part of it may be overlooked.
3. Reports.
An essential part of the work is a written report on each
experiment completed. This should be handed in within a
week after the work is finished. In preparing the report
the writer has to make clear to himself the purpose and
bearing of each part of the work and examine critically the
value and accuracy of the final result. This exercise is as
valuable as the experimental work itself. The report
should be as brief as possible, consistently with giving the
following information :
The purpose of the experiment (including the definition
of the leading terms, such as coefficient of friction, mechani-
cal equivalent of heat, etc.) ;
A brief statement of the method used ;
A statement (tabulated if possible) of the observations
and readings made :
An outline of the calculation of the final result (omit-
ting the details of the numerical work) ;
A criticism of the reliability of the result ;
Brief answers to the questions appended to the directions.
4. Errors.
A perfectly accurate experimental result is impossible;
but some estimate can usually be formed as to the magni-
tude of the possible error and this is frequently of the
greatest value. An experimental result of unknown reli-
ability is often of very little value. Hence an estimate of
ERRORS OF OBSERVATION. 3
the accuracy of a measurement is very desirable in an
account of the work.
Inaccuracy may arise from several different causes
(i) errors of observation, due to the inherent limitations
of the observer's powers of observing and judging; (2)
instrumental errors, arising from imperfections in the work
of the instrument maker in constructing and subdividing
the scale used by the observer; (3) mistakes, such as the
mistaking of an 8 for a 3 on a scale; (4) systematic errors
due to faultiness in the general method employed.
Instrumental errors may be decreased by using more
accurate instruments or by calibrating the scales of the instru-
ments used, that is, ascertaining and allowing for the errors
in their graduation. This is frequently a difficult operation
and unsuited for an elementary course. We shall, therefore,
usually assume that the accuracy of the instruments is
such that the instrumental errors are less than the errors of
observation.
Mistakes in reading can be eliminated by care and repeti-
tion. Systematic errors are apt to arise when some indirect
method of arriving at a result is adopted, a direct method
being difficult or impossible. For example, the length of
a wave of light cannot be measured directly and a method
depending on diffraction or interference is usually employed
(Exp. XXXIX). A careful study of the method used will
often enable us to eliminate such errors by improving
the details of the method, or, where this cannot be done,
some estimate of the uneliminated errors can often be formed.
5. Errors of Observation.
Different methods of estimating the magnitude of errors of
observation may be employed, the choice depending on the
nature of the measurements. In many cases the quantity
can be measured several times and the mean taken, it
being probably more accurate than a single observation.
In other cases circumstances do not permit repetition and a
4 INTRODUCTION.
single observation must suffice. In either case the observer
can, from the circumstances of the case, say with a high de-
gree of probability that the error cannot be greater than a
certain magnitude. This we shall call the "possible error"
of the measurement. It does not strictly mean the greatest
possible error, since a greater error might be theoretically
possible but very improbable.
(a) When Only a Single Observation is Made. For example,
a liquid, the temperature of which is varying slowly, is kept
well stirred and the temperature is observed by means of a
thermometer graduated to degrees. The temperatur at a
certain time is noted as being between 36 and 37 and the
observer, estimating to o . i of a division, records the tem-
perature as 36.3; but he does not trust his estimate closer
than o . i ; that is, he considers that the real temperature may
be as high as 36 . 4 or as low as 36 . 2. He therefore states
the temperature as 36.3 with a possible error of 0.1, or
36.3o.i. The actual error may, of course, be less than
o. i; the latter is only a reasonable estimate of the limit of
error of observation.
(b) When Several Different Observations of a Quantity
are Made. The mean of a number of observations of a
quantity is more trustworthy than a single reading, for
observations that are too large are likely to counterbalance
others that are too small. Greater confidence can be
placed in the mean when the separate readings differ but
little from the mean than when they differ greatly. The
average of the differences between the mean and the
separate readings is called the mean deviation. It can be
shown (as indicated in 9) that when ten observations are
made, the probability that the actual error is greater than
the mean deviation is very small, about i in 100, while if
15 observations are made it is reduced to i in 1000. Even
if only 5 observations are made (which is rather too small a
number) the probability is only i in 15. Hence, when a
quantity is measured several times, the average deviation
may be taken as a measure of the possible error.
POSSIBLE ERROR OF A CALCULATED RESULT. 5
6. Possible Error of a Calculated Result.
A piece of laboratory work usually calls for the measure-
ment of several different quantities and the calculation of a
result by some formula. Knowing the possible errors of
the separate quantities we can deduce the possible error
of the result, but the method will vary with the nature of
the arithmetical operations.
(a) Possible Error of a Sum or Difference. The possible
error of a sum or difference is the sum of the possible errors
of the separate quantities, for each possible error may be
either positive or negative.
Example. A bulb containing air (Exp. VI) weighs
2o.i425g. 0.0002 g. and after the air has been pumped
out it weighs 20.0105 g. o.ooo2g. Hence the weight of the
air is o . 1 3 2 o g. o . 0004 g. Since it is sometimes erroneously
assumed that a derived result must be accurate to as high a
percentage as the measurements from which it is deduced,
it should be noticed in the above that, while the separate
weights are found to 0.001%, the weight of the air is only
ascertained to 0.3%.
(b) Possible Error of a Power. If a measured quantity
x is in doubt by p per cent (p being small), the nth power
of x is in doubt by up per cent. For
x\ i-
IOO/ I \ IOO
squares and higher powers of p/ioo being neglected.
Example of (a) and (b).
T = 3. 506 .005 and/ = 2.018 -003. (Exp. VII). What
is the possible error of T 2 t 2 ? T 2 = 12. 2 9 and since T may be in
error by 1/7 %, T 2 maybe in error by 2/7 % or .04. Hence
T 2 = 12 . 29 .04. Similarly 2 = 4.07 .01. Hence T 2 t 2 =
8. 22 .05.
(c) Possible Error of a Product or Quotient. The percent-
age by which a product or quotient is in doubt is the sum of
6 INTRODUCTION.
the percentages by which the separate quantities are in
doubt. For if the quantities be
/ P \ / <1 \
x( i ) and y{ i I
V ioo/ \ ioo/
their product is
x ( I JL\J 1 A\^ xy L P^}
\ IOO/ \ IOO/ \ IOO/
and their quotient is
p/ioo and q/ioo being assumed small. It is evident that
a similar statement applies to any number of products and
quotients.
Example of (b) and (c) .
The diameter of a sphere (Exp. Ill) is measured by
a vernier caliper and found to be i . 586 cm., but the vernier
only reads to 1/50 mm.; so the possible error is .002 cm. or
1/8 of i%. The sphere is weighed in a balance such that i
mg. added to one pan does not cause an observable
change of the pointer, while 2 mg. does, and the weight is,
therefore, 16.344 g. with a possible error of .002 g. or
1/80 %. The calculated value of the density is 7 . 827 ; but the
volume may be in error by 3/8% and the mass by 1/80%.
Hence the density may be in error by 3 / 8 + 1 / 80 % , or practi-
cally 3/8%. Hence the proper statement of the density
is 7 . 83 with a possible error of .03 or 7 . 83 . 03.
7. General Method for the Possible Error of a Result.
The above rules for sums, differences, powers, products,
and quotients will usually suffice for finding the possible
error of a result calculated from the measurements of
several quantities. But when several of these operations
are combined, or when the formula for calculation contains
GENERAL METHOD FOR THE POSSIBLE ERROR. 7
one of the quantities more than once, the effects of the
several errors may be difficult to trace by these means. The
following general method is always applicable. It may be
carried out by simple arithmetic, but is simplified by an
elementary use of the calculus.
To find to what extent the possible error in one of the
quantities affects the result, we may calculate the result
assuming all the quantities to be quite accurate and then
repeat the calculation after changing one of the quantities
by its possible error. The difference in the result will be the
effect sought. If we do the same for each of the other
quantities, the final possible error of the result will be the
sum (without regard to sign) of the parts due to the separate
quantities.
This, however, is equivalent to differentiating the whole
expression, first with regard to one quantity, then with
regard to a second and so on and finally adding the partial
differentials. It will be seen from the following examples
that the process is much simplified by taking the logarithm
of the whole formula before differentiating.
(i) Time of Vibration of a Pendulum (Exp. VII). If in
time T a pendulum makes n fewer vibrations than the
pendulum of a clock that beats seconds and if t is the time
of a single vibration,
T
t = T-n
Taking logarithms,
log t = log T-log(T-n)
Hence by differentiating,
&_sr sr
t T T-n
This means that if T be changed by a small quantity, ST
the consequent change, Bt, in t is given by the formula. If
8 INTRODUCTION.
the possible error of T be 2 seconds, by putting 8T = 2
the value of $t will be the possible error of t. If T be 862
seconds and n be 17,
& 17X2
This indicates one of the advantages of taking logarithms.
It gives us at once the ratio of $t to t, or (multiplied by 100)
the percentage by which t is in doubt.
(2) Specific Heat by the Method of Mixture (Exp. XIX).
Let r = 95 be the initial temperature of the specimen,
2 = 25 that of the water, and let t = 45 be the final tempera-
ture of the mixture, and let the possible error of each ther-
mometer reading be o. 2. The formula for calculation is
^
M(T-t)
We shall consider how far the possible errors in the ther-
mometer readings affect x, leaving the consideration of the
other terms (the errors of which are likely to be much smaller)
to the reader.
log x=log(m+m l s) +log(tt ) logM log(T-t)
Proceeding as in (i) above, we find the effects of the possible
errors of T, t , and / respectively as follows :
$x SjT o. 2
= 0.4%
T t 50
& 0.2
/-/ 20
= 1 .O
70
Total = 2. 8%
This example will show a second advantage in the method of
taking logarithms. It separates the various terms and so
simplifies the process.
OF A MEAN. 9
8. Some General Notes on Errors.
The statement of a possible error should contain only one
significant figure. (A zero that serves only to fix the decimal
point, such as the zeros in 0.0026, is not a significant
figure). Thus in the last example in 6, 3/8% of 7.83 is
0.0293, which shows that the second decimal place in 7.83
is in doubt by 3. Hence it would be superfluous to add
figures to show that the third and fourth decimal places are
also in doubt.
Measurements sometimes seem so accurate that one is
tempted so say that "the possible error is practically zero
and need not be considered." This is never literally true.
One factor may be so accurately determined, compared with
other factors, that the effect of its possible error on the result
might seem to be negligible ; but only a calculation can show
this and the calculation will frequently show the opposite.
An illustration occurs in the first example of 6, considered
in connection with the other measurements required to
determine the density of air in Exp. III.
The consideration of possible errors is of great importance
in deciding what care need be expended in determining the
various factors in a complex measurement and what are the
best conditions for obtaining an accurate result. This
applies more especially to advanced and difficult measure-
ments, but illustrations will occur in this book. (Exp. XIX.)
But, as one of the purposes of this course is to teach the most
exact use of the measuring instruments, measurements
should usually be made as accurately as the instruments
will permit.
9. "Probable Error" of a Mean.
There is another method of indicating the reliability of
measurements which possesses some advantages over the
one that we have explained, though it is not so generally
applicable. When a large number of observations of a
quantity have been made, we can, by means of formulas
10 INTRODUCTION.
deduced from the mathematical Theory of Probability, cal-
culate the probability that the mean is not in error by more
than a given amount. When a coin is tossed up it is an even
chance whether it will come down a head or a tail; the
chance or probability of its being a head is, therefore, i in 2
or 1/2. Now the "probable error" of the mean of a num-
ber of readings is defined as a magnitude such that it is an
even chance whether the error is greater or whether it is
less than this magnitude. In other words, the probabil-
ity of the error exceeding the "probable error" is 1/2.
One formula for calculating the "probable error" is the
following :
average deviation
#=0.84^
A/number of observations
This method is useful as a method of indicating the reli-
ability of measurements when each of all the quantities that
occur in the experiment can be measured several times.
For when each mean and its "probable error" has been
found we can calculate the "probable error" of the final
result. For further details we shall refer the reader to other
works (e. g., Merriman's "Least Squares"). We shall not
have frequent occasion to refer to " probable errors," since in
most cases some of the quantities that have to be determined
cannot be measured more than once.
In justification of the use of the mean deviation as a
measure of the possible error, we may note that by the above
formula for e, when 10 observations have been made, the
mean deviation equals 3 . 6 <?. Now a reference to tables
of the probability of errors (e. g., Smithsonian Tables) shows
that the probability of an error greater than 3.8 e is about i
in 100.
10. Limits to Calculations.
By the above methods the possible error in any calcula-
tion from experimental quantities may be deduced. The
magnitude of the possible error in any calculation indicates
PLOTTING OF CURVES. II
how far it is useful and desirable to carry the calculation.
A calculation should be carried as far as, but not farther than,
the first doubtful figure. This rule must be applied not only
to the calculation of the final result, but also to each inter-
mediate step. When a calculation is carried too far, useless
and very unscientific labor is expended, and when it is not
carried far enough very absurd results are often obtained.
In addition and subtraction a place of decimals that is
doubtful in any one of the quantities is doubtful in the
result.
In multiplication and division (performed in the ordinary
way) decimal places that have not been determined are
usually filled up by zeros. 'Any figure in the result that
would be altered by changing one of these zeros to 5 is
doubtful.
ii. Notation of Very Large and Very Small Numbers.
Partly to save space and partly to indicate at once the
magnitude of very large' or very small numbers, the following
notation is used. The digits are written down and a decimal
point placed after the first and its position in the scale
indicated by multiplying by some power of 10. Thus
42140000 is written 4.2i4Xio 7 and .00000588 is written
5.88X10-. This also enables us to abbreviate the multi-
plication and division of such numbers. Thus 42 140000 X
.00000588 is the same as 4. 214 X 5-88X 10 and 42140000-7-
.00000588 is the same as (4. 214-=- 5. 88) X io 13 .
12. Plotting of Curves.
It is assumed that the general method of the represen-
tation of the connection between two related quantities by
means of a curve is familiar to the reader from his work in
Graphical Algebra or elsewhere. Attention may be called
to the following points:
i. Mark experimental points clearly by crosses or circles
surrounding the points.
12 INTRODUCTION.
2. The curve should be drawn so as to strike an average
path among the points; it does not have to pass through even
one point.
3. Abscissas may be drawn to any scale and ordinates to
any scale. Record the main division of the scales along
each axis. (Do not record the individual observations.)
4. For convenience, all the abscissas may be diminished
by the same amount before plotting, and the same is true of
ordinates.
5. Ordinates and abscissas should be drawn to such scales
that the curve occupies a large part of the paper.
6. Curves should be drawn carefully and neatly by means
of curve forms.
MECHANICS.
13. The Use of a Vernier.
The vernier is a contrivance for reading to fractions of the
units in which scales are graduated. It is a second scale
parallel to the main scale of the instrument and so divided
that n of its units equal n i or n -f i of the units of the scale.
If 5 is the length of a scale unit and v that of a vernier unit,
in the first case
nv=(n i)s or s v s-^-n;
in the second case
nv=(n + i)s or v s = s + n.
Hence the unit of the vernier is less or greater than that of
the scale by one-wth of a scale unit.
If we did not have a vernier there would be something in
the nature of an index to indicate what division of the scale
should be read in making a certain measurement and
fractions would be estimated by eye. The zero of the vernier
is taken as such an index, the whole number of scale divisions
being the number just below
the zero of the vernier, while
the fraction of a scale division
is determined with the vernier. A |J I I IJ I I I I ,|
If the wth division of the FlG T
vernier coincides with the scale
division, the zero of the vernier must be m wths of a scale
unit from the scale division just below it. Thus the use of
a vernier divided into n parts is equivalent to subdividing
the scale unit into n parts.
If no vernier division exactly coincides with a scale
division there will be two vernier divisions nearly coincident
MECHANICS.
FIG. 2.
with scale divisions, and one can often estimate fractions
of the fraction given by the vernier. In figure i the
whole number of scale divisions is 5.2, and evidently the
third and fourth vernier divisions most nearly coincide with
scale divisions. Since the third division is
somewhat nearer a coincidence, we may call
the fraction 3 . 3 tenths, or, the full reading
will be 5 . 233.
14. Vernier Caliper.
The vernier caliper consists of a straight
graduated bar, and two jaws at right angles
to it, one of which is fixed while the other
is movable. The position of the movable
jaw can be accurately determined by means
of the scale and a vernier which should read
zero when the jaws are in contact. (If this be
not the case, allowance must be made for the zero reading) .
15. Micrometer Caliper.
The micrometer caliper is a U-shaped piece of metal in
one arm of which is a steel plug with a carefully planed face
and through the other arm of which passes a screw with a
plane end parallel and opposite to that of the
screw. A linear scale on the frame reads zero
approximately when the plug and screw are in
contact, and its reading in any other position
indicates the whole number of turns of the
screw and consequently the number of mm.
(or 1/2 mm. or 1/40 in. as the pitch of the
screw may be) between the screw and the plug.
Fractions of a turn are read on the divided
head. As contact approaches, the screw should be turned
with a very light touch and the same force used for
different contacts. Some micrometers are provided with a
ratchet head which permits only a definite, moderate
pressure.
MICROMETER MICROSCOPE. 15
16. Micrometer Microscope.
The micrometer microscope is a microscope with cross-
hairs at the focus. In one type of instrument these cross-
hairs are movable by a micrometer screw. In the other
and more common type the whole microscope is moved by
a micrometer screw (see Fig. 4). The most elaborate
instruments have both movements. The rotations of the
screw are read on a fixed linear scale while the fraction of a
rotation is read by a circular scale attached to the screw,
and thus the amount of movement is ascertained if the
pitch of the screw is known. The pitch is best determined
by reading a length on a reliable scale placed in the field of
view.
17. Comparator.
The comparator consists essentially of a pair of micro-
scopes movable along a horizontal bar to which they are at
right angles. The length to be measured is placed under the
FIG. 4.
microscopes. The eye-piece of each microscope is first focused
clearly on the cross-hairs and the whole microscope focused
without parallax on the point to be observed, so that the
image of the point coincides with the intersection of the
cross-hairs. The object is then removed and a good scale
put in its place, and a reading of the scale gives the required
length, this reading being facilitated by the use of microm-
eter screws.
1 6 MECHANICS.
18. Spherometer.
The Spherometer is an instrument with four legs, three
of which form the vertices of an equilateral triangle, while
the fourth is at the center of the triangle. The fourth leg
can be screwed up and down and the distance of its extrem-
ity from the plane of the extremities of the other three legs
can be accurately measured by means of a linear scale
attached to the fixed legs and a circular scale attached to
the movable leg. The linear scale gives the number of
complete turns of the screw and the circular scale the frac-
tion of a turn. These scales are read when the screw makes
contact with an object placed beneath.
The position of contact may be determined by noticing
that the screw turns very easily for a fraction of a turn just
after contact begins. This is due to reduced friction in the
bearing, owing to the weight of the screw falling on the
body in contact and to the back-lash of the screw, the
frame not yet being raised. The screw should
be lowered until it thus begins to turn very
easily; it should then be turned back again
until it just again begins to turn hard and a
reading then made. A less sensitive method
is to turn the screw down until the instrument
is felt to rock or wobble and make a reading;
then raise the screw until rocking just ceases,
make another reading and take the mean of
FIG. 5. "the two as the contact reading. In some
spherometers the end of the screw on making
contact raises two levers arranged to greatly magnify the
motion. The screw is lowered until the top lever comes to
some definite position, for instance, with the end opposite
to a stud in the frame of the instrument.
The zero reading is the reading of the scales when the end
of the screw is in the plane of the ends of the legs, and is
so called because the instrument is most frequently used to
determine distances above or below this plane. It may be
THE DIVIDING ENGINE. 17
obtained by placing the instrument on a very plane plate of
glass. Several zero readings should be made before and
after making readings with the object in position under the
screw, for the zero reading is likely to change from slight
disturbances of the adjustment of the instrument and ther-
mal expansion due to the heat of the hand. If the mean of
the zero readings made after reading on an object should be
decidedly different from the mean of those made before,
readings should be again made on the object and the first
zero readings discarded. After each reading, the screw
should be turned up through at least a quarter revolution,
that the readings may be entirely independent.
The unit of the linear scale may be obtained by com-
parison with a standard steel scale. If the plane of the
circular scale is not exactly perpendicular to the axis, the
linear scale may not give the correct number of turns. It
is therefore best, as a check upon the linear scale readings,
to count the number of rotations.
19. The Dividing Engine.
The dividing engine received its name from its being
originally made to subdivide scales, diffraction-gratings,
etc. An equally frequent use of the instrument is for the
accurate measurement and calibration of scales, gratings,
etc.
It consists essentially of (i) a very carefully made hori-
zontal screw, the ends of which are so supported that the
screw is free to rotate, but not to advance or recede; (2) a
nut movable on the screw and bearing against (3) a plat-
form movable along a track which is parallel to the screw;
(4) a micrometer microscope which in some instruments is
held in a support movable along a rail on the same bed
plate as the track, but in other instruments is carried on the
platform; (5) dividing gear for making scales, etc.; (6) a
divided circular scale attached to the screw with a vernier
attached to the bed plate.
1 8 MECHANICS.
If, by means of the circular scale and vernier, the rota-
tion of the screw can be read to i in a very large number,
say in i in 1,000, then, since the nut moves a distance equal to
the "pitch" of the screw (measured parallel to the axis)
when the screw is given one complete rotation, it follows
that the movement of the nut and platform can be
read to a correspondingly small fraction of the pitch of
the screw.
The object whose length is to be measured is placed on
the movable platform (in the case of instruments of the first
type mentioned under (4) above) . The microscope is focused
on one end of the length to be measured, so that the inter-
section of the cross-hair coincides with that end. By turn-
ing the screw until the movement of the platform brings the
other end of the length to be measured into coincidence
with the intersection of the cross-hairs and observing the
number of turns and parts of a turn, the length of the object
in terms of the pitch of the screw as unit is obtained. The
true pitch of the screw must itself be obtained by com-
paring it by the same method with some accurately known
length, such as a length on a standard meter.
The adjustment of the microscope consists of two steps:
(i) the eye-piece must be focused on the cross-hairs (but
the eye-piece must not be taken out lest the cross-hairs be
injured) ; (2) the whole microscope must be moved toward
or away from the object until it is seen without parallax, i. e.,
until the relative position of the cross-hairs and the image
of the object is not changed by shifting the eye sidewise.
The length to be measured must then be placed parallel
to the screw. This is attained when, by rotation of the screw,
the image of each end can be brought to coincidence with
the intersection of the cross-hairs. One of the most fre-
quent sources of error, in using a measuring instrument
on the screw principle, is back-lash or lost motion. To
avoid this the screw should always be turned in the same
direction during a measurement. In many dividing
engines back-lash is impossible because the motion of the
CATHETOMETER. 19
screw cannot be reversed, the platform can only be moved
in the reverse direction by unclasping the nut (which, for
this purpose, is a split nut held together by a clasp and
spring) .
Usually the handle for turning the screw is not attached
to the screw or circular scale, but to a separate disk rotating
co-axially with the screw. The motion of the handle in
one direction is communicated to the screw by a ratchet;
when the handle is reversed the ratchet slips freely. As an
aid to counting the number of turns of the screw in measuring
a considerable length, two detents are sometimes geared to
the screw in such a way that only a definite number of turns
can be given to the screw at a time, after which the handle
must be turned back for beginning a new number of turns.
A more complete description of the dividing engine will
be found in Stewart and Gee, I, 16.
20. Cathetometer.
The cathetometer is a vertical pillar supported on a tripod
and leveling screws, and capable of rotation about its axis;
the pillar is graduated and a horizontal telescope with cross-
hairs is borne by a carriage that travels on the pillar and
can be clamped at any desired position. A slow-motion
screw serves for accurate adjustment of the position of the
telescope.
Adjustments.* (i) The intersection of the cross-hairs,
X, must be in the optical axis of the telescope. To secure
this, focus X on some mark, rotate the telescope about its
own axis and see whether X remains on the mark. If not,
the adjusting screws of the cross-hairs must be changed
until this is attained.
(2) The level must be properly adjusted. Level the
telescope until the bubble comes to the center of the scale.
Turn the level end for end. If the bubble does not come to
* Adjustments (i) and (2) are not usually required and should not be made
without the advice of the instructor.
20 MECHANICS.
the same position, the level must be adjusted until it will
stand this test.
(3) The scale must be vertical. If there are separate
levels for the shaft, this is readily attained. If there is but
one level for telescope and shaft, this and the next adjust-
ment must be made simultaneously.
(4) The telescope must be perpendicular to the scale.
The top of the scale, T, may be regarded as having two
degrees of freedom first, parallel to the line of two level-
ing screws of the base, A and B; second, in a line through
the third leveling screw, C, perpendicular to AB. If A
and B be screwed equal amounts in opposite directions, T
will move parallel to A B. If C only be turned, T will
move perpendicular to AB.
First make the telescope horizontal and parallel to AB.
Turn the shaft through 180. It is easily seen that if the
telescope makes an angle a with the normal to the scale,
turning the scale through 180 will cause the telescope to
make an angle 2a with its former direction. Hence, with
the leveling screw of the telescope, correct half the error
in the level, and, by turning A and B equally in opposite
directions, correct the remainder. Turn the telescope to
the first position and repeat the above adjustments, then to
the second and continue as often as is necessary. Then turn
the telescope normal to AB and adjust by C. When the
adjustment is complete, turning the shaft through any
angle will not alter the position of the bubble.
Unless the cathetometer is on a perfectly immovable sup-
port, perfect adjustment is not possible and too much time
should not be spent in adjusting, provided the telescope is
accurately level at each reading.
The eye-piece of the telescope is focused (but not removed)
until the cross-hairs seem perfectly distinct and the focus
of the objective changed until the object is seen ' very
distinctly and without parallax, i. e., with no relative
motion with respect to the cross-hairs when the eye is
moved about.
BAROMETER. 2 1
21. Barometer.
Text-book of Physics (Duff}, pp. 157-158; Watson's Physics, pp.
150-154; Ames' General Physics, pp. 176-177; Crew's Physics,
pp. 165-166.
If the barometer is of Fortin's cistern form, the cistern is
raised or lowered by means of the screw at the bottom
until the mercury just meets an ivory stud near the side of
the cistern. A collar to which is attached a vernier is so
placed that the top of the meniscus of the mercury column
is tangent to the plane of the two lower edges. The height
of the barometer should be reduced to zero by the formula
h = h(i . 00016 2t)
where h is the observed height, h the height at o and /the
temperature Centigrade. For the expansion of the mercury
will increase the height in the ratio (i . + .000181*) and the
expansion of the brass scale will reduce the apparent height
in the ratio (i - .000019*) (Table VII).
The siphon barometer has two scales, graduated on the
glass tube, in opposite directions, from a common zero.
The length of the mercury column is obviously the sum of
the readings of the mercury levels in the two tubes. Since
the coefficient of linear expansion of glass is only about
0.000008, the correction formula becomes
h = h(i- .000173*).
Since the mercury may adhere to the glass to some extent,
barometer tubes should be tapped gently before reading.
22. The Balance.
Kohlrausch, 7-11; Watson's Practical Physics, 25, 26.
Weighing by a Sensitive Balance. On first using a sensitive
balance note the position, purpose, and structure of the
following parts:
The beam, The knife-edges and planes,
The pointer, The arrestment,
The pillar, The rider-arms,
The pan-supports.
22 MECHANICS.
By the sensibility of a balance is meant the amount of de-
flection of the beam produced by a given small weight.
Consider how the sensibility depends on (i) the length of
the beam, (2) the weight of the beam and pans, (3) the
distance of the point of suspension of the beam from its
center of gravity. Will the sensibility of a certain balance
be different with different loads on the pans and why?
How can the sensibility be varied with a given load? (See
references.)
Precautions in Use of Balance.
1. Note the maximum load that may be placed on the
balance and take care not to exceed it.
2. Always lift the beam from the knife e<Jges before in
any way altering the load on the pans.
3. Do not stop the swinging of the balance with a jerk.
It is best to stop it when the pointer is vertical.
4. To set the beam in vibration, do not touch it with the
hand, but taise and lower the arrestment.
5. Place the large weights in the center of the pan.
6. Make final weighings with the case closed.
7. Replace all weights in their proper place in the box
when they are not actually in use. Do not use weights
from different boxes.
8. Do not place anything in contact with a pan that is
liable to injure it.
9. Avoid, if possible, weighing a hot body.
10. Never handle the weights with the fingers, as this
may change some of the weights appreciably. Always use
the pincers.
Notice the dimensions of the weights in the box, e. g.,
50 g., 20 g., 10 g., 10 g., 5 g., 2 g., i g., i g., etc. Instead of
weights of 0.005 > - 2 g-> o.ooi g., o.ooi g., it is customary
to use a rider of o . 01 g., which can be placed on the beam at
various distances from the center. The beam is for this pur-
pose graduated into 10 divisions, which may be still further
subdivided. Thus the o.oio g. rider placed at the division
4 of the beam is equivalent to o . 004 g. placed on the pan.
THE BALANCE. 23
The zero-point of the balance is the position on the scale
behind the pointer at which, the pans being empty, the
pointer would ultimately come to rest; it must not be con-
fused with the zero of the scale. As much time would be
wasted in always waiting for the pointer to come to rest,
the zero of the balance is best obtained from the swings of
the pointer. For this purpose, readings of the successive
"turning-points" are made as follows three successive
turning-points on the right and the two intermediate ones
on the left, or vice versa; e. g.,
Turning points.
L. R.
-1-3 + 2-1
I.I -f 2. O
I .O
Mean, 1.13 +2.05
-1.13
Zero-point = +0.92^-2 = +0.46.
By taking an odd number of successive turning-points
on one side and the intermediate even number on the other
side and then averaging each set, we eliminate the effect of
the gf adual decrease of amplitude of the swing.
The resting-point of the balance with any loads on the
pans is the point at which the pointer would ultimately
come to rest, and is found in the same way as the zero-point.
If the resting-point should happen to be the same as the
zero-point, the weight of the body on
one pan is immediately found by the ~
weights on the other pan and the posi-
tion of the rider. Usually, however, this
will not be so. With the rider at a suitable division, find the
resting-point on one side of the zero-point, and then, after
.altering the rider one place, find the resting-point on the
other side of the zero. By interpolation the change of the
position of the rider necessary to make the resting-point
coincide with the zero-point is deduced. For example, the
24 MECHANICS.
zero is +o . 46 ; with the rider at 4 the resting-point is +0.51;
with the rider at 5 the resting-point is + o . 10. By changing
the rider from 4 to 5, o.ooi g. was added. To bring the
resting-point to the zero we should have added 0.05-;-
(0.510.10) of o.ooi g. or o.oooi g. approximately.
Hence the weight of the body is the weight on the pan plus
0.0041 g.
The arms of the balance may be unequal. If this be so,
the weight obtained above will not be the true weight. To
eliminate this error the body must be changed to the other
pan and another weighing made. If / be the length of the
left arm and r that of the right and if u be the counterbal-
ancing weight when the body is in the left pan and v when
it is in the right, while w is the true weight of the body then,
lw=ru, lv=rw
(The geometric mean of two very nearly equal quantities
is nearly equal to their arithmetic mean.) The ratio of the
arms of the balance may also be calculated, since
The buoyancy of the air on the weights and on the body
must be allowed for in accurate work. To the apparent
weight of the body must be added a correction equal to the
weight of the air displaced by the body and from the apparent
weight must be subtracted the weight of the air displaced
by the weights. In each case the weight of the air displaced
can be calculated if its volume and density are known.
This correction in any case is very small. A small per-
centage error in the correction will not appreciably affect
the calculated true weight. Hence approximate values of
the volumes of the body and weights may be used. In
finding the volume of the weights the density of brass
weights may be taken as 8.4. The density of air at o and
760 mm. may be taken as .0013, and its density at the
temperature of the laboratory and the pressure indicated
ADJUSTMENT OF TELESCOPE AND SCALE. 25
by the barometer may be calculated by the laws of gases.
Hence the temperature and barometric pressure should be
obtained.
23. Adjustment of Telescope and Scale.
To adjust a telescope and scale, determine approximately
the location of the normal to the mirror, either by finding
the image of one eye or the image of an incandescent lamp
held near the eye. Move the stand supporting the telescope
and scale until the center of the scale is about in line with
the normal. Look along the outside of the telescope at the
mirror and move the scale up and down, or, if this is not
possible, raise or lower the stand until you see the reflection
of the scale in the mirror. It may be a help to illuminate
the scale with an incandescent lamp. Look through the
telescope pointed at the mirror, and change the focus until
the scale is seen distinctly. Remember that the more
distant the object, the more the eye-piece must 'be pushed in,
and that the image of the scale is at about twice the distance
of the mirror.
24. Time Signals.
A convenient source of time signals for a laboratory is a
chronometer which either opens or closes a circuit containing
batteries, sounders, etc., every second with an omission
at the end of each minute.
The individual second intervals indicated by a chro-
nometer, so arranged, are likely to be somewhat inaccurate,
and therefore, when an accurate interval of one second is
required, a second's pendulum should be used with a plat-
inum point making contact with a drop of mercury, and thus,
if desired, closing an electric circuit. Since it is difficult
to set the mercury drop exactly in the center of the path,
alternate seconds are likely to be too long. Therefore, if
possible, a two seconds' interval should be employed,
alternate contacts being disregarded. If these contacts
cause confusion, a pendulum omitting alternate contacts
may be used (see Ames and Bliss, p. 486).
I. TO MAKE AND CALIBRATE A SCALE.
To illustrate the use of the dividing engine (described on
page 17) a short scale is to be engraved in millimeters on a
strip of nickel-plated steel and then calibrated by compari-
son with the average millimeter of a standard scale.
Arrange the cogs of the dividing gear so that each fifth
mm. division shall be longer than the intermediate divisions
and each tenth division still longer. Test this adjustment
on a rough test strip. Next clamp the strip to be divided
on the platform of the engine so that it is parallel to the
screw; this can be tested by observing the edge of the
strip in the microscope as the platform is advanced by the
screw. Care should be taken to clamp the pillar that sup-
ports the divider so that the point of the divider moves
perpendicular to the length of the scale. A scale of 2
or 3 cms. should then be marked out on the steel strip
and the temperature of the platform ascertained by a
thermometer.
This scale is next to be calibrated. The exact pitch of the
screw is first obtained in terms of the mm. of the standard.
For this purpose a considerable length, e. g., a decimeter,
of the standard should be measured on the engine. This
should be done for three different parts of the screw. The
agreement of the three determinations will afford some
indication of the uniformity of the screw. The scale should
then be measured mm. by mm. For the first reading the
circular scale of the screw may be set to zero when the
cross-hairs coincide with the zero division of the scale to
be measured, and thereafter the screw should be turned
always in the same direction and only arrested for a reading
of the circular scale and vernier (the total number of turns
being also noted) when the microscope shows that the middle
of a division has come to coincide with the intersection of
26
ERRORS OF WEIGHTS. 27
the cross-hairs. As this coincidence approaches, the handle
should be turned slowly, and if turned too far the reading
at that point must be omitted altogether. The handle
should also be turned slowly when contact with the detent
approaches so that the screw may not be arrested with a
jerk.
As a check on the work, the whole length of the scale
should be measured.
In calculating the true length of the divisions, allowance
must be made for the temperature of the standard which
may be taken as the temperature of the platform of the
engine. The standard is correct at the temperature marked.
From its coefficient of expansion calculate the length of its
mm. at the temperature of observation and then deduce the
pitch of the screw at the same temperature. Then from the
readings made, calculate the length of each millimeter of the
scale and, by addition, draw up a table showing the true
distance of each division from the zero division.
Questions.
1 . Enumerate the possible sources of error in the use of the divid-
ing engine for the manufacture of scales.
2. At what temperature would the whole length of your scale be
an exact number of centimeters? (Table VII.)
II. ERRORS OF WEIGHTS.
Kohlrausch, 12; Watson's Practical Physics, 27.
Weights by good makers are usually so accurate that
errors in them may for most purposes be neglected. But
when less perfect weights are to be used or when weighings
are to be made with the highest possible degree of accuracy,
the errors in the weights must be carefully ascertained.
We shall suppose that a 100 g. box of weights is to be
tested, and that a reliable 100 g. weight is supplied as a
standard, and that an accurate 10 mg. rider is supplied for
making the weighings. The weights of the box will be
denoted by 100', 50', 20', 20", 10', and so on, and the
28 MECHANICS.
sum 5' + 2' + 2" + 1' by 10". To find the six unknown quan-
tities, 100', 50', 20', 20", io r , 10", we must make six weigh-
ings and obtain six relations between these quantities.
Such a set of weighings are indicated in the following table.
Each should be performed by the method of double-weigh-
ing described on page 24.
10' =10" +a
20' =10' + 10" +b
20" =20' +c
50' =20' +20" +10' +d
ioc/ =50' + 20' +20" + 10' +e
100 = 100' +/
To solve these equations, substitute the value of 10' given
by the first in the second; then substitute the value of 20'
given by the second in the third, and so on to the last, when
the value of 10" in terms of the standard 100 and a, b, c, d,
e, f will be obtained. The calculation of the other quantities
will then present no difficulty. To standardize the box
completely the same process must be applied to 10', 5', 2', 2" ',
i', i", and similarly to the smaller weights.
III. VOLUME, MASS, AND DENSITY OF A REGULAR
SOLID.
The mass of the specimen (a sphere or cylinder) is found
by weighing on a sensitive balance (see p. 21). To eliminate
the inequality of the arms of the balance, the body should be
weighed in both pans (p. 24). The zero-point and resting-
points of the balance should be found by the method of
vibrations and the various precautions in the use of the
balance must be carefully observed. Allowance should be
made for air buoyancy (p. 24) and corrections should be
applied to the weights, if the weights have been corrected in
the preceding experiment, or if a table of corrections is
supplied.
MOHR-WESTPHAL SPECIFIC GRAVITY BALANCE. 29
The dimensions of the specimen are measured by a
micrometer caliper (p. 14) or a vernier caliper (p. 14).
If the body is spherical, ten measurements of the diameter
should be made and the average taken; if it is cylindrical
ten measurements of the diameter and ten of the length
should be made.
From the mass and the volume, the density (or mass per
c.c.) is deduced.
The ratio of the arms of the balance should also be derived
from the results of the double weighing (p. 24).
The possible error of the density determination should be
calculated as illustrated on p. 6.
Questions.
1. If the object aimed at were merely the density of the body,
which of the above measurements should be improved in precision
and to what extent would it need to be improved?
2. If the above improvement were not possible, how much of the
refinement of measurement of the other quantity might be dis-
carded ?
IV. MOHR-WESTPHAL SPECIFIC GRAVITY
BALANCE.
Kohlrausch, p. 45; Stewart and Gee, I, 92, III.
This is a convenient form of hydrostatic balance for
rinding the density of a liquid by determining the buoyancy
of the liquid on a float hung from an arm of the balance
and immersed in the liquid. Instead of weights riders are
used, the arm of the balance from which the float hangs
being graduated into ten divisions. The float is made of
such a size that when hanging in air from the graduated
arm of the balance (which is less massive than the other
arm) it will just produce equilibrium. Four riders of differ-
ent mass are employed, each one being ten times as heavy
as the next smaller. The largest rider is of such a size
that if the float hanging from the balance be immersed in
water at 15 C. the addition of the rider to the hook at the
30 MECHANICS.
end of the beam will restore equilibrium. Hence it counter-
balances the buoyancy of the water on the float. Thus it is
evident that if the water be replaced by a liquid of unknown
density at the same temperature (so that the volume of the
float is the same) and if the largest rider under the circum-
stances produces equilibrium when placed at the sixth
division, then for equal volumes, this liquid can weigh
only six-tenths as much as water, or its density is 0.6. A
second rider, one-tenth as heavy as the first, would evidently
FIG. 7.
enable us to carry the process one decimal place farther,
etc. For liquids of a density exceeding unity, another
rider equal to the largest must be hung from the end of the
beam, and still a third may be necessary for liquids of den-
sity above 2.
From the above it will be seen that (i) the balance must
be adjusted by the leveling screw on the base until the end
.of the beam is opposite the stud in the framework when the
float is suspended in the air; (2) the beaker must always
be filled to the same level, that level being such that when
the liquid is water at 15 C. the balance is in equilibrium
with the largest rider hanging above the float, and (3) the
liquid tested must be at 15 C.
MOHR-WESTPHAL SPECIFIC GRAVITY BALANCE. 31
As an exercise in the use of this balance, find what shrink-
age of volume there is in the solution of some salt (e. g.,
common salt, ammonium chloride or copper sulphate)
in water and find how the shrinkage varies with the con-
centration. Solutions may be made up by weighing out
very carefully on a sensitive balance (see p. 21), 0.5 gm.,
i gm., 4 gm., 10 gm., etc., of the salt and dissolving each in
a deciliter of water. When the density of a solution has
been found, the percentage contraction is calculated from
the sum of the volumes of the constituents before mixture
and the volume of the solution after mixture; the volume
in each case equals the mass divided by the density. The
densities of various salts are given in Table VII.
The densities found and the percentages of contraction
should be represented by curves with percentages of salt
as abscissae. If any determination of density be largely in
error it will be shown by the curve.
If time permit, determine the density at 1 5 of equivalent
solutions* of several salts having the same base, e. g.,
NaCl; 1/2 Na 2 SO 4 ; NaNO 3 ; etc., and compare with the densi-
ties of similar solutions with a different base, e. g., NH 4 C1;
1/2 (NH 4 ) 2 SO 4 , NH 4 NO 3 , etc. The difference in density
between corresponding salts should be approximately
constant (Valson's Law of Moduli). Find similarly the
difference in densities contributed by the acid radicals,
e. g., NaN0 3 and NaCl; NH 4 N0 3 and NH 4 C1, etc.
Questions.
1 . What sources of error may there be in a determination of density
by this method ?
2. How might the accuracy of the riders be tested?
3 . How might the accuracy of graduation of the beam be tested ?
4. What effect has capillarity?
5. Explain the Law of Moduli. f
* The chemical equivalent of a substance is the atomic or molecular weight
divided by the valency. Two solutions are equivalent if the number of grams
of each dissolved in one liter (or that proportion) is the same fraction of the
respective chemical equivalent.
t Phy. Chem., Ewell, p. 159.
3 2
MECHANICS.
V. DENSITY BY VOLUMENOMETER.
Gray's Treatise on Physics, I, 426.
When the density of such substances as gunpowder,
sugar, starch, etc., is to be determined, neither the method
of immersion in a liquid nor that of the direct measure-
ment of mass and volume can be employed. The method
then usually employed is that of the volumenometer. This
is a method of immersion in air instead of immersion in
water, with an application of Boyle's Law
instead of Archimedes' principle. The volume
of the body is found by placing it in a glass
vessel and noting how much the pressure in
the vessel changes when the air is allowed to
expand.
A gas-washing bottle of about 150 c.c. capa-
city, A, into which the body is to be introduced,
is connected, by heavy pressure tubing, with
an open, U-shaped, mercury manometer (see
Fig. 8). The bottle, A, is closed by the stopper
6, which should be lubricated with rubber
grease,* and forced into A to a definite mark.
DE is raised until the mercury in the burette BC is at a
division B which is carefully observed. The pressure,
P, in A is carefully determined from the difference in
mercury levels and the barometer. By the use of a
rear mirror, parallax may be avoided and a small square
will assist in reading a scale between the two arms of the
manometer. The accuracy of the readings may be increased
by using a cathetometer (p. 19). Lower DE until the mer-
cury is at a division K and again determine the pressure, p.
Let the volume between B and K be v. Let V be the
volume of A, and connecting tubing, to B. By Boyle's
Law:
FIG. 8.
* Equal parts pure rubber gum, vaseline, and paraffin. The two latter are
melted together and the rubber is cut into small pieces and dissolved in the
heated liquid,
DENSITY OF AIR. 33
Make at least six determinations of P and p, bringing
the mercury each time to the same points B and K which
should be as far apart as is convenient. Calculate V from
the mean values.
Now introduce a carefully weighed amount of the assigned
powder into the bottle, A (which may be disconnected at
e), and insert the stopper, b, to its former depth. Again
determine the pressures, P' and p r when the mercury level
is at B and K respectively. Repeat as before. If x is the
unknown volume of the powder, the previous equation
becomes
from which x may be calculated. From the volume and
mass of the powder its density is determined.
If time permit, determine the density also with a specific
gravity flask (pyknometer) . Weighings should be made
of (i) bottle empty; (2) bottle filled with a liquid of known
density which is inert toward the body, and (3) with a
known mass of the body in it, the rest of the bottle being
filled with the liquid. An equation for density can be
worked out.
The possible error in the determination of the density
is found by methods explained on pages 3,8.
Questions.
1. What sources of error remain uneliminated?
2. With a view to greater accuracy what suggestions would you
make as to the most suitable magnitudes for x and v?
VI. DENSITY OF AIR.
The density of air at atmospheric pressure, or its mass
per cubic centimeter, might be obtained by weighing a flask
containing air at atmospheric pressure and then re-weigh-
ing it after all the air has been removed by an air-pump.
The difference of weight, together with the volume of the
flask, would give the density of the air. In practice the
3
34 MECHANICS.
procedure has to be modified, because it is impossible to
completely exhaust the flask of air. The modification con-
sists in finding the pressure of the air remaining in the flask
and taking account of it.
Let D be the required density at the room temperature
and pressure, P. Let d be the density of the remaining air
when the pressure has been reduced to p. Let the weight
of the flask when filled with air be W and let w represent
its weight when exhausted to the pressure p.
W-w = V(D-d)
By Boyle's Law
D _P D-d_P-p
~d = ~p " ~D~ P
Therefore
'P-p V P-p
A convenient form of flask is a round-bottom flask from
which part of the neck has been cut off and which is closed
by a rubber stopper containing a glass tube with a glass
stop-cock. The rubber stopper will hold tighter if lubricated
with rubber grease * before insertion.
If the flask, as found, is dry, it will be better to postpone
finding its volume until the end of the experiment, as the
operation requires it to be filled with water. Moreover,
of the two weighings for finding the mass of air removed,
it is better to make the one with the flask partly exhausted
first, for the weighing with the air admitted can be made
immediately after, without handling the flask or removing it
from the balance, a point of some importance where the
difference of weight to be measured is so small. To save
delay in weighing the flask after it has been exhausted, the
zero reading of the fine balance used should be obtained
before the flask is exhausted. For the method of accurate
weighing, by oscillations, see page 23.
*See note, p. 32.
DENSITY OF AIR. 35
A Bunsen's aspirator or a Geryk pump is satisfactory
for exhausting the flask. The flask should be connected
to the aspirator or pump, through a bottle for catching any
water or mercury. An open-tube manometer connected
to the tube that joins the aspirator or pump and flask will
give the pressure.
There should be a stop-cock or a rubber pinch-cock in
the connection between the manometer and the pump or
aspirator. When a sufficiently high exhaustion has been
secured this cock should be closed for several minutes to
ascertain if there is any leakage. If not, both ends of the
manometer should be read and the stop-cock of the flask
closed. Before removal of the flask, the other cock should
be opened that the. rest of the apparatus may fill with air.
If by any chance a small quantity of water should pass into
the manometer, allowance should be made for it, the density
of mercury being taken as 13.6.
The flask is then weighed as quickly as possible on a
fine balance, the method of vibration being used. It may
be necessary to hang the flask by a fine wire to the hook
which carries the pan. This weighing is repeated with the
stop-cock open, but with the flask otherwise undisturbed.
The atmospheric pressure is obtained from a reading of the
barometer (see p. 21).
The volume of the flask may be obtained by filling it
with distilled water and weighing it on an open balance.
To get the flask just filled to the stop-cock, the stopper
(removed for filling the flask) should be thrust in with the
stop-cock open, the stop-cock should then be closed, and
any water above the stop-cock should be removed. Of
course, the stop-cock should be replaced at its original
depth, which should be marked. The density of water at
different temperatures will be found in Table V.
When the experiment is completed, place the open flask
inverted on a frame to dry, so that it may be ready for the
next person who uses it.
The density of dry air may be found in the same way,
36 MECHANICS.
the flask being several times exhausted and refilled through
a drying-tube. Similarly the density of any other gas, e. g.,
carbon dioxide, may be found by filling the flask from a
generator. The gas must be admitted to the exhausted
flask very slowly and the exhaustion and filling must be
repeated to insure the (almost) complete removal of
the air.
In reporting, deduce from your measurement of the
density of air or gas, its density at o C. and 760 mm. by
using Boyle's and Charles' Laws. Find also the possible
error of the measurement of density (p. 5).
Questions.
1 . Would the first results be affected by the presence of water in
the flask? Explain.
2 . Should the flask weigh more filled with dry air or filled with
moist air, both at atmospheric pressure? Why?
VII. ACCELERATION OF GRAVITY BY PENDULUM.
Text-book of Physics (Duff}, 117; Watson's Physics, 112114;
Watson's Practical Physics, 46-49; Ames' General Physics,
pp. 74, 91, 135; Crew's Physics, 85, 86.
The acceleration of gravity, g, is most readily obtained
from the length and time of vibration of a pendulum. The
time of vibration of an ideal simple pendulum, i. e., a heavy
particle vibrating at the end of a massless cord would be
,- T
/ being the length of the pendulum. If the bob is a ball so
large that the mass of the suspending wire is negligible,
the above formula will apply provided the radius of the
ball is negligible compared with the length of the pendulum.
If these assumptions may not be made, the pendulum must
be regarded as a physical pendulum and its moment of
ACCELERATION OF GRAVITY BY PENDULUM. 37
inertia about the suspension considered. Under these
circumstances the formula
t =
Mgh
must be used, where / is the moment of inertia of the entire
pendulum about the knife-edge, M is the total mass and h
is the distance from the knife-edge to the center of gravity
of the whole. If the mass of the suspension is negligible
it is only necessary to consider the moment of inertia
of the ball about the knife-edge. It is easily shown that
the latter formula then reduces to the formula for the
simple pendulum, provided the length of the pendulum is
taken as the distance from the knife-edge to the center of
the ball plus 2r 2 / $1 where r is the radius of the ball. Hence
to find g there are three quantities, t, I, and r, to be
measured.
A convenient form of pendulum consists of a spherical
bob into which screws a nipple through which a fine wire
is passed and secured. To the upper end of the wire is
soldered a stirrup of brass which rests on a knife-edge of
steel. A short platinum wire should be soldered to the
lower side of the bob.
For accurately measuring the length of the pendulum a
cathetometer (see p. 19), which should be carefully adjusted,
may be used. (If necessary, the measurement of length may
be postponed until the time has been observed). The
horizontal cross-hair of the cathetometer is first focused
on the knife-edge, the fine screw being used for the final
adjustment of the telescope, and the scale and vernier are
then read. The telescope is then lowered and set on either
the top or bottom of the bob, whichever is the more definite.
These readings should be repeated several times, beginning
each time with the knife-edge. If the adjustments are
imperfect, the telescope should at least be made exactly
level before each reading. The diameter of the bob may be
38 MECHANICS.
measured by means of a micrometer or a vernier caliper
(see p. 14).
For fixing the vertical position of the pendulum, two
vertical pointers may be so placed that, when the pendulum
is at rest, the pendulum suspension and two pointers are
in one plane. The eye of the observer should always be
kept in this plane in using the first two methods. The pen-
dulum is set vibrating in an arc of 3 or 4 cms. Several
attempts may be necessary to get the pendulum vibrating
exactly perpendicular to the knife-edge with the bob free
from rotation.
The time of vibration is most readily obtained with
precision when the pendulum is very nearly a second's
pendulum, i.e., when the period of a complete vibration is
very nearly two seconds. For the determination of the
period several methods are available. The first and roughest
method given below will serve for adjusting the pendulum
to the required length.
(A) In the first method for determining the period,
time is found by the relay (p. 25) and the number of
vibrations in three minutes is counted, fractions of a vibra-
tion being roughly estimated. This is repeated several
times. Or a stop-watch or stop-clock may be used, but it
should be rated by comparison with a chronometer or
standard clock. The stop-watch is started as the pendulum
crosses the plane of observation and "one" is counted the
next time the pendulum crosses the plane in the same
direction. The watch is stopped on the 5oth vibration;
and the whole repeated five times. The mean time divided
by 50 will give a fair value for the period.
(B) A second and much more accurate method of obtain-
ing the time of vibration is the method of coincidences. This
consists in finding the rate at which the pendulum gains or
loses as compared with a standard clock or chronometer.
It is applicable only when the periods of pendulum and clock
or chronometer are nearly the same or when one is nearly
an exact multiple of the other. The method receives its
ACCELERATION OF GRAVITY BY PENDULUM. 39
name from the fact that what is observed is the "coincidence
interval" or the interval between the moment when a
passage of the pendulum through the vertical coincides
with some signal from the clock to the next time when such a
coincidence occurs.
In a coincidence interval, the pendulum must gain or
lose one vibration as compared with the chronometer or
other time standard. If n such coincidence intervals occur
in T sec., the number of vibrations of the pendulum during
this time is (Tri). Hence if / is the time of one vibration,
Tn
and the period of a complete vibration is
27
Tn
A convenient form of signal is given by the chronometer
and relay described on page 25. It is advisable to have the
coincidence interval something between 30 seconds and 3
minutes, and, if necessary, the length of the pendulum should
be changed for the purpose.
After the coincidence interval has been roughly deter-
mined by a few observations, the following modification of
the method will give it much more accurately. Calling the
time of the first coincidence zero seconds, observe the sec-
ond on which the next coincidence occurs and then the next,
until four have been observed. Then, after allowing a
considerable number of coincidences to pass unnoted, but
keeping note of the time, observe the number of the seconds,
counted from the original coincidence, upon which four more
successive coincidences occur.
From the first set of coincidences, three estimates of the
coincidence interval will be obtained and three others from
the second set, the mean of all giving an approximate esti-
mate. Then let the time of the first coincidence of the first
set be subtracted from the time of the first of the second
40 MECHANICS.
set, also the time of the second coincidence of the first set
from that of the second of the second set, etc. These dif-
ferences give four estimates of the time, T; of some unknown
integral number, n, of coincidence intervals. If the mean
of these four estimates be divided by the mean time of a
single coincidence interval as already found, the quotient
will be n plus or minus a small fraction. This fraction is
due to inaccuracy in the estimates of the coincidence intervals
and should be dropped. The period / of the pendulum may
now be calculated. The plus sign in the denominator is
used if the pendulum is the faster.
The following aid to the observation of coincidences is
suggested. Keeping the eye constantly in the proper plane
for observation, make a dot on a piece of paper at each click
of the- relay. When there appears to be coincidence, pro-
long the dot into a stroke. To avoid recording every click,
a cross may be used instead of a dot for marking a minute,
and the clicks may be passed unrecorded until the next
minute, or coincidence. There may be several successive
clicks during which there appear to be coincidences, in
which case several successive strokes should be made and
the mean taken. From these dots, strokes, and crosses, the
times of coincidence may be deduced. Or, a dial indicating
seconds may be employed, the second when there first
appears to be a coincidence being observed and the second
when there first appears to be no coincidence. Since the
clock cannot be observed immediately, the ticks are counted
until the clock is observed and then subtracted ; minutes must
be noted and recorded if they are not recorded on the clock.
(Q A third method consists in modifying the second
method so that coincidences of two sounds are observed.
The pendulum is made to actuate a sounder or telephone
each time it passes through the vertical and a coincidence is
observed when the sounder and relay strike together. A
block of wood with a narrow trough filled full of mercury
is placed in a mercury tray and is adjusted beneath the
pendulum so that the platinum wire on the under side of the
COEFFICIENT OF FRICTION. 41
bob just touches the mercury when the pendulum is at rest,
and crosses the narrow trough at right angles when the
pendulum is in motion,. A wire soldered to the knife-edge
is connected in series with several batteries, a sounder or
telephone, and the mercury trough. The final adjustment
of the mercury trough is made with the leveling screws of
the mercury tray. Care should be taken not to spill the
mercury.
From the possible errors in the measurements of / and t
deduce the possible errors in the value found for g
(see p. 7).
Questions.
1. Does the friction of the knife-edges and of the air increase or
decrease the value of g?
2. Why should coincidence be observed exactly, for the plane
containing the position of rest?
3. What would be the result of increasing the arc of vibration to
10 cm.? (Table III.)
4. Why should the top reading of the cathetometer always precede
the bottom reading?
5. Design, if possible, a scheme of electrical connections such
that the sounder will only operate when there is a coincidence.
VIII. COEFFICIENT OF FRICTION.
Text-book of Physics (Duff), 126130; Watson's Physics, 96-100;
Ames 1 General Physics, p. 118; Crew's Physics, 117; DanielVs
Physics, pp. 176-184.
The coefficient of friction of two surfaces is the ratio
of the force of friction opposing the incipient or actual
relative motion to the force pressing the two surfaces to-
gether. The force requisite to start the motion is greater
than that required to sustain the motion, i. e., the "coeffi-
cient of static friction" is greater than that of "kinetic fric-
tion." Moreover, the coefficient of kinetic friction is not
quite constant, but varies somewhat with the speed.
(A) The coefficient of static friction of one surface on
another may be found by means of a block of the former
resting on a slide of the latter. One end of the slide is gently
42 MECHANICS.
elevated by a screw until the block just fails to stand sta-
tionary on the slide. The tangent of the angle which the
slide then makes with the horizontal equals the coefficient
of static friction (see references). The tangent may be
measured by some simple method, using meter-stick, plumb-
line and level or square. Several entirely independent
adjustments for this angle and measurements of the tan-
gent should be made, the adjusting screw being each time
turned some distance down so that the influence of the pre-
vious setting may be avoided. The friction may vary some-
what from point to point, and if so, different points should
be chosen for the separate trials.
The accuracy of the determination of the tangent should
be calculated to see whether the possible errors will account
for the variations of the coefficient. Such, however, will
probably not be found to be the case.
(B) The coefficient of kinetic friction may be determined
by the same apparatus if we can find the acceleration
with which the block moves down the slide when the latter
is tilted beyond the angle of repose. For, if the acceleration
of the block is a and its mass m and the angle of inclination
of the slide i, then the component of gravity down the slide
is mg sin i and the pressure on the slide is mg cos i. Hence,
if JJL is the coefficient of friction, by Newton's second law,
m a m g sin i t a m g cos i
a
and, = tan ^ ;.
g cos i
This process will give the mean coefficient of friction for the
range of speeds through which the block passes, but for the
low speeds in question the coefficient does not vary much.
The acceleration, a, is found by a method frequently
employed in physical measurements. A tuning-fork (fre-
quency of 50 or less) is fastened in a clamp attached to a
support above the slide. A stylus of spring brass with a
steel needle point is attached to one prong and just behind
this stylus is a second stationary stylus which is attached to
COEFFICIENT OF FRICTION. 43
the support. A long and narrow glass plate is covered with
the washing compound called "Bon Ami" by transferring,
with a wet cloth, a little of the paste from the cake to the glass,
and then spreading it out in a thin layer. The block is
then raised to the top of the slide and secured by a trigger.
The support that carries the fork is raised and lowered and
the fork is adjusted in the clamp until each stylus touches
the coated glass, making with it an angle of about 45, the
stylus on the fork being exactly in front of the other stylus.
The frame-work is then lifted until neither stylus touches
the glass. The fork is set in vibration by drawing the prongs
together with the fingers and releasing them, or by with-
drawing a wooden wedge, and is then adjusted until its
stylus vibrates an equal amount on each side of the other
(stationary) stylus. The frame-work is then lowered until
the styli touch the glass and the block is immediately re-
leased by the trigger.
A wave line should be obtained with a straight line ex-
actly in the center, the amplitude of the wave line on each
side of the straight line being several millimeters. Since
in any measurement the effect of inaccuracies at the ends is
less important the greater the quantity measured, we meas-
ure the distance passed over during several vibrations of
the fork. This distance divided by the time in which it
was traversed, i. e., by the period of the fork multiplied by
the number of vibrations, gives the average velocity of the
block during this time. If T be the period of the fork and
x the distance passed over in n complete vibrations of the
fork, the average velocity is oc/nT. Similarly we find
the average velocity for the next n complete vibrations.
The average acceleration will be the difference between
these average velocities divided by their separation in time
or nT; for since each velocity is the average we may con-
sider it as belonging to the middle of the time for which
it is the average. From several successive groups of n
vibrations several values of the acceleration are obtained
and the mean taken.
44 MECHANICS.
It remains to determine the period of the fork. Two
methods will be described, (a) The fork is clamped be-
side a small electro-magnet connected through a battery
with a pendulum which closes the. circuit every second (see
p. 25). To the armature of the electro-magnet a stylus is
also attached. A plate of glass covered with " Bon Ami" is
clamped on a movable block so that each stylus rests upon
it. The electro-magnet and fork may have any relative
position which may be convenient, but the styli should not
be far apart. The fork is set vibrating and the block with
the glass is drawn along, the fork making a wave line and
the other stylus a straight line broken (or notched) every
second. With a square, lines are drawn at right angles to
the glass through the beginnings of alternate second sig-
nals and the number of complete vibrations, estimated to
tenths, is counted between the lines.
(6) This is known as a stroboscopic method and depends
upon the persistence of vision. The fork is watched through
holes in a disk revolving at a con-
stant speed. The holes are equally
spaced in concentric circles, the
number per circle increasing with
the radius. The speed of the disk
is varied until the fork appears
stationary when viewed through
FIG. 9. the holes of a particular circle. If
there are m holes in the circle, and
if the disk revolves n times per second, the frequency of the
fork is obviously mn. By varying the speed and using other
holes, additional determinations may be made. The speed
of the disk is obtained by determining, with a counter, the
number of revolutions in a given time.
(C) Another method of finding the coefficient of kinetic
friction is to make the slide horizontal and find the force
required to keep the block in uniform motion after it has
been started. For this purpose a braided cord is attached
to one end of the block, passed over a pulley at the end of
COEFFICIENT OF FRICTION. 45
the slide, and attached to a scale pan, to which weights are
added. In this case the weight of the pan and weights
must not be taken as the force acting on the block, for some
force is required to overcome the friction of the pulley.
The amount required must be found by a separate experi-
ment. Two pans are attached to the ends of the cord
hanging over the pulley and sufficient equal weights are
placed on the pans to make the pressure on the pulley the
same as in the main experiment where the parts of the cord
were at right angles. The additional weight on one scale pan
requisite to keep the whole in constant motion when started
is the force needed to overcome the friction of the pulley,
and is, therefore, the correction required.
With this apparatus we may also test whether the coeffi-
cient of friction varies when weights are added to the block.
The correction for friction of the pulley does not need to
be re-determined experimentally, but may be calculated
from the former determination, on the assumption that the
friction of the pulley is proportional to the pressure on it.
The possible error in the results of the first and last
methods is easily determined. The most, accurate way of
finding the possible error in method (B) is by means of
formulae deduced by the differential calculus (see p. 6),
but a much simpler and a sufficiently accurate method is
the following: Note that an overestimate of i will increase
the value of fj. and the same will be the effect of an under-
estimate of a. Hence the coefficient should be recalculated
with tan i and cos i increased and a decreased by their
possible errors and the change found in the, coefficient may
be taken as the final possible error. The possible error of a
may be taken as its mean deviation and the possible errors
of tan i and cos i may be deduced from the measurements
from which they were obtained.
Questions.
1 . In the second method, why is it desirable that the straight
line be exactly in the middle of the wave line?
2. In the third method, what error would be introduced if the cord
from the block was not exactly horizontal ?
46 MECHANICS.
IX. HOOKE'S LAW AND YOUNG'S MODULUS.
Text-book of Physics (Duff), 168, 171, 173; Watson's Physics,
172, 173; Ames 1 General: Physics, pp. 144, 145, 153, 154;
Crew's Physics, 126-129.
H coke's Law states that, for small strains, stress and
strain are proportional. Young's Modulus, E, is the con-
stant ratio of stress to strain for a stretching strain, the
stress being taken as the force per unit cross section and
the strain as the stretch per unit of length, or, if F is the
whole force, A the area of cross section, Lthe whole length,
and / the increment of length,
(A) The quantity most difficult to measure is /, the small
increase of length. If a wire be supported at one end and
force applied to the other end, there is danger that the
support may yield slightly, and a slight amount of
yielding will cause a proportionally large error in
the estimate of the small increase in length. The
peculiarity of the first method described below is
the means adopted to eliminate the yield of the
support. The increase of the length of the wire
under experiment is found by comparison with
another wire under constant stretch attached to
the same support as the former wire. One wire
6 carries a scale and the other a vernier opposite
the scale. If there be any doubt which is vernier
(see p. 13) and which is scale, comparison should
be made with an ordinary steel scale. The screws
FIG. 10. by means of which the wires are clamped to scale
and vernier should be adjusted until scale and
vernier tend to lie in one plane. A light rubber band may
then be slipped over scale and vernier to keep them together.
The stretch may be produced by means of lead weights.
The value of these weights should be determined by a
HOOKE'S LAW AND YOUNG'S MODULUS. 47
platform balance. To produce a suitable stretch it may
be advisable to add two or more weights at a time. We shall
suppose that two are added, but the description can readily
be modified to suit any number. The greatest weight
should not be more than half that required to break the
wire. (A copper wire o.oi sq. cm. section will break at 40
kgs. ; brass, 60 kgs. ; iron, 60 kgs.) Suppose, then, two
weights are added at a time and each stretch observed.
When the maximum number has been added the weights
should be removed in the same order, readings being again
taken as they are removed. The whole series of observa-
tions should be repeated at least three times. Such readings
should always be arranged in tables having in a line or
column all the readings for a particular pair of weights.
The length of the wire may be measured by means of a
long beam compass and the diameter should be measured
at least a dozen times at different places and in different
directions by means of a micrometer caliper (see p. 14).
Before calculating, the dimensions should be expressed in
centimeters and the weights in dynes. First find the mean
value of / for each pair of weights when added and when
removed and then the value of F -=- / for each of these values
of / and the respective F's. Find the mean value of F + l
and the greatest percentage deviation from the mean. This
will give the percentage deviation from Hooke's Law since
F-r-l should be a constant, A and L being practically con-
stant. The final value of Young's Modulus should be
stated in the notation explained on page n.
The possible errors of the different quantities measured
may be taken as the mean deviation in each case. The per-
centage error of the final value of E will be, as is readily seen
from the formula, the sum of the percentage errors of F, L, I,
and twice the percentage error of the radius (see p. 6).
(B) Young's Modulus may also be found by means of
the flexure of a bar. For, in bending (within limits) one
side of a bar is stretched and the other compressed (nega-
tively stretched) , and so Young's Modulus is the only constant
4 8
MECHANICS.
that need be considered. The amount of bending might be
deduced from the sag of the center or end of the bar, but a
much more delicate method is the following optical one:
A bar of rectangular cross section is laid on two knife-
edges and at each end is attached an approximately vertical
mirror in mountings that admit of considerable adjustment.
A vertical scale, nearly in line with the bar, is reflected
from the farther mirror into the nearer and thence into a
telescope also nearly in line with the bar. When a weight
is attached to the center of the bar, the bar is bent and
another part of the scale is reflected into the telescope.
This arrangement serves to determine the angle of bending.
For suppose the difference of the scale-readings on the
horizontal cross-hairs of the telescope be D cms. (Fig. n)
and let the distance between the two mirrors be p and the
distance of the scale from the farther mirror q, then, if the
change of inclination of each mirror be i,
D
"f Q n * _
Lctll l>
For a consideration of figure 11 will show that d l = p tan zi;
But since i is a small angle tan 22 = 2 tan i and tan
., = q tan 42.
^ = 4 tan i.
tan
FIG. ii.
From tan i, the weight R in dynes applied at the center, the
length of the bar between the knife edges, /, the breadth, b,
and the thickness, a, Young's Modulus, E, is obtained, by
the equation
a s b tan i
HOOKE'S LAW AND YOUNG'S MODULUS.
49
PROOF.
Let the y axis coincide with the radius from the center of curvature
of the bar to the center of the bar, and let the x axis be the tangent
to the central axis at this point. Designate distances from the elas-
tic central axis, LOM (Fig. 12), along other radii by z. The elastic
central axis remains unchanged in length. The curvature at any
point P on LOM is the rate of change of the directions of the tangent.
The angle the tangent line at P makes with the #-axis is a small one
and may be taken as dy/dx (which is really the tangent of that
angle) . The rate of change of the direction of the curve at the point
x, y, is therefore d 2 y/dx' 2 , which therefore equals the curvature. But
the curvature also equals i/r, r being the radius of curvature. Hence
Now consider two strips of the beam distant z from LOM. By
the bending these strips are changed in length in the proportion z/r or
2 d 2 y/dx 2 . (For, consider figure 13; the proportional change of
length is
G'H'-GH
GH
FIG. 12.
FIG. 13.
By definition of Young's Modulus, if a force F applied to a rod of
cross section A and length L produce an extension /,
EAl
F =
L '
where E is Young's Modulus. The stress in a strip of width b and
thickness dz is obtained by putting bdz for A and z d 2 y/dx 2 for l-i-L,
which gives
Hence the moment about P of the restoring force in the strips z is
50 MECHANICS.
The moment about P of the stress in the whole cross section is the
integral with reference to z of the above expression for values of z
from o to \a or
For equilibrium this must equal the moment of %R about P or
d 2 y^ 6R n
dy 6R \lx x 2
' dx Ea 3 b( 2 2 j
At a point of support
Hence by substitution
Rl 2
4 a 3 6 tan i'
(By integrating again, the value of y at a point of support or the
deflection of the beam is obtained. This is left as an exercise for
the student.)
The adjustment of the apparatus is most readily made
as follows. Place the telescope and scale nearly in the line
of the mirrors and, glancing above the telescope, set the
farther mirror so that the nearer mirror is seen by reflection
and then the latter so that the scale is seen. Then adjust
the eye-piece of the telescope so that the cross-hairs are as
distinct as possible and finally focus the telescope until the
scale is seen. The bar must not be strained beyond the
limits of elasticity. For adjustment of the telescope and
scale, see p. 25. Equal weights, perhaps 100 grams at a
time, should be added, but this process should be stopped
when it is found -that the scale-reading no longer changes in
the same proportion as the weights. Determine carefully
by several readings, with and without this maximum weight
attached, the change of scale-reading. The width and
thickness of the bar may be measured by a micrometer
caliper (see p. 14), a number of readings of each at different
points being made.
In calculating, use this weight for R and the average of
the changes of deflection for D and take the mean deviation
RIGIDITY. 51
as the measure of the possible error of D, a, and b. The
percentage possible error of tan i is deduced from the possible
errors of D and 2p+4q. The possible error in the latter
term is twice the possible error in p plus four times that in q.
(C) A simple optical method may also be employed for
finding the extension of a wire. In this method, one side
of a small bench carrying a vertical mirror is supported by
the end of the wire and the other by a fixed bracket. The
deflection of the mirror when weights are added to the wire
is read by a scale and telescope. The details of the method
may readily be worked out by anyone who has followed the
preceding methods.
(D) (Searle's Method.) The extension may also be deter-
mined from the change of position of a level supported by
the two wires. The lowering of the stretched wire is com-
pensated by a micrometer screw which therefore reads the
extension. For details, see Watson's Practical Physics, 45.
Questions.
1. How closely is it worth while to measure the length of the wire
in the first method ?
2. Which of the first two methods is the more accurate and what
is the chief weakness of the other ?
3. In the second method why is nothing said as to the distance
of the mirrors beyond the knife-edges? Might they be placed inside?
4. Reduce your results to tons and inches.
X. THE RIGIDITY (OR SHEAR -MODULUS).
Text-book of Physics (Duff}, 119, 170; Watson's Physics, 171, 174,
175; Ames' General Physics, pp. 151153; Crew's Physics, 131,
132; Duff's Mechanics, 117, 130, 131.
The rigidity of any material is the resistance it offers
to change of shape without change of volume. It is meas-
ured by the ratio of the shearing stress to the shear pro-
duced. In the twisting of a wire or rod, within moderate
limits, there is no change of volume. Hence this affords
a means of finding the rigidity of the material. The con-
stant or modulus of torsion of a particular wire is the couple
52 MECHANICS.
required to twist one end of unit length of the wire through
unit angle, the other end being kept fixed. If it be denoted
by r and if the length of the wire be L the couple required to
twist the wire through unit angle is r/L. If now to the wire
be attached a mass of moment of inertia, /, and the wire
and the mass be set into torsional vibrations, the time of a
semi-vibration is, by the principles of Simple Harmonic
Motion (see references) ,
If t, L and / be found r can be deduced. From the modulus
of torsion of the particular wire the rigidity n of the material
of which the wire consists can be deduced ; for
2T
Proof.
Suppose unit length of the wire to be twisted through unit angle.
The vibrations are due to the restoring couple at the lower end pro-
duced by the twist. Let the cross section of the end be divided into
concentric rings and let the radius of one ring be x and its width dx;
its area is z-xxdx. Relatively to the fixed end it is displaced through
unit angle. Hence the linear displacement (supposed small) of the
ring whose radius is x is x times unit angle or simply x. This is by
definition the shear and hence the shearing stress is nx. This is
the restoring force per unit area of the cross section. Hence the
restoring force of the ring whose radius is x is 2xnx 2 dx. The effect
of this force in producing rotation depends on its moment about the
axis or 2nnx*dx. The moment of the restoring force of the whole end
section is the sum of expressions like 2xnx 3 dx for values of x between
o and r or ^nnr*. This is by definition the modulus of torsion, r and
gives us the above equation. It should be noticed that it is not a
constant for the material of the wire, but depends on the dimensions
of the particular wire.
The length, L, may be measured by means of a long
beam compass which is afterward compared with a fixed
brass scale. The radius, R, may be measured by a microm-
eter caliper (see p. 14), measurements being made at a great
many different places and the mean taken.
The moment of inertia, /, of the wire and attached mass
RIGIDITY. 53
might be roughly obtained by calculation, but it is better
to apply an experimental method that is used in other cases.
This consists in adding to the vibrating mass, of unknown
moment of inertia, another mass of such regular form that
its moment of inertia can be accurately calculated, and
finding the times of vibration before (t) and after (T) adding
this mass. If the original moment of inertia be / and the
added moment of inertia i:
whence
One of the simplest forms of added inertia is that of a solid
cylinder of circular cross section vibrating about an axis
through the center of the axis of the cylinder and at right
angles to it. The vibrating mass may then be in the form of
a hollow cylinder in which the solid cylinder may be placed.
If / be the length and r the radius of the solid cylinder of
mass m:
/I 2 r*\
=m ( +- .
\ia 47
The quantities / and r can be obtained with sufficient pre-
cision by measurement with a steel scale divided to mm.'s,
and m may be found by a platform balance. In the above
formula for i, it is assumed that the axis of rotation is per-
pendicular to the axis of the cylinder. That this may be so
the carrier must be carefully leveled. This may be done
by supporting close under it a rod that is carefully leveled
by a spirit-level and comparing the carrier as it swings
with the leveled rod. The end of the vibrating mass should
be provided with an index, such as a vertical needle. A
stationary vertical wire is placed in front of this index
when the latter is in the position of rest. The body is set
54 MECHANICS.
vibrating through an angle of between 60 and 90, all
pendulum vibrations being carefully suppressed.
The time of vibration may be found by much more accurate
methods than simply timing a certain number of vibrations.
The most common methods for accurately timing vibrations
are the "method of coincidences" and the "method of
passages." The former is especially useful for finding the
time of vibration of a pendulum whose half period is approx-
imately one second (Exp. VII). The method of passages
will be found suitable for the present experiment. It con-
sists in noting as accurately as possible the time of every
nth passage of the vibrating system through its mean posi-
tion or position of rest. The value to be chosen for n is a
matter of convenience when two observers work together, one
counting the seconds and the other noting the passages, or
when a single observer has a chronometer in front of him.
But a single observer noting time by a clock circuit and
sounder should choose for n an odd number such that n
semi- vibrations occupy a little more than a minute. (It is
supposed that there is a minute signal, such as the omission
of a tick; see page 25.)
The passages are observed as follows: After a minute
signal, the seconds are counted until a passage occurs, for
example, from left to right. The second and fraction of a
second of this passage is recorded. The succeeding passages
in each direction are counted until the minute signal,
after which the seconds are again counted until the passage
occurs from right to left, for which the second and fraction
of a second is recorded. Obviously, if n has been properly
chosen, the passage just recorded is the nth. The succeeding
passages are counted until the next minute ignal, after
which the second and fraction of a second of the 2wth
vibration (from left to right) is recorded.
The following suggestion may aid in counting the seconds
and estimating fractions of a second. The observer should
keep counting seconds (not necessarily out loud) along
with the clock; when the number of the second is of two or
RIGIDITY. 55
more syllables, the accent should be thrown on one syllable
whose sound should coincide with the tick; thus, eleven,
thirteen, fourteen, etc., twenty-one, twenty-to?, etc. The
passage will usually occur somewhere between two ticks.
To estimate at what point of time between the two seconds
the passage takes place, the indications of the eye may be
used to reinforce those of the ear. Suppose A (in Fig. 14)
to be the mean position of the index on the vibrating body,
then if B and C be its positions at the fifth and sixth
ticks, respectively, and if BA be six-
'* > tenths of the distance BC, it is evident
< tf'Vv' that the true time of passage is 5.6
seconds. With practice the eye can
become very expert in making such
judgments, and, for the purpose of attaining such skill, the
method should be used from the beginning, although at first
not much reliance can be placed on the judgment.
For simplicity of description we shall suppose that n is 5,
but the proper substitutions must be made if n has any
other value. When the approximate time of 5 vibrations
has been obtained by observing a few passages, all of the
passages need not thereafter be observed in order to ascertain
when 'each fifth passage is due, for this can readily be foreseen
by adding to the time of the last observed passage the
known approximate time of 5 vibrations. Further assistance
is obtained by recording the time of the o, loth, 2oth, etc.,
in a second column, the first column being headed "left
to right" and the second "right to left." In this way such
a record as the following is obtained:
M. S.
l/U JX.
M. S.
JX. LU
M. S.
M.
S
(0) . . . .
(SO) ...-
($)'-
(55)..
(10)
(60)
(15)
(65)
(20)
(70)
(25)
(75)
(30)
(80)
(35)
(85)
(40)
(90)
(45)
(95)
56 MECHANICS.
from which the time of vibration is calculated thus :
M. s. M. s.
(50)- (o) .... (55)- (5) ....
(60) (10) (65) (15)
(70) (20) (75) (25)
(80) (30) (85) (35)
(90) (40) (95) (45)
Mean of 50 vibrations . ...
Final mean of 50 vibrations = . . . .possible error. . . .
Final mean of one vibration = . . . . possible error ....
To find the possible error in the value found for n, first
eliminate r and i from the equation given above and express
n in terms of the quantities observed L, /, T, t R, m. (r 2 / 4
is so small compared with I 2 / 12 that the effect of the possible
error in the former may be neglected.) T and t come in
only in the form (T 2 t 2 ) and the possible error in this may
be found by methods stated on page 5.
Questions.
1. To increase the accuracy of the result, which quantity would
have to be measured more closely ?
2. What sources of error are there other than those referred to in
the text?
XL VISCOSITY.
Text-book of Physics (Duff), 196-198; Watson's Physics, 161;
Ames' General Physics, pp. 139, 168.
A solid has rigidity; that is, it offers a continued resist-
ance to forces tending to change its shape. A liquid has no
rigidity and offers no continued resistance to forces tend-
ing to change its shape; that is, the smallest force if given
time will produce an unlimited change in the shape of the
liquid. But the rate at which a liquid changes its shape
under a given force is not the same for all liquids. Some
liquids change very slowly and are called viscous liquids,
others change rapidly and are called mobile liquids. The
VISCOSITY. 57
action of both can be stated in terms of a property called
viscosity.
The viscosity of a fluid may be defined as the ratio of the
shearing stress in the fluid to the rate of shear. From
this general definition a simpler definition can be readily
deduced. A shear consists essentially in the sliding of layer
over layer and the shearing is the force per unit area re-
quired to produce the shear. Hence we have the following
equivalent definition: "The coefficient of viscosity is the
tangential force per unit of area of either of two horizontal
planes at unit distance apart, one of which is fixed while
the other moves with unit velocity, the space between the
two being filled with the liquid." (Maxwell.)
(A) The flow of liquid through a capillary tube is essen-
tially of the nature of sliding of layer over layer. The
cylindrical layer in immediate contact with the tube remains
fixed or at least has no motion parallel to the axis of the
tube, and the immediately adjacent layer slides over it, the
next layer slides over the second, and so on up to the center
of the tube. (In a tube of greater than capillary bore this
is not so, for there are eddies in the motion. This distinc-
tion is in fact the best definition of the term capillary.)
Thus, if we measure the force causing flow through the
tube and the rate of flow, we shall be in a position to deduce
the coefficient of viscosity of the fluid. In fact, if M be the
mass of liquid of density d that flows in time t, through a
vertical tube of length / and radius of bore r, and if h be the
vertical distance from the level of the liquid in the reser-
voir above the tube to the lower end of the tube, the coeffi-
cient of viscosit is
Proof.
Suppose all the liquid in a capillary tube of length / and radius r
to be solidified except a tubular layer of mean radius x and thickness
dx. If there be a difference of pressure p (per unit of area) between
the two ends, the solid will attain a steady velocity such that the
58 MECHANICS.
viscous i
ends. I
ity that
viscous resistance just equals the whole difference of pressure on its
ends. Hence it follows from the definition of the coefficient of viscos-
} xdx
pxx
Hence, v = r ,
2/73
If q be the volume of the core that flows out per second,
. .
2/7}
Suppose now another layer liquefied. There will follow a further
flow represented by the same expression but with a different value
for x. Let the process be continued until the whole is liquid, then
the whole flow per second, Q, will be the sum of all the values of q
for values of x between o and r. Hence
If the tube be vertical and the flow be due to gravity only, instead of
p we must put gdh. If M be the mass of density d that flows out in
time t,
In the above it was tacitly assumed that the liquid adheres to the
tube without any slip. If there were any slip the outflow would be
increased by it and the above expression would not hold. Poiseuille
and others verified the above formula in all cases, thus showing that
no slip occurs. (A more formal proof of the above equation is given
in Tait's "Properties of Matter," 317).
A piece of capillary tubing should be chosen whose bore
is as nearly as possible circular in section. This can be
tested by examining the ends under a micrometer micro-
scope (see p. 15). If the section is found to be nearly cir-
cular the principal diameters of the bore should be measured.
This should, however, only be regarded as a preliminary
measurement, serving as a test of the circularity of the bore
and a check on the following more satisfactory method.
The mean radius of the bore can be best determined by
weighing the amount of mercury that fills a measured length
of the tube. For this purpose the tube should be first
cleaned by attaching it to the end of a rubber tube, at
the other end of which is a hollow rubber ball, and thus
drawing through it and forcing out a number of times (i)
chromic acid; (2) distilled water; (3) alcohol, and finally
drying it by sucking air through it. Then draw into the tube
VISCOSITY.
59
T
a column of clean mercury and measure its length as accu-
rately as possible by a comparator (see p. 15).
The mass of the mercury should next be ascertained by
weighing it with great care in a sensitive balance (for full
directions see pp. 21-24). The mercury should not be
dropped directly on the scale pan, but into
a watch-glass or paper box placed on the
scale pan. From these measurements and
the density of mercury at the temperature
of observation (see Table VII) the diameter
of the bore is obtained. It may be noted
that since it is r 4 that is used in the formula
for viscosity and r 2 that is obtained directly
from the mercury measurements of the bore
the value of r need not be deduced. The
length of the tube may be measured by the'
comparator as already described.
The tube is then attached vertically by a
rubber connection to a funnel and the mass
of water that flows through the tube in a
given time found by weighing a beaker (i)
empty and (2) containing the water that
has passed. The time is obtained by ob-
serving a clock ticking seconds or a chronometer. It is
evident that the greater the whole time the less the per-,
centage error in time due to errors in observing the time
of starting and stopping, and so, too, the greater the whole
mass the less the percentage error in weight. Hence the
time and the mass should be sufficiently great to make the
percentage errors in them less than those in / and r 4 . To
prevent evaporation from the beaker it should be covered
by a sheet of paper pierced by a hole through which the
tube passes. While the liquid is flowing the temperature of
the water in the funnel should be noted.
The value of h is the mean of its values at the beginning
and end of the flow. These values are best obtained by a
cathetometer (p. 19). For this purpose a very simple form
FIG. 15.
60 MECHANICS.
of instrument may be used. A vertical scale is placed near
the apparatus for viscosity and the cathetometer (a telescope
that may be leveled, movable along a vertical column that
.may be made truly vertical) placed so that its telescope
(leveled to horizontality) may be turned, so that the inter-
section of its cross-hairs coincides alternately with the image
of the water surface and that of the scale. This gives the
level of the surface of the liquid on the vertical scale. The
level of the lower end of the vertical tube is obtained in the
same way, whence h is obtained.
The viscosity of alcohol may be measured by the same
means, particular care being taken to prevent evaporation.
The possible error of the result is readily calculated from
the possible errors of the separate measurements. The
possible error of r is not needed, but that of r 4 should be
deduced directly from the determination of r 2 .
Questions.
i. Could the radius be found satisfactorily by measurements
with a micrometer microscope? Explain.
-2. What mass of this liquid would flow through a tube i mm. in
diameter and i meter long, under a constant head of 2 meters ?
3. Two square flat plates of 20 cm. edge are separated by i mm. of
this liquid. What force would be required to move one with a
velocity of 30 cm. per second, the other being at rest?
XII. SURFACE TENSION.
Text-book of Physics (Duff), 206214; Watson's Physics, 155160;
Ames' General Physics, pp. 182-190; Crew's Physics, 149-160;
Poynting and Thomson, Properties of Matter, Chap. XIV.
The height to which liquid rises or is depressed in a
capillary tube depends on the surface tension of the liquid,
the angle of capillarity, and the radius of the tube. From
measurements of the height, h, and radius, r, the surface
tension is deduced if the angle of capillarity is known, for
(see references)
rdgh
T =
2 cos a
SURFACE TENSION. 6 1
d being the density of the liquid and g the acceleration of
gravity. In the case of perfectly pure distilled water the
angle of capillarity a or the angle at which the surface of
the liquid meets the glass, is zero and so cos a= i.
It is important that the capillary tube be quite clean.
The cleaning should be performed with chromic acid and
distilled water. The height of the water in the tube can be
measured in two ways. One method is to place a scale
etched on mirror glass behind the tube. The mean level
of the meniscus-shaped surface of the liquid in the tube
and the ordinary plane surface of the liquid in the vessel
should be read. A preferable method is to measure the
distance between the two surfaces with a cathetometer
(see p. 19).
To make certain that the inner surface of the tube is
wet by the water and that the angle of capillarity is zero,
the tube should be thrust deeper into the liquid and then
withdrawn before the levels of the surfaces are read. This
should be repeated and the height read several times,
different parts of the scale being used, but the part of the tube
in which the liquid rises remaining the same. If the motion
of the liquid in the tube is sluggish or uncertain, the tube
should be more carefully cleaned. Finally, the point to
which the liquid rises in the tube should be marked on the
tube by a sharp file.
The tube should then be carefully broken at the point
marked and its diameter should be carefully measured by
means of a micrometer microscope (see p. 15). If the
section of the bore is not circular, the greatest and least
diameters should be carefully measured and the mean
taken, but if they differ very much the result will not be
satisfactory.
The whole should be repeated with as many tubes of
different sizes as time will permit. The temperature at
which the work is performed should be stated.
If time permit, determine also the surface tension of an
assigned solution.
62 MECHANICS.
Make an estimate of the possible error for the results
obtained by one of the tubes.
(Apparatus for determining the surface tension at different
temperatures is described in Findlay's Practical Physical
Chemistry, p. 78; and E well's Physical Chemistry, p. 117.)
Questions.
1. Are the errors of measurement sufficient to explain the dif-
ferences between results with different tubes?
2. What other sources of error may there be?
3. How could the surface tension of mercury be obtained in an
analogous way?
4. How high would this liquid rise in a tube o. i mm. in diameter?
HEAT.
25. Radiation Correction in Calorimetry.
Watson's Practical Physics, 82; Ostwald's Phys. Chem. Meas., p.
124-127; Poynting and Thomson, Heat, Chap. XVI.
A body which is above the temperature of surrounding
bodies falls in temperature at a rate that is proportional to
the excess of its temperature above that of its surround-
ings. This is Newton's Law of Cooling* If the mean
excess of the body's temperature in any time be known and
also its rate of loss of temperature at some particular ex-
FIG. 16.
cess, its mean rate of loss of temperature is readily deduced,
and this multiplied by the time for which the mean is taken
will give the whole loss of temperature.
Consider the case of the heating of a vessel containing
water by the passage of steam into the water. If a curve
(Fig. T 6 from o upwards) showing the rise of temperature of
63
64 HEAT.
the water be drawn and the same continued after the water
has reached its highest temperature, the latter, or straight
line part of the curve, will give the rate of loss of temperature
at the highest temperature attained. Let us denote this
rate by r (degrees per minute) . If the excess of temperature
when the temperature is highest is t and if the mean excess
during the whole time of rise of temperature is i' then the
mean rate of cooling was by Newton's Law rt f \t and this
multiplied by the whole time of rise of temperature, T (min-
utes), gives the whole loss of temperature. Hence the final
(highest) temperature must be corrected by addition of rTt'/.t.
If the curve of rffee of temperature is a straight line, t' is half
of t and the correction is rT/2. When the curve is not
approximately a straight line the whole time T must be
divided into a number of intervals (each perhaps of 30 sec.)
and t f must be obtained by averaging the excesses in these
intervals.
When the calorimeter is cooled below the temperature
of the room (e. g., by adding ice to the water) the calor-
imeter gains temperature by radiation from the surround-
ings; but the above method will still apply except that we
shall have to do with rates of warming instead of rates of
cooling and the correction of the final temperature will be
subtractive.
If the calorimeter starts below the temperature of the
room and is heated above it, the correction must be made in
two parts as above. In this case we must find the initial
rate of warming (before the hot body is placed in the calor-
imeter) and also the final rate of cooling (after the highest
temperature was attained). The correction will also be in
two parts when the calorimeter starts above the room tem-
perature and ends below it.
If the main rise of temperature is closely represented by
a straight line, it is easily shown* that the correction amounts
to the algebraic average of the initial and final rates multi-
plied by the whole time that the calorimeter is heating or
* Ewell's Physical Chemistry, p. 84.
THE BECKMANN THERMOMETER. 65
cooling. In fact, if the water is T l minutes below the
temperature of the room and T 2 minutes above the room
temperature, the radiation correction is (T 2 r 2 T 1 r 1 )/2 and
this differs from (T i +T 2 )(r 1 +r 2 )/2 by (T l r 2 T 2 r l )/ 2, which
is zero, since under these circumstances the rate is propor-
tional to the time that the water is above or below the room
temperature. This fact is particularly useful in cases where
the surrounding temperature is indefinite (Exp. XXVII, for
example) .
In every calorimetry experiment where the temperature
changes, this radiation correction must be applied, and there-
fore the initial and final rates of change of temperature must
be determined. The rate may usually be found with sufficient
accuracy by reading the temperature every minute for five
minutes. In very accurate work, more careful methods
must be applied.
26. The Beckmann Thermometer.
Watson's Practical Physics, 102; Findlay, Practical Physical Chem-
istry, pp. 114117; Ostwald, Phys, Chem. Meas., pp. 119120.
The Beckmann thermometer is used for determining
changes in temperature. The bulb is large and the stem is
small so that a small change of temperature is shown by a
large change in reading. The amount of mercury may be
varied, and the temperature corresponding to a particular
reading will vary with the amount of mercury in the bulb
and stem. There is a reservoir at the end of the stem into
which surplus mercury may be driven by warming the bulb.
A gentle jar will detach the mercury in this reservoir when
sufficient has been expelled. If one desires to study high
temperature changes, the bulb is warmed until the thread
of mercury extends to the reservoir, when the mercury in
the reservoir is joined to it. The bulb is then allowed to
cool until sufficient mercury has been drawn over, when the
thread is detached from the mercury in the reservoir by a
gentle jar. Several trials are often necessary before the
5
66 HEAT.
proper amount of mercury is secured. In an improved type
of Beckmann thermometer, two reservoirs are provided, and
the first has a scale which tells the amount of mercury
required in that reservoir for different ranges of temperature.
Beckmann thermometers are delicate and expensive and
must be handled with the greatest care.
XIII. THERMOMETER TESTING.
Watson's Practical Physics, 59-69; Edser, Heat, pp. 23-36; Text-
book of Physics (Duff), pp. 189, 190; Watson* s Physics, 177-
182; Ames' General Physics, pp. 220-224; Crew's General Physics,
249-252.
The readings of a thermometer gradually change for a
long time after the thermometer has been filled. The cause
of this is the gradual recovery of the bulb from the effect
of the very great heating to which the glass was subjected
when the thermometer was made. The shrinkage is rapid
at first and slower afterward, but may continue for years.
Hence the necessity for re-determining, from time to time,
the so-called "fixed points" of a thermometer, namely, the
reading in melting ice, and that in steam at standard, pres-
sure. When the thermometer is first graduated it is usually
done by determining the fixed points and dividing the
distance between them into 100 equal parts laid off on the
stem. This assumes that the bore is uniform or that, by
calibration of the bore, the variations of the bore are deter-
mined and allowed for in a table of corrections to be applied
to the readings of the thermometer in order to obtain the
true temperature. Usually the variations of the bore are
too small to have any appreciable effect except in cases
where extreme accuracy is aimed at. Nevertheless, every
thermometer needs to be carefully examined in this regard.
Let us suppose that on the scale laid off on the stem the
true readings in ice and steam have been obtained and for
the moment let us suppose that the bore is quite uniform.
To see how to make corrections for other points on
the scale we must consider the elementary definition of
temperature.
Temperature on the mercury scale is defined by the
expansion of mercury (relatively to glass). Let f ]00 be
68 HEAT.
the volume of a mass of mercury at the temperature of
steam under a pressure of 76 cm. and let V Q be its volume
at the temperature of melting ice. The degree is defined
as the rise of temperature that would produce an expansion
of (^ 100 ^o)/ IOO > an d T above zero is, therefore, the rise of
temperature that will produce an expansion of T(v 1Q<) v ) / 1 oo.
Hence if at T the volume of the mercury be v,
v v f
IOO
.'. T ' = 100.
This definition depends only on the expansion of mercury
and the expansion of the particular glass used and is other-
wise independent of the size and shape of the thermometer.
Now regard the thermometer tube under test as simply a
graduated cylinder of constant cross section containing
mercury. Let the height of the mercury as read on the
scale when the thermometer is in melting ice be a; when it is a
steam at 76 cm. let it be b, and when it is at the temperature
T, let it be t. Then
v v ta
Hence
J. \V \MJ ,
b a
where T is the true temperature when the reading of the
thermometer is t. By this equation values of T for values
of t for every five degrees should be calculated. Having
thus drawn up a table of true temperatures we subtract the
scale-reading from the true temperature and thus get a
correction (positive or negative), which added to the scale-
reading gives the true temperature.
This is on the assumption that the bore is sensibly uni-
form. The only quite satisfactory method of testing this
THERMOMETER TESTING. 69
is to calibrate the bore by measuring the length of a short
thread of mercury at different positions in the tube. This
process requires considerable time and the following will
usually suffice: Two thermometers for which tables of
true readings have been drawn up as above, are compared
at regular intervals (say every five degrees) between zero
and 100 by being used simultaneously to measure the tem-
perature of a body. If, after corrections, the readings of
the thermometers are not sensibly different, this shows that
the bores of both must be practically uniform. If they do
differ appreciably, then the bore of one or both must be
variable. If they be compared with a third thermometer,
the one with the variable bore will be detected and it must
be then calibrated.
Testing Zero-point. A calorimeter consisting of a small
copper vessel inside of a larger is suitable for holding the ice.
Both vessels should be washed in ordinary tap water.
The space between the two vessels should be filled with
cracked ice, and the inner vessel filled with cracked ice and
then distilled water poured in until the vessel is filled to the
brim. The thermometer having been washed clean, is
inserted in the inner vessel, just sufficient of the stem being
exposed to admit of the zero being observed. When the
reading has fallen to i the reading should be observed
every minute until it is stationary for five minutes. This
stationary temperature, read to o . i degree, is the true zero
point, or a in the above equation.
Sources of Error.
(1) Impurity in the ice or water.
(2) The presence of water above o near the bulb of the
thermometer.
Testing Boiling-point. The form of boiler used for this
test consists of a vessel for boiling water surmounted by a
tube up which the steam passes, this tube being enclosed in
another down which the steam passes to an exit tube and a
pressure gauge (see Fig. 17). Half fill the lower part of the
vessel with water. Push the thermometer to be tested
HEAT.
]
through a cork in the top until the boiling-point is only a
degree or two above the cork, but take care that the
bulb of the thermometer does not reach down to the
water. Apply heat, adjusting it carefully as boiling begins,
so that the pressure inside, as indicated by the pressure
gauge, shall not materially ex-
ceed atmospheric pressure.
Some excess is, of course neces-
sary, if there is to be a free
flow of steam. What excess is
permissible may be deduced from
the consideration that a rise of
pressure of i cm. (of mercury
column) corresponds to a rise of
boiling-point of 0.373 (see Table
XIII). If water is used in the
pressure gauge, a pressure of i
cm. of water column would cor-
respond to only 0.03 rise of
steam temperature. If the ther-
mometer be graduated to degrees
only, an error of 0.03 in finding
the boiling-point is negligible.
Read the barometer and reduce
the height to zero degrees (p. 21).
To the boiling-point thus
found a correction must be ap-
plied, for the difference between the atmospheric pressure
at the time and that of a standard atmosphere (76 cm. of
mercury). Find from Table XIV the true temperature of
the steam at this pressure, and the difference between the
boiling-point observed on the thermometer and this tem-
perature. Since this temperature is always within a few
degrees of 100, the thermometer will have practically the
same error at 100. Therefore b in the above equation
may be taken as 100 plus or minus the difference between
the observed boiling-point and the true boiling temperature.
FIG.
TEMPERATURE COEFFICIENT OF EXPANSION. 71
Comparison of Two Thermometers. The most satisfac-
tory method is to immerse the thermometers in steam above
water boiling under a pressure that can be regulated. A
simple means that is sufficient if the thermometers are of
the same length and graduated to degrees only, is to use
the thermometers simultaneously to find the temperature of
a block of good conducting material (copper or brass) im-
mersed in a vessel of water the temperature of which can
be gradnally raised by a burner. The thermometers should
be thrust in holes close together in the block and before
each reading the burner should be removed and the water
well stirred for a minute so that the temperature of the
block shall become uniform.
Questions.
1. Which should be determined first, boiling-point or freezing-
point, and why?
2 . How much error might there be in determining the boiling-point
if only the bulb were immersed in the steam?
3. Why is there no need to take account of barometric pressure
in finding the zero-point?
XIV. TEMPERATURE COEFFICIENT OF EXPANSION.
Edser, Heat, pp. 39-61; Text-book of Physics (Duff}, pp. 194-197;
Watson's Physics, 184, 185; Ames' General Physics, pp. 229
233; Crew's General Physics, 263-265.
For measuring the thermal expansion of a body, choice
may usually be made from a variety of methods. The par-
ticular method chosen will depend on the form of the speci-
men. The expansion of a metal rod may be measured by
means of a spherometer or by means of two reading micro-
scopes focused on definite marks near the ends of the speci-
men. The expansion of a wire is best measured by an opti-
cal lever method. The expansion of a solid of irregular
form can be found by a hydrostatic method, namely, by
weighing it in a liquid at different temperatures, it being
supposed that the density of the liquid at different tempera-
tures is known.
HEAT.
(A) Expansion of a Metal Rod.- The rod is supported
at the lower end on a firm point and is heated by being en-
closed in a tube through which steam is passed from a
boiler. A spherometer (see p. 16) is so supported that the
end of the screw can be brought down on the flat end of the
rod. The spherometer is supported in the
hole, slot, and plane method, so that its
position is definite and not liable to be dis-
turbed by thermal expansion of the sup-
porting surface.
The rod is first measured by means of an
ordinary meter scale divided to mms. It is
then accurately placed in position in the
heating tube, the end of the rod projecting
through corks. Through the cork at the
upper end should also pass a glass tube for
the entry of the steam, while a similar tube
at the lower end serves to drain off the water.
At least six readings of the spherometer
scales should be made at the room tempera-
ture. Then pass steam into the jacket
about the rod. Every few minutes read the
temperature of the interior as given by two
thermometers at different heights and read
the spherometer. When the temperature
has become constant, make at least six read-
ings of the spherometer and several readings
of the thermometer. Always estimate tenths
of the smallest division. From the differ-
ence in spherometer readings, the length, and the change
in temperature, calculate the coefficient of expansion.
(B) Expansion of a Wire. For this an optical lever
method is most suitable. A mechanical lever or system of
levers is sometimes employed for magnifying small mo-
tions. A ray of light reflected from a mirror that is tilted
by the expansion serves' the purpose of a long index arm
much better, inasmuch as it has no weight itself and may
FIG. 18.
TEMPERATURE COEFFICIENT OF EXPANSION. 73
be taken as long as we wish. The wire is hung vertically,
the lower end being solidly clamped, and the upper end
carrying a sleeve on which rests one leg of a small three-
legged bench, on which a mirror is mounted. The other
two legs rest on a fixed bracket. The wire is enclosed by a
tube through which a current of steam is passed from a
boiler and into which two thermometers are thrust to read
the temperature. A drainage tube at the lower end allows
the escape of water. The wire is prolonged above the mir-
ror and is attached to a spring by which the wire is kept
stretched. The image in the
mirror of a vertical scale is _ __ _____ ---- T
observed by a reading telescope r"~~ ---- - ------
(see p. 25 for adjustments), ^fr-lL^
and the change of reading, d,
on the horizontal cross-hairs
of the telescope, produced by the expansion is noted. Let
the length of the wire between the clamp and the support
of the bench be /, and let the length of the bench between the
point of the movable leg and the line of the other legs be a.
Let the distance of the scale from the mirror be L, and the
change of temperatures be (t 2 /J . Then, remembering that
a ray of light reflected from a mirror turns through twice the
angle that the mirror turns through, it is easily seen from
the figure that the expansion is ad/2L and the coefficient
of expansion is
ad
The most difficult quantity to determine with a high degree
of precision is a. It may be measured by means of a mi-
crometer microscope or a dividing engine (see p. 17). A
simpler and more accurate method is to place the optical
lever so that the movable leg is on the (vertical) screw of
a micrometer caliper (p. 14), while the other legs are on
a fixed support and then focus the telescope and scale on
the mirror. When the screw is turned the movable leg
74 HEAT.
is raised a known amount. From this, the distance between
the mirror and the scale, and the scale-readings, a is deduced.
This calibration may be avoided by placing two legs
of the mirror bench upon the collar attached to the wire
and resting the third leg directly upon the micrometer screw.
The extension may also be measured by Searle's combina-
tion of level and micrometer screw (see end of Exp. IX).
XV. COEFFICIENT OF APPARENT EXPANSION OF A
LIQUID.
Edser, Heat, pp. 64-71; Text-book of Physics (Duff), pp. 198-201;
Watson's Physics, pp. 211-213; -Awes' General Physics, pp. 233-
235; Crew's General Physics, 266, 267.
The object of this experiment is to determine the coeffi-
cient of apparent expansion of some salt solution with refer-
ence to glass. A vessel holds M grams of liquid at t and
m grams at a higher temperature, t'. Let V be the volume
of the vessel at the lower temperature. Since we are
considering the apparent expansion, i. e., the expansion
with reference to the vessel, we may consider V to be also
the volume of the vessel at the higher temperature. The
volume of i gram at t is therefore V/M and at *', V/m.
The increase in volume is
m M \ Mm
The coefficient of apparent expansion, e, is this apparent
expansion divided by the original volume V/M and the
range of temperature (t f t) or .
M-m
m(t'-t)'
A glass bulb with a re-curved capillary stem is used.
To fill the bulb with a liquid, warm it with the hand or by
playing a flame about some distance beneath it. Remove it
from the source of heat and plunge the end of the stem into
the liquid. As the air in the bulb cools liquid will be drawn in.
COEFFICIENT OF APPARENT EXPANSION OF A LIQUID. 75
To expel liquid, warm the bulb gently, keeping it so turned
that the stem is filled with the liquid ; when the liquid ceases
to come out, invert it so that the stem is highest, and allow
it to partially cool. Repeat until all the liquid is expelled.
Clean the bulb by drawing in a little distilled water, or,
if the interior be foul, first use chromic acid. Finally rinse
the interior with alcohol. Remove the alcohol and dry the
interior, if necessary playing a flame about some distance
beneath.
To determine the density of the (cold) salt solution,
thoroughly cleanse a tall measuring glass and a suitable
hydrometer (variable immersion). Pour enough of the
salt solution into the measuring glass to float the hydrometer,
read the density, and pour the solution back into the bottle.
Weigh the bulb very carefully on a sensitive balance
(see pp. 21-25). Support the bulb in a clamp stand, clasping
the stem between half corks. Fill a small beaker with the
salt solution and support it so that the end of the stem dips
into the solution. Warm the bulb, playing a Bunsen flame
beneath. Never allow the flame for an instant to remain
stationary beneath the bulb, and until the bulb contains con-
siderable warm liquid, do not allow the flame to touch the bulb,
and then only where there is liquid. Alternately warm the
bulb and allow it to cool a little until the bulb is filled.
When it is partly full 1 it may be best to gently boil the liquid
in the bulb. When the bulb is almost full the liquid can be
made to expand to fill the entire stem. Then allow it to
cool completely while it draws over liquid from the beaker.
When the bulb is cooled to the temperature of the room,
support it in a copper vessel in which water is kept at a
constant temperature, a few degrees warmer than the room.
When the temperature has been kept constant for five
minutes (by the addition of small amounts of hot or cold
water, if necessary) and has been frequently stirred, read
the temperature (as always estimating tenths). Remove
any liquid adhering to the end of the stem, remove the bulb
from the bath, dry the exterior, and weigh. Handle the
76 HEAT.
bulb carefully with a cloth about it so that no liquid may
be expelled. Weigh a small, clean, dry beaker. Support
the bulb again in the copper bath with the beaker beneath
the end of the stem, to catch any liquid expelled. Heat the
water in the bath to boiling. When the temperature has
been constant for live minutes, read the temperature, catch
on the side of the small beaker any liquid adhering to the
end of the stem, remove the bulb from the bath, dry the
exterior, and weigh. Weigh the small beaker with the
liquid contained. Carefully remove the liquid from the
bulb and stem as described above.
The difference between the two weights of the bulb when
filled with liquid gives the weight M m of liquid expelled.
The difference between the weight of the flask dry and after
being in the second bath gives the final weight of liquid in the
bulb. The expelled liquid is saved simply as a check and is
not used at all if the above difference be slightly greater. 1
A specific-gravity bottle may be substituted for the bulb,
but is not as satisfactory.
Questions.
1. Why is double weighing unnecessary?
2. Why is M m determined more accurately from the difference
of the two weighings than from the weight of the liquid expelled.
3. How might the coefficient of expansion of a solid, attainable
only in the form of small lumps, be found by an extension of this
method ?
4. How might the absolute expansion of a liquid be found by the
above apparatus?
XVI. COEFFICIENT OF INCREASE OF PRESSURE OF
AIR.
Edser, Heat, pp. 106 in; Poynting and Thomson, Heat, pp. 45-49;
Text-book of Physics (Duff), pp. 188, 203-205; Watson's Pracr
tical Physics, 78, 79; Watson's Physics, 195198; Ames'
General Physics, p. 240; Crew's General Physics, 269.
If the volume of a mass of gas remains constant while
its temperature is raised, its pressure increases according
to the law
COEFFICIENT OF INCREASE OF PRESSURE OF AIR. 77
in which P is the pressure at o C., P the pressure at the
temperature t, and a is a constant called the coefficient of
increase of pressure. If the pressure were kept constant
and the volume allowed to increase, the law of increase of
volume would be similar, and it is found that the constant
a is practically the same in both cases.
It is, however, difficult to keep the volume exactly con-
stant, for the containing vessel will expand when heated
(the volume of the vessel would also increase because of the
increase of pressure to which it is subjected, but this may
be neglected since it is extremely small). If p is the ob-
served pressure at temperature / and P the observed pres-
sure at o
/> = P (i+a'*),
where a' is the coefficient of apparent increase of pressure
(seeExp. XV).
To correct for the expansion of the vessel, we must
suppose the final volume of the gas compressed in the pro-
portion in which the capacity of the vessel expanded. The
law of expansion of the vessel is
V = V Q (l +b t),
where b is the coefficient of cubical expansion of the vessel.
To get the pressure P that would keep the volume of the
gas absolutely constant, we must multiply p by (i +6 t),
.'. P=P (i+a'/)(i+6/)
And so the true coefficient of increase of pressure a is ob-
tained from the apparent coefficient of increase of pres-
sure a', by adding the coefficient of cubical expansion of the
vessel, or,
a = a f + b.
The air (or gas) is enclosed in a bulb to which is con-
nected a mercury manometer. The pressure indicated by
the manometer is obtained from readings of the mercury
levels on a scale between the two columns, or, preferably,
with a cathetometer (p. 19).
HEAT.
If the true increase of pressure of dry air is desired the
air must first be carefully dried. To fill the bulb with dry
air it may be connected through a drying tube (containing
chloride of calcium) with an air-pump and the bulb several
times exhausted and refilled with air sucked through the
drying tube. (If the bulb be already filled with dry air
the process will be unnecessary.)
The bulb is then connected to the manometer. The
bulb is first immersed in a bath of ice and water as nearly
as possible at o, and the movable column
of the manometer is adjusted until the
mercury in the other column is at a definite
point, as high as possible without entering
the contraction where connection is made
with the bulb. The temperature and pres-
sure are read as carefully as possible, at
least six times, when both have become
quite steady, the manometer being read-
justed before each reading.
The bath of ice and water is now replaced
by one of water at about 10 and the movable column is
readjusted until the mercury in the stationary column is at
the former point, that the volume of the gas may remain
constant. The temperature and pressure are read when
they have become steady. The water is then heated to
about 20 and the observations are repeated. Readings are
thus made at intervals of about i o until the water boils.
The pressure and temperature when the water is boiling
should be read at least six times, the mercury level in the
stationary column being adjusted to the constant point before
each reading. It is at the initial temperature (near o) and
the final temperature (near 100) that the most reliable
observations are obtained, and it is upon these that the
most reliable estimate of the coefficient of expansion is
founded. (The readings at intermediate temperatures are
made in order to test the law of expansion.) If the two arms
of the manometer are of different radii, there will be a con-
FlG. 20.
COEFFICIENT OF INCREASE OF PRESSURE OF AIR. 79
stant difference of level due to capillarity. This should be
read when the bulb is disconnected and allowance should be
made for it at other times. Read the barometer (p. 21) and
the temperature of the barometer and of the mercury in the
manometer.
Tabulate from your observations (a) the temperatures;
(b) the differences in level of the mercury columns; (c) these
differences reduced to zero degrees; (d) the pressures as
calculated from (c) and the barometer heights (reduced to
zero degrees).
The test of the law of expansion is made by plotting
the curve of pressure and temperature, the former as ordi-
nates, the latter as abscissas. This should be nearly a
straight line. The averages for the first point (o) and the
last (about 100) are to be taken as fixing the straight line.
The divergence of intermediate points from the straight line,
while not sufficient to invalidate the conclusion that the in-
crease of pressure is linear, will illustrate the difficulty of
keeping the temperature at intermediate points constant
for a sufficient length of time for the air in the bulb to come
wholly to the temperature of the water.
Calculate from these two average pressures and tem-
peratures, the coefficient of apparent increase of pressure (a') ,
and, obtaining the coefficient of cubical expansion of the
glass (6) from Table VIII, find the true coefficient of increase
of pressure (a) . (Remember that the coefficient of cubical
expansion is three times the coefficient of linear expansion.)
If time permit, increase the range of temperature by
observations below o in a freezing mixture and above 100
in heated oil.
Questions.
1 . Why must (a) the air be dry ? (b) a capillary connect the bulb
and the manometer ?
2. What would be the percentage error if the expansion of the
bulb was neglected?
8o
HEAT.
XVII. PRESSURE OF SATURATED WATER VAPOR.
Poynting and Thomson, Heat, Chap. X; Edser, Heat, pp. 220-228;
Text-book of Physics (Duff), pp. 226-230; Watson's Physics,
216-218; Ames' General Physics, pp. 264-269; Crew' s General
Physics, 279-281.
The object of this experiment is to find the pressure of
saturated water vapor at different temperatures. By pres-
sure of saturated water vapor at a given temperature, or,
as it is often called, maximum pressure of water vapor, or,
equilibrium pressure, is denoted the pressure of
the vapor above water in a closed vessel at the
given temperature after a steady state has been
reached. A liquid continues to give off vapor
from the surface, or, "evaporate," as long as the
pressure of the vapor above the liquid is less
than the saturated vapor pressure, independent
of the total atmospheric pressure above the
liquid. After the pressure of the vapor reaches
the saturated vapor pressure for that tempera-
ture, the total quantity of vapor in the atmos-
phere above the liquid remains constant, since
for any vapor given off from the surface an equal
quantity is condensed.
There are two chief methods of finding the
saturated vapor pressure, the static method and
the kinetic method.
(A) In the static method some water (or other liquid) is
introduced into the space at the top of a barometric column
which is surrounded by a bath, the temperature of which
can be varied. The pressure of the vapor is found by
measuring with a cathetometer (p. 19) the height of the
mercury column and subtracting this from the barometric
reading, each being reduced to zero (p. 21). By varying
the temperature of the bath, the vapor pressure at various
temperatures is obtained.
(B) In the kinetic method the quantity measured is the
FIG. 21.
PRESSURE OF SATURATED WATER VAPOR.
8l
temperature of the steam above water boiling under different
measured pressures. When a liquid boils, bubbles of vapor
are formed throughout the interior of the liquid. In forming
these bubbles, the vapor overcomes the pressure of the
atmosphere above the liquid, therefore the pressure of the
vapor must equal the atmospheric pressure, and obviously
the vapor in the bubbles is saturated. Hence, in measuring
the atmospheric pressure above a liquid boiling at a known
temperature, we find the saturated vapor pressure of the
liquid at this temperature, and this is Regnault's method,
which method is followed in this experiment.
FIG. 22.
In Regnault's apparatus the total pressure above the
surface of the liquid can be kept very constant. As the
liquid is heated, the vapor is condensed in a Liebig con-
denser, and as the pressure of vapor distributed through
several conducting vessels is the vapor pressure correspond-
ing to the vessel at lowest temperature, the pressure exerted
by the vapor cannot exceed the maximum pressure corre-
sponding to the temperature of the tap water, and is there-
fore very small. As the temperature of the boiler changes,
the temperature of the air in the boiler varies, but a large
air reservoir surrounded by water is connected between the
condenser and the manometer and air-pump or aspirator,
which makes the volume of the air in the boiler small com-
6
82 HEAT.
pared with the total volume of air in the system, and thus
the increase of pressure due to the heating of the air in the
boiler is small.
The boiler should be about two-thirds full of water. Fill
with water the small tube running down into the boiler
(which tube is closed at the bottom), and insert in this
tube through a cork one of the thermometers tested by the
observer. Draw out any water which may be in the air
reservoir by means of the stopper underneath. Fill the
surrounding vessel with water. (Rubber stoppers should
be lubricated with rubber grease (note p. 32) before inser-
tion.) Exhaust the air from the system to the highest
vacuum attainable by means of a Geryk pump or aspirator.
Close all the cocks through which connection is made
to the aspirator and let the system stand a few minutes to
see if there is any leakage. If not, start a gentle stream of
water through the condenser, and place a Bunsen flame
under the boiler. Read the barometer and its temperature
(see p. 21).
When the temperature as registered by the thermometer
in the boiler becomes very steady, read it, and at once
record the two extremities of the mercury column of the
manometer. Let in a little air by first opening and then
closing a cock near the air-pump, and then opening and
closing a cock nearer the apparatus. Increase the pressure
at first by about 15 mm., gradually increasing the steps
and when near atmospheric pressure change the pressure
by about 1 2 cm. The reason for the difference in pressure
in the steps is that it is better to have the steps represent
about equal changes of temperature, for instance, about 5.
From the corrected barometer reading and the differences-
in height of the mercury columns, calculate the pressures.
Tabulate pressures and temperatures and also plot them,
making temperatures abscissas and pressures ordinates.
Ramsay and Young's method for measuring the vapor
pressure of a small quantity of liquid is described in Wat-
son's Practical Physics, 94.
HYGROMETRY. 83
Questions.
1. State precisely what two quantities you have observed in the
second method and what relation they bear to the pressure and temp-
erature of saturated vapor.
2. What condition determines whether a liquid will boil or evapo-
rate at a given temperature?
3. What was the actual vapor pressure above the boiling liquid?
(Table XIII.)
4. What determines (a) the lowest temperature, (b) the highest
temperature for which this apparatus is applicable?
XVIII. HYGROMETRY.
Poynting and Thomson, Heat, pp. 209-215; Davis, Elementary Me-
teorology, Chap. VIII; Robson, Heat, 7375; Watson's Prac-
tical Physics, 9597; Text-book of Physics (Duff), pp. 239242;
Watson's Physics, 220, 221; Ames 1 General Physics, pp. 265-
268.
Three methods will be used for studying the hygrometric
state of the atmosphere. The first method (A) determines
the dew-point, the second (B) determines, indirectly, the
actual vapor pressure, and the third (C) determines the
relative humidity.
(A) Regnanlt's Hygrometer. A thin silvered-glass test-
tube is half-filled with ether. The test-tube is tightly
closed by a cork through which passes a sensitive ther-
mometer which gives the temperature of the ether. Two
glass tubes also pass through the cork, one extending to the
bottom, the other ending below the cork. An aspirator
gently draws air from the shorter tube. The ether is
evaporated by the air bubbles and the entire vessel cools.
The silvered surface and the thermometer are watched
through a telescope and the temperature is read the moment
moisture appears on the metal. The air current is stopped
and the temperature of disappearance of the moisture is
observed. This is repeated several times and the mean
is taken as the dew-point. The detection of moisture is
facilitated by observing at the same time a similar piece
of silvered glass which covers a part of the test-tube, but
which is insulated from it. The temperature of the air
84 HEAT.
should also be carefully determined, preferably with a ther-
mometer in a similar apparatus where there is no evapora -
tion.
An arrangement of two small mirrors at right angles
so placed as to reflect light from the two tubes into the
telescope will facilitate the comparison.
(B) Wet and Dry Bulb Hygrometer. Two thermometers
are mounted a few inches apart. About the bulb of one is
wrapped muslin cloth to which is attached a muslin wick
dipping in water. The other is bare. The temperatures
of both are read when they have become steady. The
temperature of the first thermometer will be lower than that
of the bare thermometer, on account of the evaporation of
the water. From the difference of temperature of the two
thermometers and the temperature of the bare thermometer
the actual vapor pressure may be determined with the aid
of empirical tables (see Table XV). For more accurate
apparatus, see references.
(C) Chemical Hygrometer. Fill three ordinary balance
drying vessels with pumice. Saturate two with strong
sulphuric acid and the third with distilled water. Weigh
very carefully the two which have the acid and then connect
them to an aspirator, with the water absorption vessel
between them. After a gentle stream of air has passed
through for a considerable time, disconnect and weigh the
sulphuric acid vessels. The ratio of the gains in weight
will obviously be the relative humidity. Observe also the
temperature of the air.
If not directly determined, calculate from your observa-
tion, by each of the three methods, the dew-point, the
actual vapor pressure, the relative humidity, and the amount
of moisture in the atmosphere per cubic meter. Tabulate
your results. Table XIII gives the vapor pressures of water
at different temperatures.
SPECIFIC HEAT BY THE METHOD OF MIXTURE. 85
XIX. SPECIFIC HEAT BY THE METHOD OF MIXTURE.
Edser, Heat, pp. 122-136; Text-book of Physics (Duff), pp. 208211;
Watson's Physics, 200-201; Watson's Practical Physics, 82-
84; Ames' General Physics, pp. 250-252; Crew's General Physics,
252.
The specfic heat of a substance is the number of calories
required to raise the temperature of one gram of the
substance one degree centigrade, or the number of calories
given up by one gram in cooling one degree centigrade.
In the method of mixture a known mass (M) of the sub-
stance, heated to a known temperature (T), is immersed in a
known mass of liquid (m) of known specific heat (for water
= i), at a known temperature (/ ), and the unknown mean
specific heat (x) of the substance is deduced from these data
and the temperature (t) to which the mixture rises. Water
is the liquid employed unless there would be a chemical
reaction on immersion.
The liquid must be contained in a vessel which is also
heated by the immersion of the hot body. The heating of
the vessel is equivalent to the heating of a certain addi-
tional quantity of water. This equivalent quantity of water
(e) is called the water equivalent of the vessel. It is practi-
cally equal to the mass of the vessel (mj multiplied by the
specific heat (5) of the material of the vessel. Theoretically
it may be obtained by noting the temperature of the vessel
and pouring into it a known mass of water at a known
temperature and noting the final temperature. This is an
inverted form of the method of mixture applied to finding
the specific heat of the vessel. But as we shall presently
see, it is the method of mixture applied under very unfavor-
able conditions and will not usually give a very satisfactory
result. Another method will be recommended below.
The equation for finding the specific heat is obtained by
equating the heat given up by the hot body to that taken up
by the water and containing vessel. Hence
Mx(T-t)=(m+e)(t-t ). (i)
86 HEAT.
Sources of Error.
(1) Loss of heat while the hot body is being transferred
to the water.
(2) Loss of heat by radiation, conduction, or evapora-
tion while the mixture is assuming a uniform temperature.
(3) Errors in ascertaining the true temperature includ-
ing errors in the thermometers.
Choice of Best Conditions. As the accuracy of this de-
termination depends largely on the selection of suitable
conditions, we shall consider how these may be chosen so
that unavoidable errors in the separate measurements may
affect the result as little as possible.
By taking the logarithms of both sides of (i) and dif-
ferentiating partially, we obtain (see pp. 7, 8.)
(2) (3) (4)
f&cl _8M [~S*~| Jm_ |~8*1 Be
[He \M~~W [Hc\m~ m +e [x\e~m+e
(5) (6) " (7)
M * T [H * ry (T-w
[x\T = T-t' UJ'o *-* ' Up (T-t)(t-t Y
The left-hand side of (2) stands for "the ratio that the
possible error (8#) in x, due to the possible error (8M) in
M, bears to x," and so for the other equations.
M and m can be measured with great precision; hence
(2) and (3) are negligible. From (4) it is seen that the
water equivalent of the calorimeter must be found with
some care. From (5) and (6) it is seen that* the ranges
T t and t t Q should be as great as possible (see, how-
ever, "sources of error" above). This is also consistent
with the indications of (7), for although (T t ) enters
the numerator, the product of T t and t t Q is in the
denominator. Moreover, it is seen from (5), (6), and (7)
that if equal errors are made in observing T, t, and t , the
effect of the error in / may equal the sum of the effects of
the errors in T and t . Hence the necessity of determining
t with special care. But, allowing an unavoidable error in
SPECIFIC HEAT BY THE METHOD OF MIXTURE. 87
t, how can its effect be made as small as possible by prop-
erly choosing the quantities, M, m, t , Tf Let us suppose
T / is taken as great as possible under the circumstances.
How can (T t)X (* * ) be made as great as possible? The
sum of T t and / 1 is T t Q , a fixed quantity. Hence their
product is, by algebra, a maximum when they are equal or
t is midway between T and t Q . But it is seen from (i) that
this also requires MX and m + e to be equal. Hence we see
that for the best results, T and t should be as far apart as
possible and the heat capacity of the specimens should be as
nearly as possible equal to the heat capacity of the water and
the vessel that contains it.
The logical procedure, then, would be to roughly de-
termine x by the method of mixture, using any convenient
values of M and m, and with this rough value for x, calcu-
late what ratio of M to m would best satisfy the above con-
dition. Moreover, it is seen from (4) above that m should
be as large as is consistent with other conditions. Then we
should proceed to arrange a new experiment to be per-
formed under the more favorable conditions for precision.
We now see why it is not easy to determine the water
equivalent of the vessel directly. Its heat capacity is small
compared with that of the water that would fill it, and so
the change of the temperature of the water would be small
and difficult to determine accurately. If a much smaller
quantity of water were used, a large part of the surface of
the vessel would be left uncovered, and its temperature
could not be determined. Hence it is better to determine
the specific heat of the material of the vessel, using the ordi-
nary method of mixture and a mass of the same material as
the vessel. Then multiplying the mass of the vessel by its
specific heat, we have its water equivalent.
It is desirable that the body should have such a form
that it and the water in which it is immersed should rapidly
come to a common temperature. Filings, shot, thin strips,
wire or small pieces would best satisfy this condition.
Larger solid masses are more rapidly (and therefore with
88 HEAT.
less loss of temperature) transferred from the heater to the
water, and to give the water ready access to them they may
be perforated with holes, through which, by moving the
mass up and down in the water, the water may be made to
circulate. The following directions apply primarily to this
latter form, but may be readily adapted to the other forms.
Two forms of heater will be here described, (i) The
steam heater. A copper tube large enough to admit the
specimen is enclosed, except at the ends, by an outer copper
vessel which is to act as a steam jacket to the inner vessel.
Steam from a simple form of boiler is admitted to the
jacket through a tube near the top of the jacket and escapes
through an outlet near the bottom. If the body to be
heated is a solid mass, it is suspended in the heater by a long
string that passes through a cork that closes the upper
end of the heater. (If the specimen is in the form of shot
or clippings they are placed in a dipper that fits into the
heater.) A thermometer passed through the cork or cover
of the dipper is pushed down until it comes into contact with
the body tested. The lower end of the heater is also closed
with a cork.
Such a steam heater ultimately brings the specimen to a
very steady temperature, but it has the disadvantage of
heating very gradually. If the boiler which supplies steam
to the jacket has a closed tube extending from the top into
the interior of the boiler, of slightly larger diameter than
the specimen or dipper, either of the latter may be placed
therein and rapidly heated to about the steam temperature,
when they are transferred to the steam heater.
(2) The Electric Heater. A metallic tube is heated by
a strong current of electricity passing through a coil of wire
of high resistance that surrounds the tube. The current
can be varied by changing a variable resistance in circuit
with the heating coil. With an alternating current the
resistance may be an inductive resistance or choking coil
consisting of wire surrounding a soft-iron wire core. A low
resistance allowing a high current is used until the tern-
SPECIFIC HEAT BY THE METHOD OF MIXTURE. 89
perature rises to the desired point (perhaps near 100) and
then the current is reduced to the strength that will keep
the temperature constant, as indicated by a thermometer
hung in the heater. The body is introduced into the heater
exactly as in the case of the steam heater. The proper
method of varying the resistance can only be learned by
some practice.
The two thermometers used should be those for which
tables of corrections have been obtained earlier.
The calorimeter may be prepared while the specimen is
being heated. It consists of a smaller copper or aluminum
can highly polished on the outside and enclosed in a larger
one brightly polished on the inside, but well insulated from
it by corks or cotton-wool. A wooden cover fits over both
vessels and has holes for thermometer and stirrer and an
opening giving access to the interior vessel. A convenient
form has a trap-door which slides open in two halves,
exposing the entire inner vessel. A screen with sliding or,
preferably, double swinging doors, should separate the
calorimeter from the boiler, heater, etc.
The "water equivalent" of the receiving vessel means
the water equivalent of the inner vessel together with the
stirrer, if one be used. It is advisable that the stirrer
should also be of copper or aluminum. (A stirrer is, how-
ever, not necessary when the specimen is in the form of one
large block) .
At certain times, in the manipulation of this experiment
the co-operation of two persons is desirable, and, for
economy of time, two determinations should be made
simultaneously, two heaters, two specimens, and two
calorimeters being used. One specimen should peferably
be of the same material as the calorimeter, so that the water
equivalent may be determined.
The body whose specific heat is to be determined is
weighed to o . i gm. and placed in the heater along with a
thermometer. The inner vessel of the calorimeter (including
the stirrer) is weighed to o.i gm. Water near the tern-
90 HEAT.
perature of the room is poured into it until it is judged that
when the hot body is immersed it will be completely covered
and the water will rise to within a couple of centimeters of
the top of the vessel. The vessel and water are then
weighed to o . i gm. The inner vessel is now replaced in the
outer and the cover adjusted and closed.
When the temperature of the specimen has remained
constant for ten minutes, it may be assumed that the hot
body is throughout at the temperature of the heater. The
next steps require two persons, and as it is important that it
should be carried out promptly and neatly, it should be
carefully considered before being performed. One person
constantly stirs the water in the calorimeter and reads the
temperature, to tenths of a degree, every minute for five
minutes. He then opens the cover and slides the calorimeter
beneath the heater. The other observer has meanwhile
made a careful final observation of the temperature of the
heater and removed the lower stopper. As soon as the
calorimeter is in position, he lowers the hot body, without
splash, into the water. The calorimeter must then be
immediately removed and the cover closed.
One observer should then keep the mixture stirred by
moving the body up and down with the aid of the string
and note the temperature at as short equal intervals as
possible (perhaps every 15 sees.) while the other records
the readings. After the highest temperature has been
reached, the readings are continued every minute for at
least five minutes.
Simple and obvious modifications of the above procedure
are required if the specimen is in the form of shot or clippings.
After rough calculations of water equivalents and specific
heats, the observers should exchange duties and repeat
the whole, using masses that most nearly accord with the
conditions laid down in the considerations stated above.
Before calculating final results, make corrections of the
temperatures, T, t, t , according to the tables of corrections
already obtained for the thermometers used.
RATIO OF SPECIFIC HEATS OF GASES. 91
Plot all the temperature observations of the final deter-
mination, and correct for radiation according to the direc-
tions given on pages 63, 64. The possible error of the result
should be calculated as explained on pages 7, 8.)
Questions.
1. How might the present method be adapted to find the specific
heat of a liquid?
2. Considering evaporation, loss of heat when transferring the
hot body, and any other sources of error that may occur to you, is
your result more probably too high or too low ?
XX. RATIO OF SPECIFIC HEATS OF GASES.
(Clement and Desormes* Method.)
Edser, Heat, pp. 3 1 7-3 2 5 ; Text-book of Physics (Duff) , pp. 211-212,
267, 268; Watson's Physics, 259-260; Watson's Practical
Physics, 105; Ames' General Physics, pp. 252-256.
The gas is compressed into a vessel until the pressure has a
value which we will designate by p v The vessel is then
opened for an instant, and the gas rushes out until the
pressure inside falls to the atmospheric pressure, p Q . This
expansion may be made so sudden that it is practically
adiabatic and the temperature of the gas will therefore
fall. After the vessel has been closed for a few minutes, the
gas will have warmed to the room temperature, /, and the
pressure, p 2 , will be above that of the atmosphere. Consider
one gram of the gas. During the adiabatic expansion, its
volume changed from v l to v 2 , according to the adiabatic
equation for pressure and volume (see references)
W Po
Since the initial and final temperatures are the same, and
since the volume remains v 2 while the gas is warming and
the pressure is rising from p to p 2 , by Boyle's Law
92 HEAT..
Hence f, the ratio of specific heats, is given by the equation
A large carboy is mounted in a wooden case and may be
surrounded with cotton batting. The neck is closed with
a rubber stopper through which passes a T-tube connected
on one side with a compression pump (e. g., a bicycle pump),
FIG. 23.
and on the other side with^a manometer containing castor
oil.* A large glass tube, which may be closed by a rubber
stopper, also passes through this large stopper. A little
sulphuric acid in the bottom of the carboy keeps the air
dry. A very fine copper wire and a very fine constantin wire
* The density of castor oil is about .97, but it should properly be deter-
mined (Exp. VIII).
RATIO OF SPECIFIC HEATS OF GASES. 93
pass tightly through minute holes in the stopper and meet at
the center of the carboy, in a minute drop of solder.
The air in the carboy is compressed until the difference in
pressure is about 40 cm. of oil (=Pi po). The tube connect-
ing with the pump is closed, and, after waiting about 15
minutes to allow the air inside to regain its initial tempera-
ture (as shown by the pressure becoming constant), the ends
of the oil column are carefully read. The carboy is now
carefully surrounded with cotton batting, which may have
been removed to facilitate cooling. The air inside is momen-
tarily allowed to return to atmospheric pressure by removing,
for about one second, the rubber stopper from the glass tube.
After waiting until the air inside has assumed the room
temperature (shown by the pressure becoming constant),
the final pressure p 2 is determined. The cotton-wool had
better be removed during this stage.
Connect the wires to a calibrated galvanometer (Exp.
LVIII), apply the initial compression p lt and observe the
reading of the galvanometer when it has become steady.
Remove the stopper as before (for not over one second) , re-
place the stopper, and observe the galvanometer reading.
The proper reading to record is the fairly steady deflection
which is attained immediately after the stopper is removed.
There are liable to be rapid fluctuations which should be
disregarded, and of course the temperature does not long
remain steady, owing to heating or cooling from the outside.
Record as before the final pressure p 2 . Repeat several times,
starting with the same initial pressure p v Record the
temperature of the room, t, and p , the height of the baro-
meter (p. 21).
Calculate y, the ratio of specific heats by the above equation .
Calculate the change of temperature from the mean of the
galvanometer deflections and the constants of the thermo-
couple and galvanometer.
Compare the result with T l T where T l is + 273 and
r o is calculated from the adiabatic equation for temperature
and pressure (see references).
94 HEAT.
Unless exceedingly fine wire is employed (preferably
No. 40, B. & S.), the heat capacity of the wire is relatively
so great that the thermocouple will not show the full change
of temperature.
Draw a curve with volumes as abscissae, and pressures
as ordinates, which will represent the changes in this
experiment.
(Let specific volumes, i. e., volumes of one gram, be abscissae.
Calculate from Table VI and the laws of gases the specific volumes
corresponding to the room temperature and p , p lt and p 2 , and draw
the corresponding isothermal. Draw the horizontal line correspond-
ing to p . Draw a vertical through the point corresponding to p ? on
the above isothermal. The intersection of these two straight lines
will evidently be p , v 2 ).
Questions.
1. Do you see any objection to an initial exhaustion of the gas
in place of the compression?
2 . What are the advantages and disadvantages of a large opening ?
Short time of opening? Castor oil manometer?
3. How would an aneroid manometer be preferable in this experi-
ment to a liquid manometer?
XXI. LATENT HEAT OF FUSION.
Edser, Heat, pp. 145-149; Text-book of Physics (Duff), p. 225;
Watson's Physics, 211; Watson's Practical Physics, 88;
Ames' General Physics, pp. 260, 261; Crew's General Physics,
286.
The latent heat of fusion of a substance is the number of
calories required to melt one gram of the substance. The
most common method of measuring it is a method of mixture
similar to that used in finding the specific heat of a solid.
A known mass of the solid at its melting-point is placed in a
known mass of the liquid at a known temperature, and the
temperature of the liquid observed after the solid has com-
pletely melted. Allowance must be made for the water
equivalent of the calorimeter and correction must be made
LATENT HEAT OF FUSION. 95
for the effect of radiation to or from the calorimeter while
melting is taking place. The error due to radiation may be
made small by having the liquid initially as much above
the temperature of its surroundings as finally it falls below.
Thus loss and gain by radiation will approximately balance.
Nevertheless, since the calorimeter will probably not be the
same length of time above the temperature of the sur-
roundings as below, there will be a residual error for which
correction must be made.
The calorimeter consists as usual of an inner can polished
on the outside to diminish radiation, and enclosed in an
outer can polished on the inside. The space between the
two cans may be filled by cotton-wool to prevent air currents,
and still further prevent communication of heat. The inner
can is weighed, first empty and then half-filled with warm
water about 1 5 above the room temperature. It is then
placed in the outer can as described above and covered by a
wooden cover having holes for thermometer and stirrer
and a hinged cover giving access to the inner vessel.
The temperature is carefully noted each minute until it
has fallen to about 10 above the room temperature. In the
meantime, ice is broken to pieces of about a cubic centimeter
in volume. These pieces are carefully dried in filter paper.
A careful observation of the temperature of the water in
the calorimeter having been made and the time noted, a
piece of ice is dropped in without splashing and kept under
water by a piece of wire gauze attached to the stirrer.
The temperature is noted every half-minute as the ice melts,
the water meantime being kept stirred. The rate at which
ice is dropped in is regulated simply by the rate at which
it can be dried and the temperature and time noted.
The process is continued until the temperature has fallen
to about 10 below the room temperature. Then the
addition of ice is discontinued and the temperature of the
water further noted every minute for four or five minutes.
Finally, the weight of the inner can and its contents is
obtained in order that the mass of the ice may be deduced.
96 HEAT.
After the proper weight of ice has been ascertained, the
experiment should be repeated with a single piece of approxi-
mately this weight. As there are considerable sources of
error that cannot be eliminated, the whole determination
should be repeated as often as time will permit.
All the temperature observations should be plotted
against the time and the radiation correction determined
as described on pages 63 and 64. In reporting, consider
the possible error of your result so far as it depends on the
possible error of your weighings and observations of tem-
perature. State also any other sources of error that may
have affected your result.
Questions.
1 . What advantages are there in the use of one large lump over an
equal mass of small ones?
2. Why must the water in the inner vessel be pure?
3. Is it preferable to have the air about the calorimeter moist or
dry? Explain.
XXII. LATENT HEAT OF VAPORIZATION.
Ames' General Physics, p. 269; Watson's General Physics, 214; Crew's
Physics, 287; Watson, Practical Physics, 89-91; Edser, Heat,
pp. 150-9; Text-book of Physics (Duff), p. 231.
The latent heat of vaporization of a substance is the number
of calories required to change one gram of the substance
from liquid to vapor. The usual method of measuring
it is a method of mixture. A known mass of vapor, at a
known temperature, is discharged into a known mass of
liquid, at a known initial temperature, and the final tem-
perature is noted. The same precautions are necessary
as in finding the latent heat of fusion. The arrangement
of the calorimeter is also the same. To minimize radiation
the water should be initially as much below room tempera-
ture as it finally rises above, say 15. The initial rate of
warming should also be obtained, and also the final rate of
cooling.
LATENT HEAT OF VAPORIZATION.
97
Several different forms of boiler have been devised for
the purposes of this determination. Two will be briefly
described.
In Berthelot's boiler the delivery tube passes out through
the bottom of the boiler, which is heated by a ring burner
that surrounds the tube. Thus the tube is so far as possible
jacketed by the boiling water. The usual form of this
boiler is somewhat fragile, but a
good substitute may be made
from a round-bottomed boiling
flask the neck of which has been
shortened (Fig. 24).
FIG. 24.
FIG. 25.
In the electrically heated boiler the heating of the water
is produced by a coil of wire that is immersed in the water
and is heated by a current of electricity. The current must
be kept regulated by a rheostat, so that boiling proceeds at
a moderate rate.
The chief difficulty is in delivering the steam dry. Con-
densation is apt to take place in the delivery tube. This
can be reduced by inserting a trap in the delivery tube be-
tween the boiler and the calorimeter. The trap should,
from time to time, be cautiously heated by a Bunsen burner
to prevent condensation, but in general, it is better to dis-
pense with the trap and make the exposed part of the
7
98 HEAT.
delivery tube as short as possible and carefully cover it with
cotton-wool.
If the delivery tube simply passed to a sufficient depth
beneath the water, the steam would be delivered at greater
than atmospheric pressure, as the pressure of a certain depth
of water would have to be overcome. Hence it is better
to let the delivery tube pass into a condensing-box im-
mersed in the water. The latter must also be open to the
atmosphere by another tube. To prevent any escape of steam
by this tube it may be closed by a little cotton-wool. The
amount of steam that has been condensed is obtained by
weighing the condensing-box (well dried) before it is placed
in the calorimeter, and again with the contained water
at the end of the experiment. The temperature of the steam
is deduced from the barometric pressure. A pressure gauge
attached to the boiler affords a means of estimating how far
the pressure differs from atmospheric pressure.
For the best results, certain precautions must be ob-
served. The delivery tube must not be connected to the
coridensing-box until steam has begun to pass freely, and
as dry as possible, from the tube. Connection should not
be attempted until the temperature of the water has been
carefully ascertained and care has been taken that every-
thing is ready for making a deft and prompt connection.
After the temperature of the well stirred water in the inner
calorimeter has been read every minute for five minutes, the
connection is made. The temperature is read every half-
minute, the water meantime being kept well stirred by a
stirrer (which should be of the same material as the calor-
imeter and condenser in order to simplify the calculation of
the water equivalent). The flame of the ring-burner must
be regulated so that the steam does not pass too rapidly. This
may be gauged by the rate of the rise of the temperature
of the calorimeter, which should not exceed 4 or 5 per
minute. In finding the subsequent rate of cooling, the
boiler should be disconnected from the condenser and the
tube leading to the condenser should be closed by plugs of
LATENT HEAT OF VAPORIZATION. 99
cotton-wool to prevent evaporation; but in subsequently
weighing the condenser the wool should not be included.
The whole determination should be repeated as many times
as possible.
A formula for the calculation of the latent heat may
be readily worked out. Account must be taken of the
water equivalent of calorimeter, condenser, and stirrer. The
correction for radiation is made by the method stated
on pages 63, 64.
The possible error of the result, so far as it depends on
the readings made, should be calculated, and other possible
sources of error should be mentioned.
Questions.
1 . State the advantages and disadvantages of a rapid flow of steam
2. Explain why the latent heat should vary with the atmospheric
pressure.
3. Must the boiling water be pure? Explain.
XXIII. LATENT HEAT OF VAPORIZATION.
Continuous -flow Method.
Ames' General Physics, p. 269; Watson s Physics, 214; Crew 1 s General
Physics, 287; Watson, Practical Physics, 89-91; Edser, Heat,
pp. 150-9; Text-book of Physics (Duff}, p. 231.
The apparatus for this method may be readily constructed
from a Liebig's condenser. Water enters at D and leaves at
C through T-tubes connected to the condenser by short
rubber tubes. Superheated steam enters at A through a
T-tube and the condensed water drops into a covered beaker
E. The steam is superheated as it flows through a glass
tube FE. This is first covered with asbestos over which a
heating coil of wire* is wrapped, the coil being covered by
a second layer of asbestos. AB and FE are mounted on
a wooden frame and A B is thickly covered with cotton-
wool to prevent radiation. Thermometers 7\, T 3 , T 2 , T 4 ,
* "Nichrome" wire (supplied by the Driver-Harris Co., New York) is very
suitable.
IOO
HEAT.
give the respective temperatures of the superheated steam,
the outflowing water, the inflowing water, and the water of
condensation. The supply of water may come from the
water mains, if this is sufficiently constant in temperature.
FIG. 26.
It is, however, much better to have a supply of from 5 to
lo gallons in an elevated tank and keep the flow constant by
an overflow regulator as indicated in figure 29 (Exp.
XXVIII).
The boiler to supply the steam should be large enough to
allow of a flow for two hours without refilling (one to two
LATENT HEAT OF VAPORIZATION. IOI
liters will suffice). The current in the superheating coil
should be regulated by a rheostat so that the superheated
steam is at about 105. Some time should be spent in test-
ing adjustments to obtain a suitable current and a rate of
flow of water that will give a rise of temperature of about
20. The tank should be connected with the water service
so that it can be readily filled. The water as it comes
from the mains will probably be below room temperature
and this is an advantage, since with a suitable rate of flow
of the steam the water that drops into E will differ but little
from room temperature and will suffer little loss of heat by
radiation. This will require a proper regulation of the
burner that heats the boiler. The burner should be sur-
rounded by a shield of sheet-iron or asbestos to prevent
fluctuations caused by air-currents.
The thermometers 7\, T 2 , and T 3 should be read once a
minute (e. g., T 2 20 sec. after T l and T 3 20 sec. after T 2 ) . From
the mean of each of these readings, the temperature of E,
the mass of water that flows out at C, and the mass of the
water that drops into E, the latent heat can be calculated.
The specific heat of the superheated steam may be taken as
0.5. A formula can be readily constructed to express the
fact that the heat given up by the steam and condensed water
equals the heat carried off by the current of water.
Questions.
1 . Why does not the water equivalent of the condenser need to be
considered ?
2. How could you find the amount of error due to conduction of
heat from the superheater to the water in the condenser?
3. How could you find the amount of error due to radiation from
the condenser?
4. What other sources of possible error are there in this method?
IO2
HEAT.
XXIV. THERMAL CONDUCTIVITY.
Edser, Heat, pp. 416-430; Watson, Practical Physics, 106, 107;
Text-book of Physics (Duff), pp. 216-220; Watson's Physics,
236238; Ames' General Physics, p. 288; Crew's General Physics,
254, 256.
The thermal conductivity of a substance is the amount
of heat transmitted per second per unit of area through a
plate of the substance of unit thickness, the temperature of
the two sides differing by i and the flow having become
steady. If K be the thermal conductivity, and if a plate
of thick-ness / and area A be kept with one side at a tem-
perature t, and the other at a lower temperature, t', the
number of calories that will flow through the plate in time
7, after the flow has become steady, will be
KA(t-t')T
I
whence K can be derived if the other quantities are observed
or measured.
Thermal conductivity is in general difficult to measure
satisfactorily. The following very simple method cannot
be relied on to closer than a
few per cent., but it only re-
quires a small portion of the
time that the more accurate
methods call for.
A rod or wire of the substance
to be tested is inserted at one
end into a heavy block of metal,
which is heated to a constant
high temperature in a bath,
through the bottom of which
the rod passes. At its lower
end the rod is screwed into a heavy block of brass or
copper of mass M and specific heat s, which is initially at a
very low temperature. Heat is thus conducted by the rod
from the bath to the lower block. If the latter neither lost
FIG.
THERMAL CONDUCTIVITY. 103
nor gained heat by convection or radiation, and if there
were no losses from the sides of the rod, we could calculate
the conductivity of the rod from its dimensions and the
mass, specific heat, and rise of temperature of the lower
block. The loss of heat from the surface of the rod is
almost wholly prevented by enclosing it by a glass tube,
which does not come into direct contact with the rod, and
wrapping the glass tube with cotton wool and paper.
To allow for radiation or convection to or from the lower
block the experiment is modified as follows: The block
is enclosed in a vessel surrounded by a water-jacket, through
which water at a constant temperature, t', circulates. Now,
the rate at which the lower block receives heat through the
rod, when the former is at the temperature t f , is the mean
of the rates at which it receives heat when it is n degrees
below t', and when it is n degrees above t' '. For let R, R lt
and R 2 represent the rates of conduction of heat (flow of
heat in one second) to the lower block at temperatures t',
t' n, and t f +n, the upper end being at temperature /.
Then
KA(t-t')
R
I
KA[t-(t'-n)]
I
KA[t-(t'+7i)]
whence
Again, when the lower block is at the same temperature
as the jacket, it neither receives heat from nor gives heat
to the jacket. And when it is n degrees below it gains
heat as rapidly as it loses heat when it is n degrees above.
Thus by taking the mean rate as above, the effects of radia-
tion to or from the block are eliminated. In fact, adding
a to R lt to allow for the gain by radiation, and subtracting
104 HEAT.
a from R 2 , to allow for loss by radiation, would leave
unchanged.
The same would hold true for any other pair of temper-
atures equidistant from the temperature of the jacket.
If the rates of rise at two temperatures equidistant from
the temperature of the jacket be r t and r 2 , by what has been
said the rate at the temperature of the jacket would be
i( r i+ r 2)- Hence, the rate at which the body must be
gaining heat is Ms^(r 1 +r 2 ). Hence, by the definition of
thermal conductivity,
or,
A(t-t')
The lower block should be cooled initially to about 12
below the temperature of the water that circulates through
the jacket by being placed in a bath of ice and water (or
snow). When taken out, it must be carefully dried. The
jacket may be kept at a constant temperature by water
passing and repassing through it between two large vessels,
which are alternately raised and lowered about every five
or ten minutes. The temperature of the water should be
frequently read by a thermometer (which may conveniently
pass through a large cork that floats on the surface of the
water). If the temperature of the water should show a
tendency to rise or fall, a small quantity of cooler or warmer
water, respectively, may be added. If the vessels be large
and the temperature of the room does not vary widely,
there should be no difficulty in keeping the water constant to
within . 2 for a sufficient length of time.
The hot bath is in the form of a trough, which is heated
at one end, while the conducting rod passes into the tank at
the other end. To prevent direct radiation from the burner
to the rod, thick screens of wood and asbestos are interposed.
The temperature of the lower block should be read at least
THE MECHANICAL EQUIVALENT OF HEAT. 105
every minute by means of a thermometer passing through
the cork or fiber cover and inserted into a hole in the block,
the unoccupied space in the hole being filled with mercury.
The readings of temperature will, from various causes, be
slightly irregular. They should therefore be plotted in a
curve, and the irregularities eliminated by taking the more
correct values from the curve. The temperature of the
upper block should be read frequently by a thermometer
thrust into it. Small quantities of boiling water should be
added frequently to the bath to compensate for evaporation.
To gain some idea of the amount of reliance to be placed
on the result, the mean rate of rise for each pair of degrees
equidistant from the temperature of the jacket should be
obtained, and the final mean of all taken in calculating.
It is, however, to be noted that the temperatures nearest
the jacket temperature should give the best results, since
there the radiation is a minimum, and therefore any defect
in the method of correcting for radiation a minimum.
(For comparing the conductivities of poorly conducting
substances Lees and Chorlton's apparatus is quite satis-
factory. Directions for its construction and manipulation
are given in Robson's Heat, page 135.)
XXV. THE MECHANICAL EQUIVALENT OF HEAT.
Griffith, Thermal Measurement of Energy, Chap. Ill; Edser, Heat,
Chap. XII; Text-book of Physics (Duff), pp. 259-262; Watson's
Physics, 250-251; Ames 1 General Physics, pp. 203-205;
Crew's General Physics, 289; Rowland, Physical Papers, pp.
343-476.
The mechanical equivalent of heat is the number of units
of mechanical energy that, completely turned into heat,
will produce one unit of heat, or, in the c. g. s. system, the
number of ergs in a calorie. The apparatus here described
is a copy of that used in the University of Cambridge,
England, and the following description and introduction
is partly taken from that issued to students in that university.
io6
HEAT.
In this apparatus mechanical energy is expended in working
against friction, thus producing heat, which is measured
by the rise in temperature of a known mass of water.
A vertical spindle carries at its upper end a brass cup.
Into an ebonite ring concentric with the cup there fits
tightly one of a pair of hollow truncated cones. The second
cone fits into the first, and is provided with a pair of steel
pins which correspond to two holes in a grooved wooden
FIG. 28.
disk, which prevents the inner from revolving when the
spindle and the outer cone revolve. A cast-iron ring,
resting on the disk and fixed by two pins, serves to give a
suitable pressure between the cones. A brass wheel is
fixed to the spindle, and, by a string passing round this
wheel and also round a hand-wheel, motion is imparted to
the spindle. A pair of guide pulleys prevents the string
from running off the wheel. Above the wheel is a screw
cut upon the spindle. This screw actuates a cog-wheel of
100 teeth, which makes one revolution for every 100 revo-
lutions of the spindle.
THE MECHANICAL EQUIVALENT OF HEAT. 107
To the base of the apparatus one end of a bent steel rod
is attached; the rod can be fixed in any position by a nut
beneath the base. The other end of the rod carries a cradle,
in which runs a small guide pulley on the same level with the
disk. The cradle turns freely about a vertical axis. A fine
string is fastened to the disk and passes along the groove in
its edge; it then passes over the pulley and is fastened to a
mass of 200 or 300 grams. On turning the hand- wheel it is
easy to regulate the speed so that the friction between the
cones just causes the mass to be supported at a nearly
constant level. To prevent the string from running off
the guide pulley, a stiff wire with an eye is fixed to the
cradle and the string is passed through this eye. It also
passes through an eye fixed to the steel rod, to prevent
the weight from being wound up over the pulley.
The rubbing surfaces of the cones must be carefully
cleaned and then four or five drops of oil must be put be-
tween them; the bearings of the spindle and guide pulley
should also be oiled. The cones are then weighed together
with the stirrer. The inner cone is then filled to about i
cm. from its edge with water 2 or 3 below the temperature
of the room and the system is again weighed. The cones
are then placed in position in the machine and a thermometer
is hung from a support so that it passes through the central
aperture in the disk and almost touches the bottom of the
inner cone.
One observer, X, takes his place at the hand-wheel, and
the other, Y, at the friction machine. By working the
machine the water is now warmed up until its temperature is
nearly equal to that of the room. The index of the counting
wheel is read and the temperature of the water is carefully
observed every minute for five minutes. Immediately after
the last reading, X turns the wheel fast enough to raise the
mass until the string is tangential to the edge of the disk.
If the string be not tangential the moment of its tension
about the axis of revolution is seriously diminished. Y
stirs the water and notes the temperature at each passage
108 HEAT.
of the zero of the counting-wheel past the index; each
passage of the zero after the first corresponds to 100 revolu-
tions of the spindle. Y gives a signal at each passage of the
zero and X notes the time by aid of a watch. After Y has
recorded the temperature upon a sheet of paper previously
ruled for the purpose, he also records the time observed by
X. After about 1000 revolutions the motion is stopped
and the readings of the index of the counting wheel and of the
thermometer are recorded. Observations of the temper-
ature are continued every minute for five minutes, the stirring
of the water being continued.
The temperature observations are plotted against the
time, and the radiation correction is determined as explained
on pages 63 and 64. The heat produced is readily cal-
culated from the mass of water, the water equivalent of
the cones and stirrer, and the corrected rise of temperature.
From the initial and final readings of the counting wheel
and the number of complete revolutions the exact number
(n) of revolutions made by the spindle is deduced. The
work done is calculated as follows: When the spindle
has made n turns the work spent in overcoming the friction
between the cones is the same as would have been spent
if the outer cone had been fixed and the inner one had been
made to revolve by the descent of the mass of M grams.
In the latter case M would have fallen through innr cm.
where r is the radius of the groove of the wooden disk, which
must be measured. Hence the total work spent against
friction and turned into heat is 2nnrMg ergs. In the report,
estimate the possible error of the result as far as it depends
upon the errors of observations and measurements.
Questions.
1 . What amount of error is due to neglect of the work spent against
friction of the bearing of the outer cone ?
2. Why must the wheel be turned faster as the experiment pro-
ceeds?
3. What effect on the result has the variation of the viscosity of
the oil?
THE MELTING-POINT OF AN ALLOY. IOQ
XXVI. THE MELTING-POINT OF AN ALLOY.
Robson, Heat, pp. 77-79; Findlay, Phase Rule, pp. 220-223; Ewell,
Physical Chemistry, pp. 271-272.
If an alloy is melted and is allowed to cool while its tem-
perature is continuously observed, and a curve be then drawn
with times as abscissae and temperature as ordinates, it
will be found that at certain points the curvature abruptly
changes, the fall of temperature being decreased or even
ceasing. At the moment corresponding to such a point, the
alloy is radiating heat to the room, and the fact that its
temperature does not fall as rapidly indicates that heat is
being produced internally by some change of state of the
material. Such a point is therefore a solidifying-point of
some constituent of the alloy or of the eutectic alloy.
The assigned* metals are carefully weighed and melted in
an iron cup. A copper-constantinf thermocouple is plunged
into the liquid metal and kept there until the entire mass is
solid. A porcelain tube should cover one wire for some
distance from the junction. The terminals are connected
to a calibrated galvanometer through a resistance such that
the maximum deflection will keep on the scale. The galvan-
ometer is read every half-minute and the time of each read-
ing is noted. When the readings are commenced, the metal
should be considerably above the melting-point and the
readings should be continued for some time after the metal
is apparently solid. For calibration of the galvanometer, see
Exp. LVIII.
Plot the galvanometer deflections against the time. De-
termine the electromotive force corresponding to the galvan-
ometer deflections where the curvature changed, and from
the constants of the thermocouple, or a chart giving the
* Tin and lead are suitable metals. The changes of curvature are more
distinct if the former is in excess. The eutectic of tin and lead is composed
f 37% lead, 63% tin, and melts at 182.5 (Rosenhain and Tucker, Roy. Soc.
Phil. Trans., 1908, A. 209, p. 89).
f "Advance" Wire, Driver-Harris Co., Harrison, N. J.
1 10 HEAT.
temperature for different electromotive forces, determine
the temperature of these points.
Tabulate the observed temperatures of these transition
points and your opinion of what they represent.
Questions.
1. Explain why the second transition point is represented by
a horizontal portion of the cooling curve while the first transition
point is merely represented by a change of curvature.
2. Will the temperature of the first point vary with the initial
concentration ?
3. Will the temperature of the second transition point vary with
the initial concentration ?
XXVII. HEAT VALUE OF A SOLID.
Ferry and Jones, pp. 237-242.
(A) HEMPEL BOMB CALORIMETER (Constant-volume
Calorimeter)
A pellet of the fuel to be tested is formed in a press, a
cotton cord being imbedded with a loose end. After being
pared down to about i gm. and brushed, it is carefully
weighed. It is then suspended in a Hempel combustion
bomb, and the thread is wrapped around a platinum wire
connecting the platinum supports of the basket. The ter-
minals attached to these supports are connected with several
Edison or storage cells sufficient to just bring the wire
to a brilliant incandescence (as ascertained by a preliminary
trial) .
The bomb is charged with oxygen under at least fifteen
atmospheres' pressure, either from a charged cylinder or
produced by a retort. Bomb and pressure gauge should be
immersed in water while the oxygen is being supplied. As-
certain that the bomb valve is open and that all connections
are screwed tight. Open the cylinder valve (if a cylinder is
used) until the pressure becomes high, and then close. Lift
the bomb out of the water, loosen one of the connections,
and allow the mixture of air and oxygen to escape; then
HEAT VALUE OF A SOLID. Ill
tighten, replace in water, open the cylinder valve again until
the pressure becomes high (at least fifteen atmospheres);
close both the cylinder valve and that of the bomb, and
finally disconnect and dry the bomb. (If the oxygen is
produced in a retort, partly fill the latter with a five to
one mixture of potassium chlorate and manganese dioxide,
connect to the bomb and pressure gauge, and heat the upper
part slowly with a Bunsen burner.)
Attach the electrical terminals, place the bomb in the
special vessel containing about a liter of water, adjust the
Beckmann thermometer to read about i (see p. 65), stir
the water continually, and read its temperature every half-
minute for five minutes, estimating to tenths of the smallest
graduation. Close the electric switch; after a few seconds
open it and read the temperature of the continually stirred
water for ten minutes.
Let H be the heat value of the fuel and m the mass of
the specimen, M the mass of the water, e the water equivalent
of the bomb, t l the initial temperature, and t 2 the final tem-
perature (corrected for radiation, see p. 63.) Then
eWi-tJ
cal. pergm.
wi
The best method in practice to determine e is to repeat
the determination, using 'salicylic acid as fuel, and assum-
ing its heat value to be 5300 calories per gram.
(B) ROSENHAIN'S CALORIMETER (Constant-pressure
Calorimeter)
Phil. Mag., VI, 4, p. 451.
Instead of burning the fuel in a fixed volume of highly
compressed oxygen, the oxygen is supplied continuously
at only slightly above atmospheric pressure.
The coal is pulverized and a sample is compressed, in a
special screw press, into a pellet weighing about one gram.
This is placed on a porcelain dish which rests on the bottom
112
HEAT.
of the inside chamber. The ignition wire should be about
3 cm. of No. 30 platinum wire and the external terminals
should be connected to storage-battery terminals through a
key and a resistance such that the wire will glow^brightly.
A gasometer is charged with oxygen from a cylinder or gen-
erated from "oxone" and water. The action of the dif-
ferent valves having been studied, the apparatus should
be assembled, the upper side valve (see Fig. 29) being closed
and the ball valve lowered. Connect with
the oxygen supply through a wash-bottle,
turn on a very gentle stream of oxygen,
and pour into the outer vessel a measured
volume of water, at the room tempera-
ture, so that the combustion chamber is
just covered.
If a Beckmann thermometer (see p. 65)
is used, adjust to read between o and i
in this water. The bulb should be sup-
"~\CT_-Jj/ ported on a level with the center of the
combustion chamber. Read the tempera-
ture every half-minute for five minutes,
then increase the oxygen current, and care-
fully read the temperature and the time, close the key and
ignite the pellet with the hot platinum wire, and immediately
remove the wire. During such operations it is best to hold
the inner vessel steady by grasping the oxygen inlet tube.
Keep the water pressure in the gasometer constant, and as
combustion proceeds increase the flow of oxygen. If pos-
sible, read the thermometer every half-minute.
When combustion has ceased, move the hot wire about
to igni-te any unconsumed particles. Keep the wire hot as
short a time as possible and remove it immediately from any
combustion, otherwise it is liable to be melted.
Finally, turn off the oxygen supply, open the upper valve,
and raise the ball valve, allowing the water to enter the
inner chamber. Then force out the water by closing both
valves and turning on the oxygen. Record the highest
FIG. 29.
HEAT VALUE OF A GAS OR LIQUID. 113
temperature and the time and the temperature every
half-minute for five minutes. For radiation correction,
see p. 63, and for formulae, see (A) preceding.
To determine e, assemble the apparatus, including the
Beckmann thermometer, and pour in 1000 c.c. of water.
Determine very carefully the temperature with a 0.1
thermometer and then add 500 c.c. of water at about 50, the
temperature of which has also been very carefully deter-
mined. Determine also very carefully the final steady
temperature and from these data determine e.
For anthracite coal add sugar in the proportion 3:1.
(Heat of combustion of sugar = 3 900 calories per gram.)
Questions.
1. Calculate the heat value of (a) one kilo of this substance, (b)
one short ton in B. T. U. per lb., (c) the mechanical energy equivalent
to the latter.
2. What error would be caused by (a) an error of 20 in the water
equivalent? (b) allowing a current of 5 amperes to flow through a
platinum wire of 2 ohms' resistance for 5 seconds? (c) Neglecting
the radiation correction ?
XXVIII. HEAT VALUE OF A GAS OR LIQUID.
Ferry and Jones, pp. 243-246.
The heat value will be determined with Junker's cal-
orimeter.
(A) A measured volume (v liters) of gas under an ob-
served pressure, p, is burned in the calorimeter, and the
rise of temperature, from tj to t 2 , of a mass of M gr. of
water, is determined. The flow of water and gas is so regu-
lated that the burned gas leaves the calorimeter at approxi-
mately the temperature of the entering gas, and there
should be a difference of at least 6 in the temperature of the
in- and outflowing water. Also, the flow of water must be
sufficient to furnish a constant small overflow at the supply
reservoir (see Fig. 30). The burner should be lighted out-
side the calorimeter. When the temperatures indicatedjDn
the various thermometers have become constant, note the
gasometer reading, and immediately collect in graduates the
8
HEAT.
heated overflowing water, and also the water condensed
by the combustion of the gas. Let the mass of the latter be
m gr., and its temperature /'. Note the temperatures of the
inflowing and outflowing water every 15 sec. until two or
three liters have passed through.
Then immediately note the gaso-
meter reading and remove the
graduates.
Assuming that condensation of
the gas occurs at 100, the heat
liberated is m [536+100 2']. If
H represents the heat value of
the gas in gram-calories per liter
and v the volume, reduced to o,
and 760 mm.,
M(t 2 -t 1 )-m(6 3 6-t f )
FIG. 30. (B) To determine the heat
value of a liquid fuel, the gas
burner is replaced by a suitable lamp which is attached to
one arm of a balance. The rate at which the liquid is
consumed is determined from the weights in the pan, on the
other side, at different times. It is best to make the weight
in the pan slightly deficient and note the exact time when
the balance pointer passes zero, as the liquid is consumed.
Practically complete combustion is obtained with a "Primus"
burner, supplied from a reservoir where the liquid is under
considerable pressure. With very volatile liquids, the
opening of the burner must be large and the pre-heating
tubes must be in the cooler part of the flame.
Express your results in (a) calories per liter (b) B. T. U.
per gallon or cubic foot.
Questions.
1. Is the heat value of a gas, in calories per gram, definite? Per liter?
2. What difference would there be in the result if all the water
vapor escaped without condensing?
3. Why is no radiation correction necessary?
PYROMETRY. 115
XXIX. PYROMETRY.
Edser, Heat, pp. 339411; Bulletin, Bureau of Standards, I, 2, pp.
189-255; Watson, Practical Physics, 208-210; Le Chatelier,
High Temperature Measurements, Chaps. Ill, VI, IX.
This exercise is a study of three of the methods used in
the measurement of very high temperatures.
A hollow black body is to be heated electrically and the
temperature of its interior is to be determined by means
of a calibrated thermocouple. A platinum resistance ther-
mometer is to be calibrated, and also an incandescent lamp
is to be calibrated for use as an optical pyrometer.
Electric Furnace. The electric furnace (see Fig. 31)
consists of a thin porcelain cylinder about 15 cm. long and
FIG. 31.
10 cm. in diameter upon which is wound about 5 m. of
No. 22 " Nichrome" wire,* if a 220-volt supply is to be used.
Whatever the voltage, the winding must be such as to
consume about half a kilowatt. The ends of the cylinder
are closed by porcelain caps with proper apertures and the
whole is surrounded by many layers of asbestos.
Heating and cooling must be very gradual so that the
thermocouple and platinum thermometer may acquire
the temperature of the furnace. The highest temperature
should not exceed 1000.
*$ee note bottom of page 99,
Il6 HEAT.
Thermocouple. A platinum and platinum +10% rhodium
thermocouple should be connected to a galvanometer through
a key and such a resistance as will keep the deflection on the
scale at the highest temperature. The galvanometer with
resistance should be calibrated as a voltmeter (Exp. LVIII).
The chart or table accompanying the couple gives the
temperature of the hot junction when the electromotive
force is known. (The cool junction should be in ice and
water.)
Platinum Resistance Thermometer. The platinum resist-
ance thermometer consists of a coil of fine platinum wire
(for example, 50 cm. of No. 30) wound on a porcelain frame
and surrounded by a glazed porcelain tube. The coil
constitutes one arm of a Wheatstone's bridge. A pair of
dummy leads are connected to an adjoining arm (see figure)
and compensate for the heating of the lead wires. A
suitable switch connects the galvanometer to either the
bridge or the thermocouple.
Optical Pyrometer. The optical pyrometer consists
essentially of a lens and a miniature incandescent lamp. The
lens focuses the interior of the enclosed furnace (an ideal black
body) on the filament of the lamp. The current through
the filament is adjusted until the tip of the filament is
invisible against the image of the furnace. When this is
true, both must be emitting similar light, and therefore
they must be at approximately the same temperature. A
small eye-piece aids in observing the tip of the filament.
The incandescent lamp filament is to be calibrated; i.e., the
current necessary to heat the tip of the filament to different
temperatures is to be determined. The temperature of the
filament is determined by finding the temperature of the
furnace, by the thermocouple, when the two have the same
temperature. (The incandescent lamp circuit contains an
ammeter for measuring the current, which is omitted from
Fig. 31.)
Observations. While the furnace is slowly heating, and
also while it is slowly cooling, observe at frequent intervals,
PYROMETRY. 117
(A) the galvanometer deflection with the thermocouple,
and the resistance in the galvanometer circuit; (B) the
resistance of the platinum thermometer; (C) the current
through the filament. The three observations should be
made in succession and the times of each recorded. (D)
Calibrate the galvanometer as described in Exp. LVIII, if
the constant is not furnished. (E) If time permits, use the
optical thermometer to determine the temperature of
various distant, brightly heated bodies; e. g., melted silver,
iron, or copper. An image of the hot body is formed upon
the tip of the filament, and the current through the latter
is adjusted until the two are indistinguishable. The tem-
perature corresponding to this current is obtained from the
calibration (see (e) below).
Report, (a) Tabulate readings, (b) Plot the three sets
of readings against the time, (c) Make a second plot with
resistance as abscissae and, for ordinates, the platinum
temperatures, as given by Callender's equation,
RIRQ
pi IOO - .
where R is the extrapolated resistance at o and R 1 is the
resistance at 100. Transfer to this plot also the readings
of the true temperature, /, and determine the mean value of
Callender's difference constant, 8, by applying at several
points the equation
For pure platinum 8 is 1.50. The platinum resistance
thermometer is the most accurate, convenient method of
measuring temperature below 1000.
(d) Construct a curve which gives the temperature of the
tip of the incandescent lamp filament plotted against the
current, (e) Finally, determine by the latter curve the
temperatures of any bodies tested with the optical pyrometer
and record the results. (For a discussion of the error in
Ilg HEAT.
assuming for different bodies that the radiation is similar to
that from a black body, see the above reference to the
Bulletin of the Bureau of Standards and Haber, "Thermo-
dynamics of Gas Reactions" pp. 281-291.)
Questions.
1. Would you expect the platinum resistance thermometer to
attain a slightly higher or a slightly lower temperature than the
thermocouple? Explain.
2. Is any correction required for the absorption of the lenses
in such an optical pyrometer? Explain.
SOUND.
/^s\
XXX. THE VELOCITY OF SOUND.
Ttxt-book of Physics, (Duff), pp. 319, 320, 334-337; Watson's Physics,
287, 288, 309; Ames' General Physics, pp. 337, 363; Crew's
General Physics, 213; Poynting and Thomson, Sound, Chap. VII-
The velocity of sound in a medium can be found if the
wave-length, L, in the medium of a note of frequency n can
be determined; for v = n L. If
the medium be contained in a
tube, one end of which is closed,
the closed end must be a node
and the open end a loop. Hence
the length of the tube must be
an odd number of quarter wave-
lengths. Such a tube will reso-
nate to a fork, if the wave-length
of a natural vibration of the pipe
be the same as the wave-length
to which the fork gives rise.
Thus, if the length of pipe that
resonates to a fork of known
pitch be measured, we have the
means of finding the velocity of
sound.
A long glass tube is mounted
on a stand. Water is introduced
from the bottom, where is at-
tached a rubber tube provided
with a pinch-cock and connected
to a glass bottle. By raising and
lowering the bottle, water may be
brought to any height in the tube. The additional connec-
tions represented in figure 32, permit raising or lowering the
water level by opening pinch-cocks. When A is opened
119
FIG. 32.
120 SOUND.
the tube fills and the tube empties upon opening D. C
and B are pinch-cocks which regulate the rate of flow.
The vessels are so large, compared with the capacity of the
tube that only rarely is it necessary to transfer water from
the lower to the upper vessel.
If a tuning-fork be vibrated above the tube, resonance will
first take place when the air column is approximately one-
quarter wave-length of the fork, next when three-quarter
wave-lengths, etc. In reality, the first loop is not exactly at
the open end of the pipe, but a short distance beyond the
open end. The distance between two nodes is accurately
half a wave-length, and it is from this distance that the
wave-length is best determined.
The tuning-fork may be sounded by gently striking the
end of the fork against the knee or a block of soft wood.
The fork should be held above the end of the tube, so that
the plane of the prongs includes the axis of the tube. Each
node should be located very carefully, at least four times,
each location being tested both with the water rising
and with the water falling, and the distance of each position
from the open end noted. The mean is taken as the true
distance. The whole should then be repeated with a fork
of different pitch. Observe the temperature and barometric
pressure, and measure the diameter of the tube. With a
little practice one can often locate nodes corresponding to
the higher modes of vibration of the fork. This should
be tried and, from their wave-lengths and the velocity of
sound as already determined, the pitch of these higher sounds
can be calculated.
The pitch of the forks used may be determined by com-
parison with a standard fork by the method of beats. The
standard is mounted on a resonance box and is set in vibra-
tion by pulling the prongs together with the fingers and
then releasing them. The fork of unknown pitch is sounded
in the usual way, and the end of the shank is set upon the
resonance box of the standard. If the two forks are of
nearly the same pitch, beats will be heard. With a stop-
THE VELOCITY OF SOUND. 121
watch the time of ten, fifteen, or twenty beats, as may be
convenient, is several times determined. Dividing by the
time, we have the number of beats per second, and this
is the difference of pitch of the two forks. To determine
which fork is the higher, add a little wax to one prong of
the fork used in the experiment. Since this increases the
inertia of the fork, it decreases its pitch. If originally the
two forks have very nearly the same pitch, so that there
is only a fraction of a beat per second, very small amounts
of wax should be added. A large piece of wax might
change the pitch of the fork from above that of the standard
to below. (Time may often be saved by comparing the
fork with the standard during the necessary delays of the
work described below.)
The velocity of sound in carbon dioxide may be deter-
mined in a similar manner. The water surface is first
lowered to the bottom of the tube and the tube is filled with
the gas from a generator through a small tube lowered just
to the water surface (not below) . The generator consists of
a tube filled with marble, surrounded by dilute hydrochloric
acid. In filling the generator with marble, use only whole
pieces, carefully excluding any dust or pieces small enough
to drop through into the outer vessel. If the gas is not
evolved in sufficient quantity, add hydrochloric acid to the
outer vessel. When the air in the tube has been entirely
displaced by the gas, a lighted match introduced into the top
of the tube will be extinguished. The delivery tube is now
withdrawn from the resonance tube, while the gas still flows
out to fill the volume occupied by the delivery tube.
The water surface is now slowly raised and the nodes
located for one of the forks. Observations can only be made
with the water rising, for, when the water surface is lowered,
air enters the tube. Hence each node can be located but
once. The tube is again filled and the nodes redetermined
with the same fork.
From the distance between the nodes and the pitch of the
fork the velocity of sound is determined.
1.2 SOUND.
For each gas, find the correction for the open end; that
is, the displacement of the loop beyond the end of the tube.
Use the mean position of the highest node and one-fourth
of the mean wave-length. Find what fraction this displace-
ment is of the radius of the tube.
Calculate, from the mean values of the velocities, the
velocity of sound in each gas at o C. (see references).
Find also the ratio of the specific heats from the velocity
at o C., the standard barometric pressure (both in absolute
units) and the density as given in Table VI.
Questions.
1. Explain why (a) readings are made with both rising and falling
water (b) the plane of the prongs of the fork must contain the axis
of the tube.
2. What is the influence of atmospheric moisture upon the velocity
of sound?
XXXI. VELOCITY OF SOUND BY KUNDT'S METHOD.
Text-book of Physics (Duff), p. 338; Watson's Physics, 317; Ames'
General Physics, p. 364; Crew's General Physics, 215; Poynting
and Thomson, Sound, pp. 115-117; Watson's Practical Physics,
113-
A glass tube, A G, about a meter long and about 3 cm.
internal diameter is closed at one end by a tight-fitting
piston, C, and at the other end by a cork through which
passes a glass tube having at one end a loosely fitting card-
board disk, D (Fig. 33). The glass tube should be about a
meter long. A little dry powdered cork is sprinkled in
the tube, the stopper at G is loosened, and a current of air,
dried by passage through several drying tubes, is slowly
forced through the hollow rod of the piston, C. The stopper
at G is then replaced and the glass tube, F, is held at the
center and stroked longitudinally with a damp cloth. The
piston, C, is adjusted until the powder collects in the sharp-
est attainable ridges. These ridges will appear where the
pressure changes are least; that is, at the loops. Measure
carefully the distance between two extreme ridges and
VELOCITY OF SOUND BY KUNDT's METHOD. 123
divide by the number of segments into which the tube is
divided. This distance (between two loops) is a half wave-
length of the waves in the tube. Disturb the powder and
make a new adjustment of the piston, C, and a new measure-
ment of the half wave-length. Make a third repetition of
the adjustments and readings.
7^1
1 1
c r~
j
A C
D
FIG. 33.
6
Fill the tube with another dried gas, for example, carbon
dioxide, illuminating gas, hydrogen, oxygen, or hydrogen
sulphide, and determine the half wave-length. If n is the
constant pitch of the note emitted by the glass tube and / is
the wave-length in the gas
v i l i
v=nl .'. =~i.
^2 k
Since the velocity changes at the same rate with change of
temperature in all gases (see references), the velocity of
sound or compressional waves at zero degrees in any other
gas than air can be calculated from the ratio of the wave-
lengths at a common temperature, and the velocity in air
at zero degrees (33,200 cm. per second).
From the velocity of sound at zero degrees in the gases
other than air and the standard atmospheric pressure cal-
culate the ratio of specific heats, 7- (see references). Table
VI gives the densities of the more common gases and vapors
at zero degrees and a pressure of 76 cm. of mercury = 1013200
dynes per square centimeter.
Questions.
1. Calculate (a) the velocity of compressional waves in glass,
(b) the elasticity E. (Notice that each end of the glass rod must be a
loop, and the center a node. The density of glass can be obtained
from Table VIII.)
2. Why must the glass rod be set in longitudinal vibration?
3. Why does the powder collect at the loops?
LIGHT.
27. Monochromatic Light.
The simplest and most useful monochromatic light is the
sodium flame. Sodium may be introduced into a Bunsen
flame by surrounding the tube of the burner with a tightly
fitting cylinder of asbestos which has been saturated with a
strong solution of common salt and formed into cylindrical
shape by wrapping around the burner while still damp.
As the top of the cylinder is exhausted, it should be torn off
and the rest of the tube pushed up into the lower part of
the flame. A piece of hard-glass tubing held in the flame
will also give a good sodium light.
Elements giving red, green, blue, and violet light will be
found in Table XVIII. Salts of these elements (e. g.,
KN0 3 , SrCl 2 , CaCl 2 , LiCl) may be introduced into the outer
edge of a bunsen flame, either in a thin platinum spoon, on
copper gauze, or by a piece of wood charcoal which has
absorbed a solution. If a very intense light is not required,
a vacuum tube is a very satisfactory source (Table XVIII).
Intense light of one general color may be obtained by filtering
sun light or the light from an arc light through colored glass
or gelatine. The solutions given in the accompanying table
give much purer monochromatic light.
Light Filters (Landolt).*
Color
Thickness
of layer
(mm.)
Aqueous
solution of
j Grams per
IOO C.C.
Average
wave-length
(Angstrom units)
Red..
20
20
, Crystal violet 560
Potassium chromate
.005
10.
6560
Green
2O
20
; Copper chloride
Potassium chromate
60.
10.
5330
Blue..
20
2O
l Crystal violet
1 Copper sulphate
.005
15-
4480
* Mann, Manual of Advanced Optics, p. 185.
124
SPHERICAL MIRRORS AND LENSES. 125
28. Rule of Signs for Spherical Mirrors and Lenses.
Mirrors. Consider the side upon which the incident light
falls as the positive side of the mirror. If the object, the
image, or the principal focus is on this side, their respective
distances, (u, v, f=rj 2) will be positive; if on the other side,
negative. Therefore, the focal length (and hence radius) is
positive for concave mirrors and negative for convex. The
object distance, u, will obviously, in most cases, be positive.
The formula for all spherical mirrors is :
I I I 2
U V f Y
if the signs of the numerical quantities which are substituted
for u, v, f, and r are determined by the above rule.
Lenses. Let all the distances, u, v, f, r lt r 2 be positive
for the double convex lens, when the object is outside the
principal focus; that is, in the most common case. The
formula for all lenses is then
U V f
As an illustration of the application of this rule, consider the
signs of these distances when an image of a real object is
formed by a double concave lens. The distance, u, of the
object is obviously measured on the same side as it would
be in the standard case of the double convex lens and is,
therefore, positive. The distances / and v are, however
measured on the same side of the lens as the object, or
opposite to the standard case with the double convex lens,
and are, therefore, negative. r lt the radius of the front face,
is on the same side as the object, while in the case of the
double convex lens this radius is on the other side, therefore
r lt and similarly r 2 , is negative for a double concave lens.
XXXII. PHOTOMETRY.
Text-book of Physics (Duff), p. 353; Watson's Physics, 361-364;
Ames' General Physics, pp. 437, 442; Watson's Practical Physics,
pp. 382-387; Edser, Light, pp. 9-20; Stine's Photometrical
Measurements; Palaz' Photometry.
The intensity of illumination of a surface by a source of
light of small area varies inversely as the square of the
distance. Hence it follows that, if two lights produce equal
intensities of illumination at a point, P, their illuminating
powers, or the intensities of illumination they can produce at
unit distances, are directly as the squares of their distances
from P. This is the basis of all practical methods of com-
paring illuminating powers.
As a means of testing when two different sources of light
produce equal illumination at a point, various so-called
screens have been used. The one that has been most
extensively employed is Bunsen's grease-spot screen. It is
based on the fact that a grease-spot on paper is invisible
when the paper is equally illuminated on both sides, since
viewed from one side as much light is gained by transmission
from the farther side as is lost by transmission to the farther
side.
Another screen more perfect in some respects is that of
Lummer and Brodhun. A white opaque disk (see figure 34)
is illuminated on opposite sides by the two sources of light.
An arrangement of mirrors and lenses enables one eye to
view both sides at once. Two plane mirrors reflect rays
from the two sides into a double glass prism. This consists
of two separate right-angled prisms, the largest face of one
being partly beveled away and the two being cemented to-
gether by Canada balsam, which has the same optical density
as the glass, and therefore reflects no light. The central
rays from the left pass through the double prism to the tele-.
126
PHOTOMETRY. 127
scope while the marginal rays are totally reflected by the
beveled edge. The marginal rays from the right are totally
reflected and reach the telescope, but the central rays pass
through. Thus the eye sees a circular portion of the left
side of the opaque disk and a surrounding rim of the right
side.
To eliminate error from lack of symmetry, the lamps
compared should be interchanged in the course of the readings
or the screen should be rotated 180.
The lights to be compared are
mounted at opposite ends of a
graduated bar 3 meters long, which,
with a parallel bar and suitable sup-
ports, constitutes the photometer
bench. The screen is mounted on a
carriage movable along the bench.
Many light standards have been
employed. A candle of certain care- FlG
fully specified dimensions was long
employed, and the illuminating power of such a candle is still
regarded as the unit and called "one candle-power," but,
for practical purposes in testing, some other standard is
usually employed. The best such standard is a lamp, with
a wick of specified form and dimensions, burning amyl
acetate with a flame of specified height. (See references.)
Its relation to the "candle-power" is i c. p. = 1.14 amyl
acetate units. For most purposes an incandescent lamp
that has been standardized is the most useful standard,
especially in the study of incandescent lamps; but it must
not be used any great length of time without being re-stand-
ardized, since its illuminating power changes with prolonged
use.
The chief difficulty in comparing two different forms of
light is due to the fact that a difference of quality of the
two lights renders perfectly equal and similar illumination
of the two sides of the screen impossible. This difficulty
is still more marked in the study of arc-lights (for mechan-
128 LIGHT.
ical arrangements see Stine, p. 236), for which it is best to
use as an intermediate unit a very powerful incandescent
lamp. The latter may be standardized by comparison with
an ordinary incandescent lamp, which again is compared
with an amyl acetate standard. Before connecting a lamp
to a circuit, ascertain that the voltage is not excessive for that
particular lamp.
(A) Carefully standardize an incandescent lamp, for
use as a working standard, by comparison with either an
amyl acetate lamp or a standardized incandescent lamp.
If the latter is used, the lamps should be in parallel, that the
voltage may be the same, and a variable resistance should
also be in the circuit, by varying which the voltage across
the lamps is maintained at the value prescribed for the
standard. See that the filament of the standard is in the
marked azimuth and note the position of the filament for
which the other lamp is standardized.
In each case several settings of the screen should be
rapidly made, and then the screen reversed and several more
made. The calculations may be facilitated by Table XXL
(B) The law of inverse squares should be tested by com-
paring two somewhat different incandescent lamps (i) when
3 m. apart, (2) when 2.5m. apart, (3) when 2 m. apart
on the photometer bench. The ratio of their illuminating
powers, as deduced in the three cases, should be a constant.
(C) The horizontal distribution of candle-power about
an incandescent lamp should be studied. This incandes-
cent lamp should be connected in parallel with the working
standard and the voltage maintained at the value for which
the latter was standardized. The lamp should be mounted
on the revolving lamp-holder of the photometer, care being
taken to have the center of the filament at the same height
as the center of the screen. The lamp is first turned to the
standard position, i. e., the position in which the plane of
the shanks of the filament is at right angles to the photom-
eter bench, and a marked face of the lamp is toward the
screen. The candle-power of the lamp is to be found in
PHOTOMETRY. 1 29
this position and at positions 30 apart as the lamp is ro-
tated through 360. Two careful readings should be made
at each angle and the c. p. deduced from the mean. The
mean of all these values of the c. p. gives the mean horizon-
tal candle-power, A curve should be plotted, giving the
distribution of c. p. in polar co-ordinates. The mean hori-
zontal candle-power is more easily determined by continu-
ously rotating the lamp about a vertical axis by means of a
small motor.
(D) Efficiency of an Incandescent Lamp. Keeping the
potential of the working standard at the proper point (or
calibrating and using a lamp which may be connected to the
lighting circuit if the potential of that is constant), apply
various potentials to the lamp used in (C) at intervals be-
tween about 25% below the normal voltage to 25% above.
For each potential, determine the candle-power and current.
Calculate the watts consumed and the watts per candle-
power.
In the report, plot in three curves, with volts as abscissae,
(a) current, (b) candle-power, (c) watts per candle-power.
The scales of the three curves should be shown on the
vertical axis.
(E) Mean Spherical Candle-power. With the lamp in the standard
position of (C), find the c. p. at intervals of 30 in a vertical circle by
rotating the lamp about a horizontal axis. After this, start again
from the standard position and first turn the lamp through 45 in
azimuth (or around a vertical axis), and then, as before, find the c. p.
at intervals of 30 in the vertical circle of 45 azimuth, and so for the
vertical circles of 90 and 135 azimuth. As before, plot the curves
of distribution in polar co-ordinates.
To find the mean spherical candle-power omit any repetitions
and take the mean of the readings in the following positions :
1 . At tip i
2. At 60, 120, 240, 300 on the vertical circles of o and 90
azimuth 8
3. At 30, 150, 210, 330 on the vertical circles of o, 45, 90,
135 azimuth 16
4. 12 equidistant positions on horizontal circle 12
5. At base (o) i
Total, 38
These directions are chosen because they are nearly uniformly
distributed in space.
9
130 LIGHT.
(F) If time permit, study the differences of quality of
light given by different sources; e. g., compare an oil lamp and
an incandescent lamp using interposed colored glasses:
(i) a pair of red glasses, (2) of yellow glasses, (3) of blue
glasses. Calculate the relative illuminating powers in each
case.
The possible error may be deduced as usual from the
mean deviation of the readings in a set.
Questions.
1. Explain the deviation of the current-voltage curve from a
straight line.
2. What is the advantage in increasing the voltage applied to an
incandescent lamp? Disadvantage?
XXXIII. SPECTROMETER MEASUREMENTS.
Text-book of Physics (Duff), pp. 387, 436, 437, 440; Watson's Phys-
ics, pp. 468, 469, 493; Antes' General Physics, pp. 459, 460,
505, 506; Crew's General Physics, p. 510; Edser's Light, pp. 86-91.
A spectrometer consists of a framework supporting a
telescope and a collimator, both movable about a vertical
axis, and a platform movable about the same axis. The
platform is for supporting a prism or grating. The colli-
mator is a tube containing an adjustable slit at one end and
a lens at the other end. The purpose of the collimator is
to render light coming from the slit parallel after it leaves
the lens. (Only when the light that falls on a prism is par-
allel light, that is, light with plane wave front, does it seem
when emerging from the prism to come from a clearly de-
fined source. When it is not parallel, there is spherical
aberration.) Hence the slit of the collimator should be in
the principal focus of the lens. The telescope is for the
purpose of viewing the light that comes from the collimator,
either directly or after the light has been refracted or
reflected. Hence, since the light that comes from the colli-
mator is supposed to be parallel, that is, as if it came from
a very distant source, it follows that if the telescope is to
SPECTROMETER MEASUREMENTS. 131
receive the light and form a distinct image of the slit, the
telescope must be focused as for a very distant object (theo-
retically an infinitely distant one) .
The first adjustment is to focus the telescope. First
focus the eye-piece of the telescope on the cross-hairs and
then focus the whole telescope on a distant object out of
doors. The telescope will now be in focus for parallel rays.
Turn the telescope to view the image of the slit formed by
the collimator and adjust the slit until its image is seen
most distinctly.
That the instrument should be in complete adjustment,
it is necessary that the telescope, collimator, and platform
should rotate about the same axis, and that the optical axes
of the telescope and the collimator should be perpendicular
to this axis of rotation. For fine work spectrometers are
made with all these parts separately adjustable, but simpler
instruments have the telescope and collimator put into per-
manent adjustment by the instrument maker. In any case
the telescope and collimator should not be adjusted for level
without the advice of an instructor.
Adjustment of Prism. The refracting edge of the prism
must be made parallel to the axis of the instrument. Place
the prism on the platform with one of the faces perpendicular
to the line joining two of the leveling screws. Turn the
collimator slit horizontal and place the telescope so as to
receive the image of the slit reflected from this face of the
prism. Adjust these two leveling screws until the image of
the stationary edge of the slit coincides with the horizontal
cross-hair. Then observe the image reflected from the
other face and adjust the third leveling screw until the edge
of this image is on the horizontal cross-hair. A little con-
sideration will show that, when these two adjustments have
been made, both faces, and therefore the refracting edge,
are parallel to the axis. Restore the collimator slit to the
vertical position.
Measurement of the Angle of a Prism. Method (A). The
prism should be so placed that the faces are about equally
132 LIGHT.
inclined to the collimator. To secure good illumination, the
edge of the prism should be near the axis of the instrument.
The telescope is turned to view the image of the slit in the
two faces alternately, and the scale and vernier read when
the slit and cross-hair coincide, the slit being narrowed until
barely visible. If the scale is provided with two verniers,
to eliminate error from eccentricity, always read them both.
Half of the angle between the two positions of the telescope
gives the angle, A, of the prism, as may be readily seen by
drawing a diagram. The readings on each side should be
repeated three times.
Method (B). The following method, which is some-
times easier than the preceding, may be used if the platform
that carries the prism can be rotated and the rotation read
by a scale. Turn one face of the prism so as to reflect the
image of the slit into the telescope. Adjust the telescope
until the vertical cross-hair coincides with the slit and then
read the platform scale. Now rotate the platform until the
other face of the prism reflects the slit and again read the
platform scale. The difference of the readings is 1 80 A, as
may readily be seen by drawing a figure. The observation
should be repeated at least three times.
We are now in a position to make a final measurement
for finding the index of refraction of the glass of the prism
for any particular light of the spectrum, for instance,
sodium light (see p. 124). The only additional measure-
ment necessary is the deviation produced by the prism when
it is in such a position that it gives a minimum deviation to
the light refracted through it.
Minimum Deviation. The position of minimum devia-
tion is such that the image of the slit seen in the telescope
moves in the same direction (that of increasing deviation)
no matter which way the platform carrying the prism is
turned. There are, of course, two positions in which the
deviation can be obtained, one with the refracting edge turned
toward the right of the observer, and the other with it
toward the left. The deviation in each case is the angle
SPECTROMETER MEASUREMENTS. 133
between the corresponding position of the telescope and its
position when looking directly into the collimator, the prism
being removed. But it is not necessary to remove the
prism, for it is easily seen that the minimum deviation must
also be equal to half of the angle between the two positions
of the telescope when observing the minimum deviation.
FIG. 35.
These two positions should be observed three times success-
ively, and the mean value for the minimum deviation, D,
taken. From A and D the index of refraction may be cal-
culated by the formula
sin
If time permit, determine the index of refraction for as
many other wave-lengths (colors) as possible (see p. 124).
The possible error of the determination of the refractive
index can be calculated by means of formulae deduced by
the calculus, as explained on pp. 7, 8. A simple, but
less accurate method is to recalculate n with A and D,
increased by their mean deviations and to consider the
difference between this value and the original value as the
possible error.
It is probable that in this experiment there are other
sources of error that exceed mere error in reading the
scale; e. g., (i) The faces of the prism may not be true
planes, (2) the divided circle may not be uniform, (3) the
center of the circular scale may not coincide with the cen-
ter of the instrument, (4) the various adjustments may not
be perfect, (5) there may be difficulty in fixing the position
134 LIGHT.
of minimum deviation. These errors might be eliminated by
repeating all the adjustments and observations many times
and using different parts of the divided scale. There is no
other way of allowing for them.
Questions.
1. Give both physical and mathematical definitions of the refract-
ive index.
2. Why is monochromatic light used?
3. Why is the minimum deviation chosen?
XXXIV. MEASUREMENT OF RADIUS OF CURVATURE.
Glasebrook and Shaw, Practical Physics, pp. 339-343; Edser, Light,
pp. 116-121; Koklrausch, pp. 174-176.
The radius of curvature of a surface may be determined
from the size or position of the image which the spherical
surface, regarded as a mirror, forms of a definite object.
Method (A) below is especially applicable to the measure-
ment of the radius of curvature of convex surfaces, and
method (B) to concave surfaces.
(A) Two bright objects (see Fig. 36) are placed on a line
at right angles to the axis of the spherical surface, the
intersection of the line and the axis being at a considerable
distance A, from the surface, and each object being at a
distance L/ 2 from the axis. If the apparent distance be-
tween the images of the two objects be /, the radius of curva-
ture of the surface is
~
the + sign being used for a concave surface and the sign
for a convex.
Proof.
(For convex mirror.)
Let <i = true distance between the images, x = distance of images
from mirror.
By geometry
L A+r d_A+x . L (A+r)(A+x)
d~r-x' l~ A ' '' I "' (r-x)A '
MEASUREMENT OF RADIUS OF CURVATURE. 135
By the equation for spherical mirrors (see p 125).
\ 112
x r A r
r x_A+r
roc ~ Ar
L A + x A
.'. -y = -- =1 + .
/ X X
From ( i ) = i H
OC T
L iA
The radius of curvature should be found for both surfaces
of a double convex lens. The lens, preferably the one used
in Exp. XXXV, if that has already been performed,
is fitted in a clamp in a darkened recess. At some distance
are two vertical slits illuminated from behind by incan-
descent lamps (or the lamp filaments themselves may be
used) and between them a *>
telescope. The telescope and
the lens are adjusted until, on
looking through the telescope
toward the lens, the illumi-
nated slits are seen reflected
from the near surface of the
lens. Distinguish these im-
ages from the images produced by the rear surface of the lens
by the change of focus necessary to make one pair of images
most distinct, and then to make the other pair most distinct,
or, by observing the two images of a light held just outside
one of the slits. Remember that the telescope inverts.
A paper scale is pinned over the lens so that the upper edge
is just below the center of the lens. The telescope is focused
upon the scale, and rotated until the vertical cross-hair
bisects one of the images from the near surf ace 'of the lens,
and the scale read where crossed by the cross-hair. (Esti-
136 LIGHT.
mate tenths of millimeters as always.) A similar reading
is made for the other image. The difference between the
two readings gives the apparent distance, /, between the
images. At least six independent determinations of this
distance should be made. Measure the distance, L, between
the slits, the distance, A, from the lens surface to the line
joining the slits and substitute in the formula.
From the two radii of curvature and the focal length, if
known, calculate the refractive index, n, of the glass of the
lens by means of the formula
(B) As a concave mirror we may use one of the surfaces
of a concave lens, mounted in a lens-holder. To reduce
reflection from the other surface, the latter may be covered
by moist filter paper. The radius is determined from u,
the distance of the object, v, the distance of the image and
the formula
112
-+-=-, (seep. 125).
u v r
In locating the image, use is made of the fact that if the
eye is a considerable distance off, a real image can be seen
in space as well as a virtual image, and a wire, needle, or
pointer is moved about until there is no parallax between it
and the image; i. e., until, when the eye is moved about,
there is no relative motion of the two.
A vertical wire illuminated by a lamp, behind which is
a sheet of white paper, is a convenient object, and a second
mounted wire is moved about until it coincides with the
image of the first (see (B) Exp. XXXV). The image
should be found for at least the following three typical
positions of the object. For each position make several
settings and from the means determine u and v, and from
them determine r.
(i) Let the object be at a considerable distance from the
mirror.
FOCAL LENGTH OF LENS. 137
(2) Let the object be at the center of curvature of the
mirror. In this position the image and the object coincide.
(3) Let the object be within the principal focus. For this
position the wire locating the image must be on the other
side of the lens. This wire is moved about until the pro-
longation above the lens of the image of the
first wire coincides with what is seen of the
second wire above the lens.
In the report, sketch the relative positions
of mirror, image, and object, and state whether '
the image was magnified or diminished, erect
or inverted.
(C) If time permit, check your results with a
spherometer (see p. 16). The spherometer
should be read alternately on a plane surface
and on the lens. Let a = difference in the
two readings (see Fig. 37) and r = radius of the circle of the
legs. Then the radius of curvature of the lens is
as may be easily shown.
Questions.
1. For what lenses would the first method of determining the radius
of curvature be preferable, and when would the spherometer be
preferable ?
2. What objection is there to determining the radius of curvature
of the farther face of a convex lens, considering it a concave surface?
3. What advantages has the method used in (B) for locating real
images over the use of a screen ?
4. How could you directly determine with a screen the center of
curvature of a concave mirror?
XXXV. FOCAL LENGTH OF A LENS.
Text-book of Physics (Duff), pp. 392-398; Watson's Physics, pp. 471-
479; Ames' General Physics, pp. 470483; Crew's General Physics,
pp. 466-469; Edser, Light, pp. 110-116; Glazebrook and Shaw,
Practical Physics, pp. 343-352.
The focal length of a lens is the distance from the optical
center of the lens to the focus for rays of light from an
138 LIGHT.
infinite distance; i. e., for plane waves. If / is the focal
length, u the distance of the object from the lens, and v
that of the image, then, with the convention respecting signs
given on page 125, for all lenses
iii
v u f
(A) Real Image. (li Exp. XXXIV has preceded, use
the same lenses.) An "object," the lens, and a screen
for receiving the image of the object, are mounted so that
they can be moved along a graduated scale. A convenient
form for the object is a wire cross or gauze, mounted in a
black wooden support, and illuminated from behind by an
incandescent lamp. The lens is clamped in a wooden frame
movable along the scale. This should grasp the lens on the
sides, leaving the top and bottom clear. The distance from
the center of the lens to some point on the support must be
determined once for all and applied as a correction to the
readings. With object and screen in fixed positions that are
recorded, the lens is adjusted until the image
on the screen is as distinct as possible and
its position is then recorded. This should
be done several times, and the mean taken
for the position of the lens. Keeping object
and screen fixed and moving the lens about,
, another image will be found, for which similar
observations should be made. Calculate / from the averages
of all the values of u and v. The object and screen should
then be shifted and the observations repeated. From the
two sets of observations a mean value of / is deduced.
Study of Spherical Aberration. Determine / for the central
part of the lens by covering, with a pasteboard screen,
all but a central disk of about one-third the diameter of
the lens. Similarly determine / for the edge of the lens,
using a diaphragm covering all but the edge.
Study of Chromatic Aberration. Using the entire lens,
FOCAL LENGTH OF LENS. 139
determine / for red light by placing red glass before the lens
or object, and similarly for blue or green light.
In the report, tabulate, for comparison, the different
mean values of /.
(B) Virtual Image. In the preceding a real image was
observed, but the focal length may also be found from
observations of a virtual image. The following directions
apply to a divergent lens. A vertical dark line on a white
background serves as object. The image (between the lens
and the object) is located with a short vertical wire, which
is moved back and forth until a position is found where
the image of the dark line seen through the lens (a in Fig. 38)
appears at the same distance as the portion of the wire seen
just below or above the lens (b in Fig. 38). This is secured
when there is no relative motion of the image and this wire
as the eye is moved horizontally, i. e., the wire appears as
the prolongation of the. image of the dark line or remains
equidistant from such a prolongation, v will be the distance
from the center of the lens to this wire which locates the
image. Using a longer wire as the object and the dark line
to locate the image, this method may be applied to the
virtual image of a convergent lens.
Estimate the possible error of a typical measurement
of /. . Since practically all the error is in the location of the
lens, the distance between the object and the screen may
be considered free from error. If this distance is desig-
nated by w , the formula becomes
,
u wu j
from which a formula may easily be derived for the possible
error in / in terms of the possible error in u. The latter may
be taken as the mean deviation from the mean in the location
of the lens (see p. 4).
If Exp. XXXIV has preceded, determine the refract-
ive index of the glass from the focal length and the radii
of curvature.
140 LIGHT.
Questions.
1. What is the minimum distance between object and screen to
secure a real image? The maximum distance between object and
lens to secure a virtual image?
2. What advantage is there in covering with a diaphragm all but
the central portion of a lens? What disadvantage?
3. What is the cause of chromatic aberration?
4. What sort of a lens would show large spherical aberration?
Large chromatic aberration?
XXXVI. LENS COMBINATIONS.
Edser, Light, Chaps. VI, VII, X; Watson's Practical Physics, pp. 358-
367; Drude's Optics, pp. 44-46, 66-72; Hastings' Light, Appendix
A. Antes' General Physics, pp. 488, 493, 494.
(A) Determination of Principal Foci. Calculation from
Focal Lengths and Separation. Let two lenses of focal lengths,
/!, and/ 2 , be separated a distance d. An object at a distance
u from the first lens forms an image at a distance v deter-
mined by the equation
v ^ u uf l '
This image acts as an object for the second lens at a
distance d v. Hence the distance of the final image from
the optical center of the second lens is given by the equation
i i i i u ~fi
v' /a d ~ v / 2 du-dfi-ufi
If u is infinite, v r is the distance, V, of the principal focus
from the optical center of the second lens,
Experimental Location. Determine experimentally the
position of this principal focus by finding the position in
which an object must be placed for clear vision when it is
viewed through the combination with a telescope focused
for a very distant object. Compare with the calculated
position. Determine similarly the other principal focus.
LENS COMBINATIONS. 141
(B) Determination of Focal Length. To determine the
focal length, the position of the "principal points" must
be known, as well as the principal foci. With thin lenses,
the principal points practically coincide at the so-called
"optical center." In thick lenses or lens combinations they
may be considerably separated. The focal length is the
distance from either principal focus to the nearer principal
point. The equations for locating the image,
iii
-+-=7,
it v f
and for finding the linear magnification,
7'
M = -
u
are applicable in all cases, if u and v are measured from the
principal points.
Since it is difficult to locate the principal points, a method
is often employed for determining the focal length which
eliminates their position. Suppose that the linear mag-
nification is M lt when the distance of the object from the
principal plane is u lt and M 2 when the object is moved
until its distance is u.
If the focal length is small, the magnification should
be determined with a micrometer microscope. A carefully
graduated scale is a convenient object and the size of the
image of one or more divisions is measured for two positions
of the object a known distance apart. (Principle of Abbe's
Focometer.)
Thus determine the focal length of the combination
used in (A). Determine also the focal length of both the
objective and the eye-piece of a telescope, using the same
142 LIGHT.
one as employed in Exp. XXXVII, if that has pre-
ceded, and calculate the magnifying power for great
distances.
In the report, draw careful figures of the lens combina-
tions, representing the principal foci and principal points
as calculated from the final mean results.
(C) If the principal points of a convergent system are
close together, i. e., if the lens or lenses may be said to have
an optical center, we may use the following approximate
method: If w is the distance between object and image,
and x that between the two positions of the lens for real
images, u=(wx)/2, v=(wx)/2. Substituting these
values in the formula we get
Zf 2 X 2
4W
If time permit, try this method for the combination of
lenses.
Describe a lens combination which (i) magnifies without
distortion; (2) magnifies without chromatic aberration;
(3) inverts without magnifying. (See references.)
XXXVII. MAGNIFYING POWER OF A TELESCOPE.
Glazebrook and Shaw, pp. 358363; Watson's Practical Physics, pp.
367, 368.
The magnifying power of a telescope is the ratio of the
angle subtended at the eye by the image as seen through
the telescope to the angle subtended by the object viewed
directly. (If Exps. XXXVI and XXXVIII have already
been performed, use the telescope employed in those
experiments.)
(A) Direct Method. A minute mirror is attached to the
telescope by wax so as to make an angle of about 45 with
the axis and partly cover the aperture of the eye-piece.
The telescope is focused upon a scale. A second scale is
mounted parallel to the first and near the eye-piece, in such a
MAGNIFYING POWER OF A TELESCOPE. 143
position that the observer's eye sees, side by side, the image
of the scale viewed though the telescope and the image of
the other scale reflected in the small mirror. From the
ratio of the images of one or more scale divisions, and their
distances, the angles are calculated, and from their ratio the
magnifying power is deduced.
Find the magnifying power for at least six distances,
making several observations for each. Also determine
the angular field of view of the telescope by determining for
each distance, r, the total distance on the scale, n, visible
in the telescope. The angular field of view, in degrees, will be
The magnifying power defined above is very approximately
equal to (i) the ratio of the magnitude of the image to the
magnitude of the object when the two are in the same
plane, and, for great distances, is equal to (2) the ratio of
the focal length of the objective to the focal length of the
eye-piece.
.*
~~---K)*
---'' f>
FIG. 39.
The eye-piece is of such short focus that the angle subtended by
its image is practically the same as if the image were at infinity. For
convenience we will consider the virtual image P' Q' (see Fig. 39)
produced by the eye-piece to be at the same distance as the object
Since the telescope usually views objects at distances great com-
pared with its own length, the angle subtended by the object viewed
directly is practically P a Q = p a q, and that subtended by the image
is P' b Q f = p b q. The ratio of these two angles, which may be taken
as the ratio of the tangents, since the angles are small, = a c -r- c b = the
ratio of the focal length of the objective to the focal length of the
eye-piece; and also, since the length of the telescope is short compared
with the distance of the object, this ratio = P' Q' + P Q, or the ratio
of the magnitude of the image to the magnitude of the object.
144 LIGHT.
(B) The first approximate statement of the magnifying
power furnishes another method for determining the mag-
nifying power for different distances of the object. The
telescope is directed toward a horizontal scale. The scale
is viewed through the telescope with one eye and is also
observed with the other eye by looking along the outside
of the telescope. The eye-piece is moved in or out until the
image appears at the same distance as the scale as viewed
outside the telescope with the other eye, i. e., until there is
no parallax between the scale and its image (no relative
motion of the two as the eye is moved about). It may
require some practice to secure this. Determine the number
of divisions on the scale which, as viewed directly, are
covered by the image of one or two large divisions as viewed
through the telescope. If it is difficult to read the division
on the scale viewed directly, two black strips may be moved
along the scale until they include the image of one or more
divisions as seen through the telescope, and the distance
between these strips read off. Repeat the measurements
of (A).
(C) The second method of defining the magnifying power
of a telescope is useful in determining the magnifying power
'for very distant objects. Focus the telescope on some very
distant object. Without changing the focus, remove the
object glass and substitute for it a diaphragm with a rec-
tangular opening. The ratio of the focal length of the
objective to the focal length of the eye-piece is the ratio of a
linear dimension of the aperture of the diaphragm, L, to the
corresponding dimension, /, of the image of this aperture
produced by the eye-piece.
Since the telescope was focused for parallel rays, the distance, u,
of the object, L, from the eye-piece is numerically very nearly the
sum of the focal lengths, P+f (Fig. 40). .'.the distance of the
image, /, formed by the eye-piece, is determined by
i = _-i i -f+(F+f) F
v (F+fVf- /(F+/) /(F+/)'
Hence,
RESOLVING POWER OF OPTICAL INSTRUMENTS. 145
To measure /, a micrometer microscope (see p. 15) may
be used, the microscope being in line with the axis of the
telescope and focused upon the real image in space. L may
FIG. 40.
be measured with vernier calipers, or the same micrometer
microscope may be placed opposite the other end of the
telescope and L measured in the same way as /.
Questions.
1. Explain why the magnifying power should vary as you have
found it to do with the distance of the object.
2. Which is preferable to gain magnifying power by increasing
the focal length of the objective, or by decreasing the focal length of
the eye-piece? Why?
XXXVIII. RESOLVING POWER OF OPTICAL
INSTRUMENTS.
Text-book of Physics (Duff), pp. 421, 422; Watson's Practical Physics,
pp. 335338; Antes' General Physics, pp. 483487; Mann,
Advanced Optics, pp. 11-18; Drude's Optics, pp. 235-236; Hast-
ings' Light, pp. 7072.
The magnification obtained with an optical instrument
depends upon the focal lengths of its lenses, as has been
seen in the case of the telescope. The ability to distinguish
details of the image, i. e., the "resolving power," depends
on the diameter of the aperture through which light enters
the instrument.
If d is the distance between two details of an object at
a distance D from an aperture whose width parallel to these
details is a, they may be distinguished if
d I
D>a
10
146 LIGHT.
where / is the wave-length of the light employed. If the
aperture is circular, the equation is
d I
where a is the diameter of the aperture.
(A) Resolving Power of Telescope. Metal gauze answers
as a very satisfactory object for studying the resolving
power, as it gives a great amount of uniform detail. Since
this detail consists of rectangular lines, and the aperture of
the object glass is circular, the determination of the maxi-
mum distance at which the lines are discernible will be more
definite, if the aperture is made rectangular by placing a slit
in front of the object glass.
Determine carefully by several settings, the maximum
distance, D, at which the lines of the gauze, parallel to the
slit, are perceived. The gauze should be illuminated from
behind by monochromatic light (p. 124). Measure carefully
the distance, d, between the centers of the adjacent wires
of the gauze, and the width of the slit, a. Compare d/D
with //a; / may be obtained from Table XVIII. Repeat
with other slits and other gauzes.
(B) Resolving Power of Eye. With Porter's apparatus
the resolving power of the eye may be determined for
various apertures.
Four different gauzes, 37 .6, 27, 20, and 14, meshes to the
cm., respectively, may be viewed through four different
apertures of diameters i.oo mm., 0.65 mm., 0.53 mm.,
and 0.35 mm., respectively. The resolving power is de-
temined by finding the distance, D, from a slit of diameter
a to a gauze, of which the distance between the centers of
two adjacent wires is d, when the wires are separately dis-
cernible. From the mean position of several settings of a
particular gauze for a particular aperture, d/D should be
calculated and compared with 1.2 I /a. If ordinary light is
used, / may be taken as 0.00006 cm. Use each aperture
and gauze in succession.
WAVE-LENGTH OF LIGHT BY DIFFRACTION GRATING. 147
Questions.
(1) Upon what does the illumination of the image of an optical
rnstrument depend?
(2) When the diameter of the pupil of the eye is 4 mm., how far
away may two points be distinguished which are o. 2 mm. apart?
XXXIX. WAVE-LENGTH OF LIGHT BY DIFFRACTION
GRATING.
Text-book of Physics (Duff), pp. 423, 437; Watson's Physics, pp.
529-532; Ames' General Physics, pp. 530-537; Crew's General
Physics, pp. 488491; Edser, Light, pp. 448-458; Wood, Physical
Optics, pp. 168-180.
A diffraction grating consists of a great many lines ruled
parallel and equidistant on a plane (or concave) surface.
If the surface be that of glass, the grating is a transmission
grating; if of metal, a reflection grating. If a transmission
grating be placed perpendicular to homogeneous parallel
light from a collimator (see Exp. XXXIII) and with
the lines parallel to the slit, a series of spectra will be formed
on either side of the beam of light transmitted without
deviation. If n be the number or order of a particular
spectrum counting from the center, 6 the deviation or angle
that the rays forming the spectrum make with the original
direction of the light, a the grating space or average distance
between the centers of adjacent lines, and / the wave-length
of the light
I a sin 6.
n
The deviation 6 may be observed by placing the grating
on a spectrometer (see Exp. XXXIII where the ad-
justments of the spectrometer are described). The
position of the telescope when in line with the collimator is
read. The grating is adjusted parallel to the axis of rotation
of telescope and collimator as one face of a prism is adjusted.
For convenience in adjusting, the plane of the grating should
be perpendicular to the line of two of the leveling-screws.
This enables us to adjust the lines of the grating parallel
148
LIGHT.
to the slit by means of one leveling-screw without altering
the plane of the grating. The lines are parallel to the slit
when the spectrum of some homogeneous light, e. g., from a
sodium flame (p. 124), is as distinct as possible. When the
plane of the grating is perpendicular to the incident light, the
deviations (on opposite sides) of the two spectra of the same
order should be equal. This adjustment is also secured
when that part of the beam which is
reflected back to the collimator appears
co-axial with its object glass.
Determine first the wave-length of
sodium light. For a final measure-
ment of the deviation of any spectrum
the mean of at least three measure-
ments on each side should be taken.
The deviations of all the spectra clearly
visible should be obtained.
If the grating space be not too small
it may be obtained by measurements
on a dividing engine (p. 17), or with a
micrometer microscope (p. 15). In determining the grating
space with the dividing engine, secure the best possible
illumination of the lines. Set the cross-hair of the micro-
scope on a line and read the position of the divided head
(circular scale). Watching the lines through the micro-
scope, turn the screw, always in the same direction, until,
for example, the tenth line is under the cross-hair, and read
the circular scale. Then turn the screw until the tenth line
from this is under the cross-hair, read the scale, and so on.
Take ten such groups in different parts of the grating.
Find the average grating space from the mean. When the
grating space is very small, the wave-length of some well-
known spectrum (e. g., sodium) is assumed in order that the
grating space may be derived by reversing the process of
finding the wave-length.
If time permit, determine the wave-length of as many
other lights (colors) as possible (see p. 124).
FIG. 41.
INTERFEROMETER. 149
Questions.
1. What do you observe as regards the width of spectra of
different orders? What would this indicate as regards the disper-
sion if mixed light or light from an incandescent solid were used ?
2. What is a normal spectrum and wherein does a prismatic
spectrum differ from a diffraction spectrum? (See references.)
XL. INTERFEROMETER.
Text-book of Physics (Duff), p. 438; Watson's Physics, p. 540; Mann,
Advanced Optics, Chap. V; Wood, Optics, Chap. VIII; Michelson,
Light Waves and Their Uses.
The interferometer is an instrument for determining the
number of wave-lengths of a monochromatic light con-
tained in a given distance. For a description of the inter-
ferometer and the adjustments, see the references.
The interferometer will be used to determine the wave-
length of sodium light, assuming a knowledge of the true
pitch of the screw. This will | i
illustrate the more practical
and common, but also more
difficult, utilization of the in-
terferometer in determining a
length, assuming a knowledge
of the wave-length of the light
employed.
The light 5 (see figure)
had best be monochromatic, FlG 42
e. g., a sodium flame. Initially,
place the mirror D at approximately the same distance
from the rear face of A as the distance from this surface
to C. Adjust C until its image coincides with either of
the images from D. (There will be two images owing to
reflection from the two faces of A.) A slight adjustment
will now give the fringes (alternate light and dark bands,
preferably arcs of circles) . The observer must look at A in
a direction parallel to AD.
Move the mirror D by means of the worm, and count
150 LIGHT.
the number of fringes which pass over the field of view.
A needle in front of A may help as an index.
From the number of turns and fractional turns of the
screw and the value of the pitch of the screw, find the
distance D has moved and from this and the number of
fringes which have passed, calculate the wave-length.
Notice that the length of the path of the light changes by
twice the displacement of the mirror D.
XLI. ROTATION OF PLANE OF POLARIZATION.
Text-book of Physics (Duff), pp. 476-479; Watson's Physics, pp. 580-
582; Ames' General Physics, pp. 563-565; Watson's Practical
Physics, pp. 370377; Edser, Light, pp. 503509; Wood, Optics,
Chap. XIV; Ewell, Physical Chemistry, pp. 217-223.
Plane polarized light is obtained by passing light through
a Nicol prism. If the light be then allowed to fall on a
second Nicol prism that can be rotated, there will be two
positions of this second prism in a complete rotation in
which no light will pass through. If an optically active
substance, such as a solution of cane sugar, be then intro-
duced between the two Nicols, it will rotate the plane of
polarization of the light which falls on the second prism,
and then, to quench the light, the second prism must be
rotated through an equal angle. Thus the rotation pro-
duced by the sugar is measured.
Monochromatic light must be used and a sodium flame
is most convenient (see p. 124). The light rays must be
made parallel before they fall on the polarizing prism, other-
wise rays in different directions would pass through different
thicknesses of the sugar and would consequently be rotated
by different amounts. Parallel light may be obtained by
putting the source at the principal focus of a convex lens
through which the light has to pass before falling on the
polarizing Nicol. The light must also be parallel to the axis
of the Nicol.
ROTATION OF PLANE OF POLARIZATION. 151
The empty tube intended to contain the sugar solution is
first placed in position between the prisms and the position
of the analyzing Nicol noted, on the circular scale, when the
light is quenched. This setting will be facilitated by using
a screen to cover all but a small central part of the prism.
It may be found that the Nicol can be rotated through an
appreciable angle without the light reappearing. The best
that can be done is to take the middle of this space as the
position of extinction. The observation should be repeated
a number of times and the mean taken. The analyzing
Nicol should then be rotated through 180 and the zero
reading in that position also noted. Sugar solutions of
different strengths (which should be carefully made up and
recorded) are then introduced in succession into the tube and
the rotations they produce observed. The zero readings
should be frequently repeated. The length of the tube
should also be obtained, so that the rotation per decimeter
may be deduced. With the results obtained, a curve
should be plotted, rotations per decimeter being ordinates
and concentrations abscissae.
FIG
43-
The apparatus here described is simple but very imperfect
in its action. The sensitiveness is greatly increased by
introducing between the polarizer and the specimen a so-
called biquartz, two parallel, abutting, plates of quartz,
one with left rotatory power and the other with right. A
source of white light must be employed, for example, a
frosted incandescent bulb, and the analyzer is set for equality
of color in the two halves. There are two common colors,
but the darker is preferable.
Fig. 44 explains the color changes. R and L are the two halves
of the bicjuartz, viewed from the analyzer. The two halves are of
such a thickness (3.75 mm.) that the plane of polarization of yellow
152
LIGHT.
light is rotated through 90. Owing to the rotatory dispersion
the other colors will be rotated different amounts as shown by the
letters R (red) and B (blue). If the analyzer is set to transmit light
vibrations parallel to those which left the polarizer, the yellow light
will be omitted and each half of the biquartz will appear of a purplish
color ("tint of passage"). If the analyzer
is displaced slightly clockwise, more of the
red component on the right will be trans-
mitted and less of the blue, and therefore
this half will appear red and the other
half will appear blue.
If a dextrorotatory specimen is placed
between the biquartz and the analyzer,
the directions of vibration of the different
colors will be rotated to the positions
indicated by the dotted lines and the
analyzer must be rotated to a new position
(/') , perpendicular to the emerging yellow
vibration, in order to have the two halves
the same color. With the help of the
biquartz the analyzer can be set within
about a tenth of a degree.
The effective thickness of the
biquartz will not be correct (90
degrees rotation of sodium light),
FIG. 44. unless it is perpendicular to the axis
of the Nicol prisms. This may be
secured by using sodium light and analyzer set for extinc-
tion, and then placing the biquartz in such a position that
there is still extinction when the analyzer is rotated 90.
Repeat all the measurements with white light and the
biquartz and plot the results on the same sheet with the
preceding.
QUESTIONS.
1. How can the rotation be partially explained ? (See references.)
2. What is the chemical characteristic of substances that are
optically active in solution ?
3. Wherein does the rotation produced by a solution differ from
that produced by a magnetic field?
4. What would be the effect of using white light in the first part
of the experiment?
ELECTRICITY AND MAGNETISM.
29. Resistance -boxes.
A resistance-box consists of a number of resistance coils
joined so that each one bridges the gap between two of a
series of brass blocks placed in line on the cover of the box
within which the coils are suspended. For each gap a plug
or connector is also provided, and when the plug is inserted
into the gap the resistance at the gap is " cut out " or practi-
cally reduced to zero. The coils are wound so as to be free
from self-induction. The successive resistances are arranged
in the same order, and are of the same relative magnitudes
as the successive weights in a box of weights. By removing
the proper plugs any combination of resistances can be
obtained from the smallest to the sum of all. Before begin-
ning work, it is advisable to clean the plugs with fine emery-
cloth so that they may make good contacts, and thereafter
care should be taken not to soil them with the fingers.
One important precaution in regard to the use of the
resistance-box should be observed. If any of the plugs
are in loosely, there will be some resistance at the contact.
Hence, the plug should be screwed in firmly, but not vio-
lently. When any* one plug has been withdrawn, the others
should be tested before proceeding, for the removal of one
may loosen the contact of the others. This precaution is
especially important in making a final determination.
30. Forms of Wheatstone's Bridge.
The practical measurement of a resistance consists in
comparing it with a known or standard resistance. Wheat-
stone's Bridge is an arrangement of conductors for facilitat-
ing this comparison, and consists essentially of six branches
153
ELECTRICITY AND MAGNETISM.
which may be represented by the sides and diagonals of a
parallelogram (see Fig. 45). The unknown resistance,
R, and the known resistance, 5, form two adjacent sides.
The other two sides are formed by two conductors of resist-
ance P and Q, which, however, do not need to be known
separately, provided their ratio be known. One of the-
diagonals contains a battery
and the other a galvanom-
eter. If the ratio of P to
Q is adjusted until no current
flows through the galvanom-
eter, R:S::P:Q. (See refer-
ences under Exp. XLIV.)
Two forms of the Wheat-
stone " Bridge arrangement
are in common use. One is
called the Wire (or meter)
Bridge; the other, which uses
/r
FIG. 45.
is called a Bridge Box.
a box of adjusted resistances,
In the wire bridge the "ratio
arms" (whose resistances are P and Q) are the two parts of
a uniform wire i meter long, and the ratio of P to Q is that
of the lengths of the corresponding parts of the wire. The
known resistance, 5, may be that of a standard coil or one
of the known resistances of a resistance-box.
The Bridge Box, or "Post-office Bridge," consists of a
resistance-box with three series of resistances in line, forming
three arms of the Wheatstone Bridge, the unknown resist-
ance forming the fourth arm. The "ratio arms" consist of
resistances of i, 10, 100, 1000 (all of which are not always
necessary), so that the calculation of the ratio is very
simple. Keys for closing the battery and galvanometer
branches are also usually mounted on the box.
GALVANOMETERS. 155
31. Galvanometers.
Text-book of Physics (Duff), pp. 559-563; Hadley's Electricity and
Magnetism, pp. 273-284; Watson's Practical Physics, 170-174;
Ames and Bliss, Appendix iii.
There are two chief types of reflecting galvanometers.
In both the principle at basis is that if a magnet be placed
in the plane of a coil of insulated wire, on passing a current
through the coil both magnet and coil become subject to
forces that tend to set them at right angles to each other.
In the Thomson type the coil is fixed and the magnet
suspended within the coil is free to turn, while in the
d'Arsonval type the magnet, of a horseshoe form, is fixed,
and the coil, suspended between the poles of the magnet,
is free to turn.
The sensitiveness of the Thomson galvanometer is greatly
increased in two ways: first, two magnetic needles, forming
an astatic pair, are attached to the same axis of rotation,
second, an external control magnet is used to weaken the
restraint of the earth's magnetic force or even to overcome"
the earth's field and produce a suitable field of its own.
The chief difficulty in greatly increasing the sensitiveness
by means of the control magnet is that slight variations of
the whole magnetic field, due to outside currents or move-
ments of magnetic materials in or near the laboratory,
disturb the needle.
On the d'Arsonval galvanometer variations of the external
magnetic field have practically no effect, since its own
magnetic field is very strong. On the other hand, the
torsion of the fine suspending wire through which the
current has to pass changes somewhat with the temperature,
so that the zero reading of the galvanometer is subject to
some change. The sensitiveness can be increased by in-
creasing the strength of the magnet, but there is a limit to
this, since small traces of iron are always present in the
wire and insulation of the coil, and this, acted on by the
magnetic field, exercises a magnetic control that is pro-
portional to the square of the strength of the field. When
156 ELECTRICITY AND MAGNETISM.
an extremely sensitive galvanometer for very accurate
work is required, the Thomson type must be used.
A ballistic galvanometer is a reflection galvanometer of
either type, so made that its period of swing is very long,
so that it starts into motion only very slowly. If this con-
dition be fulfilled, and if it be subject to only very slight
damping of its motion, the galvanometer may be used for
comparing quantities of electricity suddenly discharged
through the coils of the galvanometer, for, practically
speaking, all the electricity will have passed before the
swinging system has appreciably moved from this position
of rest. In these circumstances it can be shown that the
quantity of electricity is proportional to the sine of half the
angle of the first swing, or (since the angle is very small)
practically to the deflection as read on the scale.
Two methods of reading the deflection of a galvanometer
are in common use. In one, called the English or objective
method, a beam of light reflected from the mirror of the
galvanometer falls on a scale, forming a spot of light which
moves as the needle or coil is deflected. In the other,
called the German or subjective method, the image of a scale
formed by the mirror of the galvanometer is read by a
telescope with a cross-hair.
Devices for Bringing a Galvanometer to Rest. For bringing
to rest the needle of a ballistic Thomson galvanometer a
coil is mounted on the outside of the galvanometer in front
of the lower needle. The terminals of the coil are brought
to a reversing switch by which the current from a cell can
be sent through the coil in either direction. By suitably
choosing the direction and duration of the current, the
needles and mirror may be brought to rest. (A current
in this coil affects the needle in the same manner as would a
current in one of the regular galvanometer coils, but it is
much more convenient to use a separate coil like this, which
is readily accessible and which does not interfere with the
other connections.)
The suspended system of either type of galvanometer
GALVANOMETER SHUNTS. 157
may also be brought to rest by short-circuiting the galvan-
ometer by a simple key directly connected to the terminals.
For, by Lenz's Law, the currents induced are such as to
bring the moving coil or needle to rest. If the resistance
of the coils is high, this method is slow, and the following
more rapid method may be used. A coil in which a small
bar magnet can be moved is placed in series with the short-
circuiting key. By suitably moving the magnet in and out,
currents are induced which will quickly bring the suspended
system to rest.
32. Correction for Damping of a Ballistic Galvanometer-
Kohlrausch's Physical Measurements, 51; Stewart and Gee's Practical
Physics, II, pp. 364369.
In considering the throw proportional to the charge
passing through the coils of a ballistic galvanometer, we
assume that the galvanometer is free from damping; i. e.,
that the suspended system, needles, mirror, etc., experiences
no resistance to turning. Since this is never realized, a
correction must be applied to the throw.
The correction is not of importance where we compare
throws, since the correction cancels out, but in much
work with ballistic galvanometers this correction is very
important.
Set the needle vibrating and record n + i successive turn-
ing-points. From these we obtain by successive subtrac-
tion n successive full vibrations of the needle from one
side to the other. Call the first full vibration a l and the
last a n . Then the correction by which each throw should be
multiplied is (i -M/ 2 ) where
^^loga 1 -logq M
n i
\
33. Galvanometer Shunts.
If a galvanometer of resistance G is shunted by a shunt
of resistance 5 and if C is the whole current and C l the
current through the galvanometer
ELECTRICITY AND MAGNETISM.
C G+S'
Galvanometers are frequently supplied with shunt-boxes
in which the ratio of 5: G are 1/9, 1/99, 1/999, so "that the
values of 5 :(G +S) are i/io, i/ioo,
i/iooo. Such a shunt-box cannot
easily be used with any galvanom-
eter except that for which it was
designed.
Universal shunt-boxes are now
made which can be used with any
galvanometer. Such a box consists
of a series of high, resistances con-
nected as indicated in the figure. AB
is a coil, of resistance S, connected to the galvanometer, of
resistance G. Let the current through the galvanometer
be C lt and let the whole current be C. Then as above
c,= cs
VVvNA/VV\AAAA/V j?
FIG. 46.
Now let the battery circuit be connected to A and P, where
the resistance of AP is S/n. Denoting the current through
the galvanometer by C/, and the whole current by C and
making the proper changes in the above equation,
Cy_ Sin _i 5
~C ~S/n + (S-S/n)+G ~n S+G'
i CS
nS+G
Hence when a current is connected to A and P, the galvan-
ometer deflection is i/n as great as when the same current
is connected to A and B, or the sensitiveness is i/n as great.
By subdividing AB, the values of 3, 10, 100, etc., are given
to n.
Shunting a Ballistic Galvanometer. The formulas stated
above were deduced from Ohm's Law for steady direct
currents. It can, however, be shown that shunts like the
STANDARD CELLS. 159
above may be used in the same way with ballistic galvan-
ometers through which charges of electricity are passed.
To prove this, all we need to do is to show that charges,
like steady direct currents, divide in a parallel arc into parts
inversely as the ohmic resistances. Consider any one of
several branches in a parallel arc. Let the part of the charge
that passes through it be q, and let the magnitude of the
instantaneous current through it, at time t after the begin-
ning of the discharge, be *\. The induced e. m. f. at that
moment is L^dijdt where L t is the self -inductance of the
branch. Suppose the discharge is caused by connecting an
e. m. f. to the parallel arc for a short time and then dis-
connecting it, and let the whole time of rise and fall of the
brief current be T. Then
E C T L C
-! dt-\ di v
R ijo R ijo
ET
Hence the charges through the various branches are in-
versely as their ohmic resistances. If the above proof be
carefully examined, it will be seen that it simply means
that the total quantity due to the induced e. m. f . is zero, since
the induced current in the first half of the process is op-
posite to that in the second half.
34. Standard Cells.
Text-book of Physics (Duff), pp. 584-585; Watson's Physics, pp. 806-
807; Watson's Practical Physics, 202-203; Bureau of
Standards Bulletin, Nos. 67, 70, 71; Henderson's Electricity and
Magnetism, pp. 176-182. Ewell, Physical Chemistry, pp. 334-336.
The standard Daniell cell consists of an amalgamated
zinc rod dipping into a porous cup containing a solution of
l6o ELECTRICITY AND MAGNETISM.
sulphate of zinc, which, in turn, stands in a glass vessel
containing a copper sulphate solution and a copper plate.
To amalgamate the zinc rod, thoroughly clean it with sand-
paper, dip it in dilute sulphuric acid, and rub over it a few
drops of mercury with a cloth. The porous cup should be
thoroughly cleaned inside and out. The copper plate should
be cleaned bright with sand-paper. The porous cup is half-
filled from a stock bottle with a solution of zinc sulphate
(44. 7 g. of crystals of c. p. zinc sulphate dissolved in 100 c.c.
of distilled water). The zinc rod is introduced and the
porous cup is placed in the glass vessel, which is filled,
not quite up to the level of the zinc sulphate in the porous
cup, with copper sulphate solution (39.4 g. of c. p. copper
sulphate dissolved in 100 c.c. of distilled water). The
copper plate is also placed in the outer vessel. After being
set up, the cell should be short-circuited for 15 minutes
and then allowed to stand on an open circuit for 5 minutes.
The cell should not remain set up more than a few hours.
When it is no longer needed, pour the copper sulphate solu-
tion back into the stock bottle and the zinc sulphate solution
back into its bottle, unless the zinc has turned black, in which
case throw the zinc sulphate away. The e. m. f. of the
Daniell cell, prepared as above, is i . 105 international volts,
correct to o . 2 per cent.
The Clark cell, which differs from the above in the fact
that the copper is replaced by mercury and the copper sul-
phate by mercurous sulphate, is a more constant standard
than the Daniell cell, but it needs to be treated with much
greater care, since the passage of a very small current through
it will alter the e. m. f. Hence it can be used only for null
methods and kept in circuit for the briefest time possible.
At temperature / its e. m. f. in volts is
i.433-.ooi2(/-i5).
In the cadmium cell the zinc and zinc sulphate of the
above are replaced by cadmium and cadmium sulphate.
Its e. m. f. is
1.019 . 00004(2 17).
DOUBLE COMMUTATOR.
35. Device for Getting a Small E. M. F.
In many experiments it is desirable to use an e. m. f-
much smaller than that of a single cell. To get such an
e. m. f., a box of very high resistance may be placed in
series with a constant cell and any desired fraction of the
whole e. m. f. may be obtained by tapping off from various
FIG. 47-
points; e. g., at the ends of a resistance r out of the total re-
sistance R of the box (Fig. 47). The e. m. f. thus obtained
may be found from Ohm's Law, but it must be noticed
that the resistance between the terminals of r is the resist-
ance of a parallel arc. If, however, the resistance of the
branch circuit be proportionally very large and that of the
cell proportionally very small, both may be omitted in the
calculation.
36. Double Commutator.
It is sometimes desirable to be able to reverse two parts
of a network repeatedly and at the same rate. For this pur-
pose a double commutator is con-
venient. It consists of two two-part
commutators mounted on a common
shaft; e. g., on opposite ends of the
shaft of a small motor. If, for ex-
ample, the battery used with a Wheat-
stone's Bridge be connected through
one commutator while the galvanom
eter is connected through the other,
ii
FIG. 48.
1 62 ELECTRICITY AND MAGNETISM.
an alternating current will act in the arms of the bridge,
while a direct current (or a succession of unidirectional
pulses) will pass through the galvanometer.
37. Relation Between Electrical Units.
(E.S. = Electrostatic; E.M. = Electromagnetic.)
Ampere =io- 1 C.G.S.-E.M. units of current.
Coulomb =io- 1 C.G.S.-E.M. units of quantity.
Volt =io 8 C.G.S.-E.M. units of electromotive force.
Ohm = io 9 C.G.S.-E.M. units of resistance.
Farad =io- 9 C.G.S.-E.M. units of capacity.
Microfarad = io- 15 C.G.S.-E.M. units of capacity.
Henry =io 9 C.G.S.-E.M. units of inductance.
Volt =xio- 2 C.G.S.-E.S. units of electromotive force.
Coulomb =3X10" C.G.S.-E.S. units of quantity.
Microfarad = 9 Xio 5 C.G.S.-E.S. units of capacity.
XLII. HORIZONTAL COMPONENT OF EARTH'S
MAGNETIC FIELD.
Ames 1 General Physics, pp. 609613; Watson's Physics, pp. 602-607;
Text-book of Physics (Duff), pp. 498-500; Crew' s General Physics ,
pp. 319-323; Hartley's Electricity and Magnetism, pp. 92-98;
Watson's Practical Physics, pp. 403-414; Kohlrausch's Physical
Measurements, pp. 240-247; Stewart and Gee's Practical Physics,
pp. 284-309-
In this experiment the horizontal component of the
earth's magnetic field, at a point in the laboratory, is de-
duced from the period of vibration of a bar-magnet and the
deflection of a magnetic needle produced by this same bar-
magnet when placed at known distances E and W (mag-
netically) of the needle. The dimensions and mass of the
magnet must also be obtained in order that its moment of
inertia may be calculated.
If the period of vibration of the magnet be T in the
place in which we wish to determine the horizontal com-
ponent H, its magnetic moment be M, and its moment of
inertia /, then
d)
\M#
For, when the magnet is deflected through a small angle 6, the
restoring couple is MH sind = MH6. Hence if the angular accelera-
tion at that moment is a
-MH6=Ia
and
Since M, H, and 7 are constant, the motion is simple harmonic and
T is given by (i).
If a magnetic needle at a distance d, E or W of this same
bar-magnet, in line with it and the point where H is to be
164 ELECTRICITY AND MAGNETISM.
determined, be deflected through an angle <p and, when
at a distance d^ be deflected through an angle (^
M d 5 tan (bd* tan d>\
(->) = L- l
H 2(d 2 -d, 2 )
Equation (2) is deduced from the expression for the force F pro-
duced by a magnet of magnetic moment M at a distance d in the
direction of the axis of the magnet. For, if m is the strength of either
pole of the magnet and 2 1 its magnetic length, the resultant force
due to the two poles is
By expanding the'denominator we may also write this :
F = *4
in which K is approximately a constant. (If the length of the needle
were also taken account of, this expression would remain unchanged ,
except that the value of the constant K would be different) . If, un-
der the force F and the component H of the earth's magnetic field,
a magnetic needle makes an angle with the magnetic meridian,
F = H tan 0. Hence,
M / K\ d* tan d>
If, now, the distance be changed to d lt and the deflection becomes
</> t another equation similar to the above will be obtained and the
elimination of K will give equation (2) above.
From equations (i) and (2) both H and M may be obtained
when the other quantities have been measured.
(A) To determine the period of vibration, remove all
movable iron (knives, keys, etc., included) to several
meters from the vicinity of the entire experiment. Suspend
the deflecting magnet, by means of a stirrup attached to a
single strand of silk thread, in a box which has glass ends
and sides and is surmounted by a glass tube through which
the suspension passes. Level until the thread hangs in the
axis of the tube. The magnet should be adjusted until it is
horizontal as tested by comparison with a leveled rod
attached to the outside of the box. Attach pointers to the
opposite glass sides of the box (or adjust those provided)
so that they are in line with the magnet at rest. Set the
magnet vibrating through an angle not exceeding 10.
HORIZONTAL COMPONENT OF EARTH'S MAGNETIC FIELD. 165
Check any pendulum vibrations by judiciously pressing on
the top of the glass tube. Then determine the period by the
method of passages as in Exp. X. (see p. 54).
The magnet should be vibrated as near as is convenient
to the place where the needle is deflected in the second
part, i. e., where we wish to determine H.
(B) The instrument used in the deflection part of the
experiment is called a magnetometer. It consists of a box
with glass sides in which is suspended a mirror attached to
either a small magnetic needle with a damping vane or a
small bell magnet vibrating in a copper sphere. The sphere
is placed at the center of a graduated bar upon which can
be placed the deflecting magnet. Level until the suspending
fiber is at the center of the bottom of the suspension-tube.
If the needle or the damping vane does not swing free, a
little additional leveling will be necessary.
A specially mounted large compass needle is used to
adjust the magnetometer bar perpendicular to the magnetic
meridian. By means of it a rod is placed in the direction
of the magnetic meridian, and then, by means of a square,
the magnetometer bar is made perpendicular to the rod.
Place a telescope and scale about one meter from the mag-
netometer. See that the scale is perpendicular to the
telescope. Adjust until the scale reflected from the mirror
is clearly seen in the telescope (for directions for this adjust-
ment see p. 25).
Place the magnet whose period of vibration has been de-
termined on a small wood slide near one end of the magnet-
ometer bar. Note the scale-reading on the magnetometer
bar corresponding to the end of the magnet nearer the
needle. When the needle comes to rest, record the scale-
reading against the vertical cross-hair of the telescope.
Remove the magnet several meters and read the zero.
Replace the magnet at the same distance from the needle,
but reversed, and again read the scale division correspond-
ing to the vertical cross-hair. Make two similar readings
with the magnet at an equal distance on the other side of
1 66 ELECTRICITY AND MAGNETISM.
the needle. Read the zero before or after each reading and
always estimate tenths of millimeters. Make four similar
readings with the magnet at about two-thirds the distance
on each side of the needle.
If the zero is somewhat unsteady, the following method
will be found better. Omit zero readings and obtain the
four deflection readings as rapidly as possible. Repeat this
twice so that twelve readings in all are obtained. Take
half the difference of each two successive readings as one
value of the deflection. The final result will be the mean
of all values so found. The extent to which they agree
will indicate the reliability of the mean.
Measure the distance from the center of the scale beneath
the telescope to the center of the suspension-tube of the
magnetometer (i. e., the distance to the mirror). From
this distance and the mean scale-reading for that distance,
tan 2$ is obtained (for it must be remembered that a re-
flected ray of light is turned through twice the angle that
the reflecting mirror is turned through). Since </) is a small
angle tan 2(f> = 2 tan (/> very nearly. The distances from the
needle to the near end of the magnet plus half the length
of the magnet give d and d^. At the close of the experiment,
measure the length of the magnet with vernier calipers,
and the diameter with micrometer calipers, and also weigh
it. If / be the length, r the radius, and m the mass, the
moment of inertia is :
In reporting, state the possible errors of the measure-
ments of r, /, d, d lt tan <, tan fa.
Questions.
i . How could the true length of the deflecting magnet be obtained ?
2. H having been obtained at one point in the room or building,
what would be the easiest way of finding its value at any other point ?
3. What are the other "elements" of the earth's magnetism?
4. If you have done Exp. XLIII, calculate the total force and
the vertical component.
MAGNETIC INCLINATION OR DIP. 167
XLIII. MAGNETIC INCLINATION OR DIP.
Ames' General Physics, pp. 618619; Watson's Physics, pp. 605-607;
Text-book of Physics (Duff), p. 506; Crew's General Physics, p.
312; Hartley's Electricity and Magnetism, pp. 99102; Watson's
Practical Physics, pp. 4 1 5-4 1 7 ; Stewart and Gee's Practical Physics,
II, pp. 275-284.
(A) The dip, or inclination of the earth's magnetic lines
of force to the horizontal, is found by means of a dipping
needle or magnetic needle suspended on a horizontal axis
which passes as nearly as possible through the center of
gravity of the needle, with a vertical graduated circle for
reading the angle of inclination. Such an apparatus is
called a dip-circle, and includes a level and leveling screws
for making the circle vertical, knife-edges for bearing the
axis of the needle, a horizontal graduated circle for fixing
the azimuth of the vertical circle, and an arrestment, with
Y-shaped supports, for raising and lowering the needle and
placing it so that its axis of rotation passes as nearly as
possible through the center of the vertical circle.
The zero-line of the vertical circle must first be made
vertical. This adjustment is made by means of the level-
ing screws and level just as a cathetometer is leveled (see
p. 19). The circle must then be turned into the plane
of the magnetic meridian. To attain this, advantage is
taken of the fact that if the plane in which the needle is
free to rotate be at right angles to the magnetic meridian,
the needle must stand vertically; for in that position the
horizontal component of the earth's magnetic force is par-
allel to the axis of rotation of the needle, and hence has no
moment about that axis. The circle is, therefore, turned
approximately east and west and then adjusted until the
needle is vertical. This adjustment should be repeated
several times, and each position should be carefully read
with the assistance of a vernier if one is provided. A rota-
tion of the circle through 90 from the mean position,
as indicated by the horizontal circle, should then bring the
plane of the circle to coincidence with the plane of the
1 68 ELECTRICITY AND MAGNETISM.
magnetic meridian. By raising and lowering the arrest-
ment, the needle is then placed on the knife-edges in the
proper position for indicating the dip.
A single reading of the needle in this position would give
a very imperfect value of the dip. Errors arise from
various causes: (i) the axis may not roll freely on the
knife-edges, owing to dust or friction. To remove any
dust the axis and knife-edges should be brushed with a
camel's-hair brush. The setting by means of the arrest-
ment and the readings should be made at least twice, and
both sets of readings recorded. (2) The axis of rotation
of the needle may not be exactly at the center of the divided
circle. This error may be eliminated by reading the posi-
tion of both ends of the needle, one reading being from this
cause as much too great as the other is too small. (3) The
line of zeros on the vertical scale may not be truly vertical,
and this would cause errors in the same direction in the
readings of the ends of the needle. These errors may be
eliminated by turning the vertical circle through 180 about
a vertical axis and repeating the readings, for in these read-
ings the quadrants on the other side of the zero line are
used. (4) The axis of rotation may not pass exactly through
the center of gravity of the needle. So far as the fault
lies in the fact that the axis of rotation is to one side of the
axis of figure of the needle, the error may be eliminated
by reversing the needle in its bearings and repeating the
readings; for in one position gravity will make the readings
as much too great as in the other case it makes them too
small. But gravity will also cause an error if the axis
of rotation be in the axis of figure, but not at the center of
the latter. The error will not be eliminated by reversing
the needle on its bearings, but it will be if the magnetism
of the needle is reversed and all of the preceding readings
repeated; for then the other end of the needle will be lower
and the error will be in the opposite direction. The re-
versal of the magnetism should be done under the direction
of the instructor, the method of double touch being used.
MEASUREMENT OF RESISTANCE. 169
In recording these various positions and readings, the
side of the circle on which the scale is engraved may be
called the face of the instrument, and similarly one side of
the needle may be fixed upon as its face. Thus two readings
of each end of the needle are to be made in each of the fol-
lowing positions:
(1) Face of instrument E, face of needle E;
(2) Face of instrument W, face of needle W;
(3) Face of instrument W, face of needle E;
(4) Face of instrument E, face of needle W.
The magnetism of the needle having been reversed,
readings are to be again taken in the above positions. The
final result is taken as the mean of these 32 readings.
(B) Another method of determining the dip is by means
of an earth inductor in series with a ballistic galvanometer
(p. 156). The earth inductor is first placed with the plane
of its coils vertical and perpendicular to the magnetic
meridian. It is then rotated through 180 and the throw d^
noted. Several readings should be made. The plane of
the coils is then placed horizontally and the throw d 2 on
rotation through 180 noted. The ratio of d, to d l is the
tangent of the dip.
Questions.
1. What other sources of error may there be in measurement by
the dip-circle?
2. Would you be justified in making a calculation of "probable
error" from the various readings with the dip-circle?
3. If you have performed Exp. XLII, calculate the total force
and the vertical component.
XLIV. MEASUREMENT OF RESISTANCE BY WHEAT-
STONE'S BRIDGE.
Ames' General Physics, pp. 725-727; Watson's Physics, pp. 685-687;
Text-book of Physics (Duff), p. 572; Hadley's Electricity and
Magnetism, pp. 306-310; Watson's Practical Physics, pp. 432-
437; Kohlrausch's Physical Measurements, p. 303.
The practical measurement of a resistance consists in
comparing it with a known or standard resistance. For
170 ELECTRICITY AND MAGNETISM.
resistances of medium magnitude, Wheatstone's Bridge is
usually used (p. 153).
In joining the known and unknown resistances to the
bridge, connectors should be used whose resistance is negli-
gible; that is, less than the unavoidable error that may
occur in determining the unknown resistance. In con-
necting the battery and galvanometer, no such precaution is
necessary, for their resistances do not enter into the calcu-
lation. The galvanometer may be connected to either pair
of opposite corners; but, where the greatest sensitiveness
is required, if the galvanometer has a higher resistance
than the battery, it should be in the branch that connects
the junction of the highest two of the four resistances P,
Q, R, S to the junction of the lowest two; while, if the
battery has the greatest resistance, it should occupy that
position. Two spring keys should be included in the con-
nections, one in the battery arm and the other in the gal-
vanometer arm. When testing for a balance, the battery
key should be pressed first, then the galvanometer key.
If taken in the reverse order, there might be a small deflec-
tion due to the self-induction of the various parts. These
keys should be pressed for a moment only. Except for a
final determination, it is not necessary to wait until the gal-
vanometer has quite come to rest, for a lack of balance will
be indicated by a sudden disturbance of the swing when
the galvanometer key is pressed. The pressure of the gal-
vanometer key should be brief, sufficient merely to indicate
the direction of the initial movement.
In practice, it is best to use a box-resistance as nearly
as possible equal to the unknown resistance. This comes
to the same thing as saying that the box-resistance should
be varied until a balance is attained when the parts of the
meter wire are nearly equal. The reason for this preference
is that the sensitiveness is then a maximum, or a slight lack
of balance is most easily detected by the deflection of the
galvanometer. The exact ratio of P to Q for a balance
should be very carefully ascertained. At least six settings
GALVANOMETER RESISTANCE BY SHUNT METHOD. 171
should be made ; and to secure independence of the settings,
the eye should be kept on the galvanometer-scale and the
reading of the bridge not examined until the setting has
been decided on. The mean of these six is then taken. R
and 5 should then be interchanged and six more settings
made. This interchange will serve to eliminate the effect
of lack of symmetry of the two sides of the wire bridge
and its connections.
The structure of the galvanometer to be used, its coils,
magnets, and connections, should be carefully examined and
care taken that it is thoroughly understood (p. 155).
Three unknown resistances should be measured sep-
arately and then all in parallel. From the separate resist-
ances the resistance of the conductors in parallel should be
calculated and compared with the measurement of the
same. The resistance of a wire should then be measured
and its length and mean diameter obtained. From these
data, the specific resistance of the material of the wire
should be deduced. The temperature at which the resist-
ance is measured should also be noted, and from the tem-
perature coefficient of the material (Table XXII) the specific
resistance at o C. calculated.
The possible errors of the measurements, and hence the
extent to which the calculations should be carried, may be
deduced from the mean deviation in each set of readings.
Questions.
1 . Does the battery need to be a constant one ?
2. What objections are there to allowing the battery circuit to
remain closed?
3. Why is it difficult by this method to measure very large or very
small resistances ?
XLV. GALVANOMETER RESISTANCE BY SHUNT
METHOD.
Kohlrausch's Physical Measurements, p. 325.
If a galvanometer of resistance G connected in series
with a battery of resistance B and e. m. f. E and a box
resistance R gives a deflection d and if C be the current
172
ELECTRICITY AND MAGNETISM.
C =
R+B+G
= Kd
where K is a constant for the galvanometer. If now the
galvanometer be shunted by a resistance S and the deflection
be then d f and the current through the galvanometer C',
E S
/-*/
R+B +
GS X G+S
= Kd'.
G + S
Hence
(R+B) (G+S)+GS_d
S (R+B+G) = ~d r>
and from this G is readily deduced provided B is known.
Usually a battery of such low resistance can be used that B
is negligible compared with R and may be omitted; otherwise
A ^ B must be obtained as in Exp. LI II.
The galvanometer should be connected
through a commutator and several read-
ings on both sides should be made.
If the e. m. f. of the cell supplied is too
great, a suitable fraction of it should be
employed (p. 161).
As a check, the determination of G
should be repeated, a different value for
5 being used. If the galvanometer is
very sensitive, its resistance must be found
from two readings with shunts. A suita-
ble formula is readily worked out.
If R should be very great compared with the other
resistances, the formula may be simplified. This will
usually be the case if the galvanometer is very sensitive or of
low resistance. The quantities added to R in the first two
equations may then be neglected and we get
G+S d
GALVONOMETER RESISTANCE BY THOMSON'S METHOD. 173
XLVI. GALVANOMETER RESISTANCE BY THOMSON'S
METHOD.
Hadley's Electricity and Magnetism, p. 321; Kohlrausch's Physica
Measurements, p. 328; Stewart and Gee's Practical Physics, II,
p. 140-142.
The resistance of the coils of a galvanometer may be found
by means of Wheatstone's Bridge as the resistance of any
ordinary conductor is found. This would require the use
of a second galvanometer. The second galvanometer, for
detecting when the bridge is balanced, is frequently un-
necessary. The condition for
a balance is that, when the
branch in which the galvanom-
eter is usually placed is closed
by a key, no current shall flow
through it. If a current did
flow through it, a change
would take place in the cur-
rents in the other arms. Now
the presence of a galvanom-
eter in one of these other arms
enables us to test whether any
change in the distribution of
the currents takes place on
the key's being pressed. Hence, in Thomson's method for
galvanometer resistance the galvanometer is placed in the
"unknown" arm and a spring key, K, is placed in the
branch in which, in the ordinary arrangement of Wheat-
stone's Bridge, a galvanometer is found. A diagram to
illustrate the connections is given in figure 50.
From the above it will be seen that in this method a bal-
ance is obtained when the deflection of the galvanometer
does not change on the key, K, being pressed. Two practical
difficulties are met with. The first is that the deflection
of the galvanometer before the key is pressed may be so
large that it cannot be read. When the galvanometer is of
FIG. 50.
174 ELECTRICITY AND MAGNETISM.
the Thomson type (p. 155), this difficulty may be overcome
by turning the control magnet until the deflection can be
read (the zero position of the galvanometer could, of course,
not then be read on the scale, but that is not necessary).
In the d'Arsonval type of galvanometer there is no such
way of overcoming this difficulty, and so this method is not
so easily applied to such a galvanometer. The second
difficulty is that if the battery be a variable one, the gal-
vanometer will not give a steady deflection. Hence, a
constant battery of the Daniell or Gravity type should be
used (p. 159). It may also be necessary to decrease the
current through the bridge and galvanometer by putting
considerable resistance in series with the battery, or a fraction
of the e. m. f. of the cell may be used (p. 161).
In the experiment it is better to use a bridge-box instead
of a wire bridge, for the condition for sensitiveness, that
the arms should be as nearly equal as possible, still holds,
and the resistance of a wire bridge is usually very small
compared with that of the galvanometer. Beginners some-
times find difficulty in deciding on the proper connections.
The best way is to consider what the connections would be
in the ordinary use of Wheatstone's Bridge, and then con-
sider the modifications introduced in the present method.
If possible, ratio arms of 1000 to 1000, 100 to 1000, and 10
to 1000 should be used in succession to obtain successive
approximations. The last should give the resistance to two
places of decimals (if one ohm is the least box-resistance),
but the decreasing sensitiveness may prevent the latter
ratios from giving more accurate results than the first.
If the galvanometer has more than one coil, the resist-
ance of each should be measured separately and then the
resistance of all in series. This will afford a check on the
work.
Question.
i. Describe carefully the swinging system, coils, control magnet,
and connections of the galvanometer used.
MEASUREMENT OF HIGH RESISTANCES. 175
XLVII. MEASUREMENT OF HIGH RESISTANCES (i).
Watson's Practical Physics, pp. 460-461; Henderson's Electricity
and Magnetism, pp. 66-72.
The method of Wheatstone's Bridge is not suitable for
measuring very high resistances. One method is to con-
nect the unknown resistance X, a battery of negligible re-
sistance and e. m. f. E, and a sensitive galvanometer of
resistance G in series. If the current be C,
E
C = - , giving a deflection d.
X -\-G
Now replace X by a known resistance, R, and shunt
the galvanometer by such a resistance, 5, that the deflection
is readable. By considering the total current and the part
C' of the total current that passes through the galvanometer,
we readily find that
W S"
C' --- , giving a deflection d' ,
GS G+S
+
Hence,
R(G+S)+GS _d
S(X+G) = J"
and from this X is readily deduced. G may be found as in
Exp. XLV or XLVI; but if (G+S)/S is known and G is
small compared with X and R, the resistance of the galvan-
ometer need not be determined. Many galvanometers are
provided with shunt boxes, for which S/(G+S) is o.i,
o.oi, or o.ooi.
The galvanometer should be connected through a com-
mutator, and several readings on both sides should be made
to obtain a reliable mean.
As a check, repeat the measurements with a different
value for R and a different value for S.
7 6
ELECTRICITY AND MAGNETISM.
XLVIII. MEASUREMENT OF HIGH RESISTANCES (2).
Text-book of Physics (Duff), pp. 537-538; Watson's Physics, pp.
656-657; Ames' General Physics, pp. 658-659; Henderson's
Electricity and Magnetism, pp. 71-75; Hadley's Electricity and
Magnetism, pp. 202-210; Watson's Practical Physics, pp. 569-
57 1 -
A very high resistance, such as the insulation resistance of
a cable or the resistance of cloth, paper, wood, etc., may
be measured by finding the rate at which the electricity in a
charged condenser leaks through the conductor. An
electrometer is used to find the change of potential of the
condenser and from this the rate of loss of its charge is
deduced. The Dolezalek form of Kelvin's quadrant electrom-
FIG. 51.
eter is suitable. Its needle is kept charged to a high poten-
tial by being connected to one pole of a battery of small
cells, the other pole being grounded.
To find the insulation resistance of a cable the whole of
the cable except the ends is immersed in a tank of salt water
which is connected to the earth. One of the ends is carefully
paraffined to prevent surface leakage and the core of the
other end is connected to the insulated pair of quadrants.
If the cable is sheathed with metal, immersion is not neces-
sary. Other materials, such as those mentioned, are
pressed between sheets of tinfoil, one sheet being connected
to the earthed quadrants and the other to the insulated
quadrants.
MEASUREMENT OF HIGH RESISTANCES. 177
L e t V\ = potential given the condenser on closing the
key K. The charge Q in the condenser and cable = C V^
where C is their joint capacity. Upon opening K the charge
flows through a resistance R for a time, t, ^ t being the insul-
ation resistance of the cable, the condenser, and the elec-
trometer and keys in parallel.
Since the current at the time / equals V/R^ by Ohm's law,
and also equals the rate of decrease of Q or of CV
= -c dv -
R, dt
dV _dt
'T".s;
Integrating between the limits t = o when V=V 1 and / = t
when V=V 2 we get
R t ^-434*
C lg e ~ C 1~ *
where d l and d 2 are the initial and final deflections of the
electrometer from the zero position.
The zero should be determined both before V l is found
and after V 2 is found. As it is very apt to vary slightly,
more reliable results can be attained by continuing to read
V at intervals (e. g., every half -minute) until it has fallen
to about one-half of its original value. From a curve drawn
to represent V and /, two reliable points may be chosen
to give values for V l and V 2 to be used in the calculation.
A subdivided condenser is desirable in order that a
capacity giving a sufficiently rapid fall of potential may be
chosen.
The total insulation resistance, R 2) of the other parts in
parallel with the cable are found by disconnecting the
cable and making a second set of observations as above.
12
178 ELECTRICITY AND MAGNETISM.
The insulation resistance, R, of the cable is then deducible
for
The capacity, C lt of the cable can be compared with
that of the condenser, C 2 , by the method of "divided
charge." First charge the condenser and observe its
potential by the electrometer and let the deflection be d r
Then connect in the cable and let d 2 be the new deflection.
Since the total charge Q remains unchanged,
and, since the deflections are proportional to the potentials,
C=C l 2
Questions.
1. Why should one pole of the battery that charges the needle be
grounded ?
2. Why must keys of specially high insulation be used in this
method ?
3. Calculate the capacity of the cable in electrostatic units from
rough measurements of its dimensions and reduce to microfarads
(see p. 162).
XLIX. MEASUREMENT OF LOW RESISTANCES (i).
Very low resistances cannot be measured by the Wheatstone
Bridge method, because the unknown resistances of the
connections are not small compared with the resistance
to be measured. The simplest method for low resistances
is a "fall of potential " method. A current is passed through
the resistance, the current is measured by an ammeter and
the difference of potential is measured by a voltmeter; then
the resistance is known from Ohm's Law. For very low
resistances the fall of potential will be very small and an
instrument much more sensitive than any commercial volt-
meter must be used. Instead of a voltmeter a sensitive gal-
vanometer of high resistance, or a low resistance galvan-
ometer in series with a high resistance, is used and the
MEASUREMENT OF LOW RESISTANCES. 179
value of a scale division of the galvanometer regarded as a
voltmeter is found by a separate experiment.
Let the resistance to be measured be x, and let the differ-
ence of potential at its ends when current C passes through
it be e. Then
e
x= c
C, which should be large, may be measured by an ammeter.
To find e we must know the constant, K, of the galvanometer
considered as a voltmeter; that is, the number of volts per
unit deflection. If the deflection is D
e=K.D
To find K apply to the galvanometer a small fraction of
the e. m. f., E, of a Daniell'scell (p. 159). , E
For this purpose connect the cell in
series with a very high resistance box
and a box of moderate resistances and
join the galvanometer to the ends of
one of the small resistances, r, choosing
r so that the deflection, d, will not be
very different from D. Then if the re-
sistance of the galvanometer be great
compared with r (see p. 161) and
if the total resistance in series with FIG. 52.
the battery be R, the e. m. f. acting on the galvanometer
is Er/R.
Hence
As a check on the work redetermine K using a different
value of r. From the above equations x is found.
In the first part of the experiment place a commutator
in the main circuit so that C may be reversed and the effect
of thermo-electric forces at the contacts eliminated, and
connect the galvanometer through a second commutator so
that lack of symmetry in its deflection may be eliminated.
i8o
ELECTRICITY AND MAGNETISM.
Thus D will be the mean of four readings. Exactly simi-
lar precautions should be observed in the second part.
Close the currents only for the shortest possible times
necessary to make the readings, otherwise heating may
occur and resistances (especially the unknown x) may change.
The determination should be repeated several times with
different values of C. If the work has been reliable, D should
be proportional to C. Note also the temperature of the
specimen and calculate its resistivity from its resistance and
dimensions.
Questions.
1. If r had not been negligible compared with the galvanometer
resistance how would this have appeared in the course of the work?
2. Find the equation that must replace the above if the resist-
ance of the battery is not negligible compared with R and if r is not
negligible compared with the galvanometer resistance.
L. MEASUREMENT OF LOW RESISTANCES (2).
Henderson's Electricity and Magnetism, pp. 5758; Stewart and Gee's
Practical Physics, II, pp. 177-181.
When a standard low resistance (o.oi or o.ooi ohm) is
available, a conductor of low resistance x may be connected
in series with it and a battery,
and a very sensitive voltmeter,
or a high-resistance galvanom-
eter, serving as a voltmeter,
may be used to compare the
falls of potential in x and the
standard. The resistances will
be proportional to the falls of
potential.
Connection with the battery
should be made through a com-
mutator to reverse thermal
effects at the connections, and
the galvanometer should be
connected through a second
commutator to eliminate asym-
MEASUREMENT OF LOW RESISTANCES.
181
metry of the galvanometer readings. Thus each final read-
ing will be the mean of four separate readings.
The currents should be closed for the shortest times
sufficient for the readings, to avoid heating. Note the
temperature of the specimen.
Questions.
1. What are the comparative advantages and disadvantages of this
and the preceding method ?
2. Why is a high-resistance galvanometer to be preferred?
3. Will poor contact have as much effect as in a measurement of
low resistance by Wheatstone's Bridge? Why?
LI. MEASUREMENT OF LOW RESISTANCES BY THE
THOMSON DOUBLE BRIDGE.
Watson's Practical Physics, pp. 465-469; Stewart and Gee, II,
pp. 182-187.
In Thomson's Double Bridge the errors of the contacts
in the use of Wheatstone's Bridge are avoided. Its princi-
ple is, in fact, that of the fall of potential method (Exp.
L) the direct comparison of the falls
of potential being replaced by a null
method. This method is applicable to
extremely low resistances as well as to
medium resistances.
In the diagram x is the resistance to
be measured and r a standard known
resistance; a may be made 10 or 100; "
and b may be made 100, 1,000, 10,000.
Similarly, a' may be made 10 or 100
and b f 100, 1,000, 10,000. Now it can
FIG. 54.
be shown that if there is no current in the galvanometer, and
if the ratios a/ b and a'/b' are equal,
x a a'
~r = b = l)''
hence a value of r (which is variable) and values of a, b,
a' b', are sought which give no current in the galvanometer,
and from these x is calculated.
l82
ELECTRICITY AND MAGNETISM.
The form of Double Bridge made by Hartmann and Braun
is very satisfactory. The correspondence of parts to parts
of the diagram is readily traced. The ratio of a to b can be
varied from 100 to 100 to 10 to 10,000; moreover, a and b
may be interchanged and so the ratio reversed. Similar re-
marks apply to a' and b'. Thus values of x/r varying
from 10/10000 to 10000/10 may be measured. The
variable r may be varied from o . 044 down to o, but can
hardly be read with an accuracy of i% below o.ooi.
Hence values of x between
o.ooi/ 1000 or o.oooooi and
0.044X1000 or 44, may be
measured by the bridge.
Care must be taken not to
injure the standardized bar by
scraping the contact maker along
it. The contact maker must be
raised for each movement. Do
not allow the sharp jaws of the
clamps to come down on the bar
too suddenly, for they might cut
into the bar somewhat.
Test as many as possible of the following materials:
(i) Brass. (2) Iron. (3) Copper. (4) Zinc. (5) Lead.
(6) Carbon. (7) Rail Bond,
and calculate the Specific Resistance of each.
Proof of Formula.
Regard Thomson's Double Bridge as a modified Wheatstone's
Bridge, the modification consisting in the paralleling of parts of the
arms AB and BC as indicated. Let G and D be points at the same
potential as indicated by the galvanometer. E B F is the heavy
conductor joining the unknown and the standard. Let B be a
point in it at the same potential as G. We may suppose B and G
permanently connected. Let the resistance of p and a' in parallel
be m, and that of q and b' be n. Then, by the Wheatstone Bridge
formula :
FIG. 55.
a _x + m
b~ r + n'
COMPARISON OF RESISTANCES. 183
Now if we show that
it will follow that
a = x_
b 7'
To show this, note that
a _ a' _ p
a' + p b'+q'
Now
= a'p ^b^q
m p a
Hence
x a
Questions.
1. Considering this as a modified fall of potential method, why
should a, b, a', b', be of very large and E B F of very small resistance?
2. Does the battery current need to be steady? Why?
3. Could an alternating current be used in any circumstances?
LII. COMPARISON OF RESISTANCES BY THE CAREY-
FOSTER METHOD.
Watson's Practical Physics, pp. 442-446; Schuster and Lees' Practical
Physics, pp. 307-309; Henderson's Electricity and Magnetism,
pp. 5357; Stewart and Gee's Practical Physics, II, pp. 158170.
To find very accurately the difference between two very
nearly equal resistances R and 5, connect them and two
other nearly equal resistances, P and Q, as indicated in the
diagram, where ab is a very uniform wire, which we shall
suppose to have a resistance of more than i ohm. Let the
resistance of unit length of the wire ab be p. Let the dis-
tance ad be x lt when a balance has been obtained in the
usual way. Then exchange R and 5, and again obtain a
balance. Denote the new value of ad by x 2 . Since P and Q
1 84
ELECTRICITY AND MAGNETISM.
have not been changed and the total resistance R, S, and ab
was not changed, it is clear that R -i-x 1 p = S +x^p, or
R S= (x 2 xjp. To find the value of p, replace R by a
standard i-ohm coil, and 5 by a heavy connector of neg-
FlG.
ligible resistance, and proceed as above; then p(x 2 x l ) = i.
The exchange of R and 5 is made by means of a special
key designed so that the resistance of the connections will
remain the same (see Fig. 57).
FIG. 57 .
To compare a box of unknown errors with a standardized
box, the difference between each resistance of the former
and a corresponding resistance of the latter is found by the
above method.
BATTERY RESISTANCE BY MANGE'S METHOD.
185
To calibrate two boxes, put one in place of R and replace
5 by a standard i-ohm coil and so find exactly the value
of each i-ohm unit in the box. Then replace the standard
by the other box, in position S, and compare the i-ohm
units of the second box with those of the first box. Then
compare a 2-ohm unit in one box with two i-ohms in
the other, and so on. Special care must be taken to avoid
confusion in making the calculations, and for this purpose
the box resistances may be denoted by I lf I 2 , II lt II 2 , etc.,
for one box, and I\, I' 2 , II' 2 , etc., for the other.
Questions.
1. State the formula for Wheatstone's Bridge before and after R
and 5" are interchanged and therefrom deduce the above formula.
2. Do P and Q need to be exactly equal, and why?
LIII. BATTERY RESISTANCE BY MANGE'S METHOD.
Hartley's Electricity and Magnetism, p. 322 ; Watson's Practical Physics,
p. 475; Schuster and Lee's Practical Physics, pp. 303-306.
The resistance of a battery may be determined by plac-
ing it in the "unknown arm" .R of a Wheatstone's Bridge
(p- I 53)- I n "this case there
will be a current through
the galvanometer when the
bridge battery is not con-
nected. But if P, Q and 5
be adjusted until there is no
change in the deflection when
the key of the bridge battery
is pressed, the points to
which the galvanometer is
connected will be at the same
potential so far as the effect
of the bridge battery is con-
cerned. Since, when the
FIG. 58.
adjustments are right, the bridge battery sends no current
through the galvanometer, this battery may be removed and
the key alone will serve to test the adjustment of P, Q, and S.
1 86 ELECTRICITY AND MAGNETISM.
If the deflection of the galvanometer is too great to be
readable, the control magnet (in the case of a Kelvin gal-
vanometer) may be used to bring the needle back, or the
galvanometer may be shunted or a resistance put in series
with it. Most cells vary slightly in resistance and e. m. f.
when on closed circuit; hence, the keys should not be
pressed longer than is necessary.
In Lodge's modification of Mance's Method a condenser
is placed in series with the galvanometer. There will then
be no continuous current through the galvanometer; but, if
the adjustments of P, Q, and 5 are not right, on pressing
the key by which the adjustment is tested the galvanometer
will be momentarily deflected.
Questions.
1. Why is there a slow movement of the galvanometer needle
when the keys are kept pressed ?
2. Should the condenser be of large or small capacity? Would a
Ley den jar do?
LIV. TEMPERATURE COEFFICIENT OF RESISTANCE.
Text-book of Physics (Duff), p. 564-566; Ames' General Physics, pp.
731-732; Watson's Physics, pp. 681-682; Hadley's Electricity
and Magnetism, p. 294; Henderson's Electricity and Magnetism,
pp. 95-101.
The resistance of most solids increases as the tempera-
ture rises; carbon is one of the exceptions, for its resistance
decreases. For moderate ranges of temperature the resist-
ance is approximately a linear function of the temperature or,
if ^ be the resistance at o and R that at t,
R=R Q (i+at)
The constant a is called the temperature coefficient of the ma-
terial. It may be defined as the change per ohm, referred
to the resistance at o, per degree change of temperature.
The change of resistance can be most conveniently studied
by the box form of Wheatstone's Bridge (p. 154).
(A) For finding the temperature coefficient of a wire such
as copper, a length sufficient to give several ohms resistance
TEMPERATURE COEFFICIENT OF RESISTANCE. 187
should be used. The determination of the temperature
coefficient does not require that the dimensions of the
specimen should be known, but the specific resistance of
the specimen might as well be determined at the same time.
Hence the length and mean diameter of the wire should be
carefully measured. The wire should then be soldered to
heavier lead wires and immersed in a bath of oil, and its
resistance determined at intervals of about 10 as the tem-
perature is raised. The thermometer should be placed inside
the coil so as to be as nearly as possible at the temperature
of the latter. It will be an improvement if the coil and
thermometer are in a test-tube that is immersed in the
bath, the opening of the tube being closed with cotton-wool.
To keep the temperature constant, while measuring the
resistance, would be difficult. The following method will
be found to give much better results: Having measured
the resistance at the temperature of the room, adjust the
known resistance of the bridge so that there would be a
balance if the resistance of the wire were increased 4 or 5
per cent. The galvanometer will be deflected. Now heat
the wire very slowly and the galvanometer reading will
begin to drift toward zero. When it just reaches zero, read
the thermometer and continue the process step by step.
The various resistances and temperatures should then
be plotted in a curve that should be approximately a straight
line. If exactly a straight line is obtained, the temperature
coefficient should be calculated from two reliable and widely
separated points on the curve. Let R and R' be the resist-
ances at t and t', respectively. Substituting these values in
the above equation we shall get two equations from which
R can be eliminated.
If the plotted readings give a distinct curve, the resistance
must be expressed as a quadratic function of the temperature.
R=RQ(I + at + bt 2 )
From three points on the curve three equations may be
written down and from these a and b may be calculated.
1 88 ELECTRICITY AND MAGNETISM.
(B) For finding the temperature coefficient of carbon
an incandescent lamp may be used. As it would be diffi-
cult to determine accurately the temperature of the filament
in the exhausted bulb by the preceding method, water may
be used for the bath and two careful determinations of the
resistance made, the first being while the water is at about
the temperature of the room, and the other when the water
is boiling. In each case the final determination of the
resistance should not be made until the temperature of the
filament has become constant, as is indicated by its resist-
ance becoming quite constant. The leads, where they are
immersed in the water, should be carefully insulated with
tape.
LV. SPECIFIC RESISTANCE OF AN ELECTROLYTE.
Watson's Practical Physics, pp. 475-486; Henderson's Electricity and
Magnetism, pp. 8084; EwelVs Physical Chemistry, pp. 54-57;
Kohlrausch's Physical Measurements, pp. 316-321.
The object of this experiment is to determine the specific
resistance of an electrolyte for instance, solutions of copper
sulphate of different concentrations. The box form of
Wheatstone's Bridge is most suitable for the purpose (p. 1 54).
A steady current from a battery and a galvanometer to
determine when there is a balance, as ordinarily used with
Wheatstone's Bridge, cannot satisfactorily be used in meas-
uring the resistance of an electrolyte, for a steady current
produces in a short time polarization at the electrodes.
This polarization leads to too high an estimate of the resist-
ance of the electrolyte, for when no current flows through
the galvanometer, the three other arms of the bridge are
balancing the potential difference necessary to overcome
the true resistance of the electrolyte plus the potential
difference required for overcoming the polarization potential
difference at the electrodes. This difficulty is obviated by
using the rapidly alternating current from the secondary of an
induction coil instead of a steady current from a battery.
SPECIFIC RESISTANCE OF AN ELECTROLYTE.
189
The time that the current continues in one direction is so
short that no appreciable accumulation can form at the
electrodes to produce an opposing difference of potential.
An ordinary galvanometer would not be affected by an
alternating current, but a telephone which is a very delicate
detector of an alternating current may be substituted.
In the simplest form of apparatus a vertical glass tube of
known cross section, which may
be found by calipers (p. 14),
holds the electrolyte. The elec-
trodes are connected to wires
that pass through the stoppers;
the upper electrode can be
raised or lowered as desired.
The resistances corresponding
to two different distances of
separation of the electrodes
should be determined. From
the difference we get the re-
sistance of a column whose
length is the difference in the
two lengths and thus eliminate uncertainty as to remaining
polarization at the electrodes and the exact ends of each
column.
Two tubes should be available for the work. In one
measurement should be made of the resistance of samples
of several solutions of different concentrations, the samples
being obtained from stock bottles. The other tube is for the
purpose of determining the temperature coefficient of a
solution. It is placed in a steam heater such as is used in
Exp. XIX (p. 88). As the heating to a steady temperature
will require considerable time, the tube should be prepared
at the beginning of the work. The resistance of the electro-
lyte in it having been determined at room temperature,
the heating may be allowed to proceed while the measure-
ments with the other tube are made.
A tube of small cross section may also be used. If its
1 90 ELECTRICITY AND MAGNETISM.
ends pass through stoppers into much larger tubes that
contain the electrolyte and electrodes, the resistance
measured will be virtually that of the electrolyte in the
small tube. The diameter of the tube may be found as in
Exp. XI (p. 58). Substituting a different length of the
same tubing and taking differences we may as before
eliminate residual polarization effects.
In measuring resistances it will probably be desirable to
use equal resistances, e. g., 100 ohms, in the ratio arms of the
bridge. It may be impossible to obtain a balance for
which there is no sound, for even though there were a
balance for steady current, there would not in general be a
balance for varying currents such as are used in this experi-
ment, owing to the inductive electromotive forces of capacity
and self-induction in the resistance coils. When there is
uncertainty as to whether a small resistance should be added
or cut out, the ear is often assisted by adding and cutting
out a larger resistance about which there is no doubt. On
comparing the change of tone on a variation of this latter
resistance with the variation of tone with the uncertain
resistance, one can often decide whether the small resistance
should be added or not. With a little practice one should
determine resistances within i per cent.
Calculate the specific resistance of each solution at each
temperature and tabulate the results. Find also the
temperature coefficient of the solution which was heated
and calculate its specific resistance at o.
Questions.
1 . Why should we expect the resistance to decrease with increased
temperature ?
2. What is supposed to be the nature of electric conduction in an
electrolyte?
3. Are the specific resistances inversely proportional to the con-
centrations ? Why ?
COMPARISON OF E. M. F/S BY HIGH-RESISTANCE METHOD. 191
LVI. COMPARISON OF E. M. F.'S BY HIGH-RESISTANCE
METHOD.
Watson's Practical Physics, pp. 430-431; Stewart and Gee's Practical
Physics, II, pp. 101-102.
The readiest method of comparing the electromotive forces
of cells is by means of a galvanometer of sufficiently high
resistance. If the deflections are (by the use of added
resistances) kept small the deflections of the galvanometer
will be closely proportional to the currents that pass through
it or i = k.d where k is a constant. Two methods may be
employed for comparing two cells. In the first, called the
"equal resistance" method, the total resistance R is kept
constant (the resistance of the cells being supposed negligi-
ble). Hence, by Ohm's Law, the e. m. f.'s are proportional
to the currents, that is, to the deflections, or
In the other or "equal deflection " method, such resistances
are used in the circuit that the cells cause equal deflections of
the galvanometer. Hence by Ohm's Law, since the currents
are equal, the electromotive forces must be proportional to
the resistances, or
EjJZi
E 2 R,
Both methods should be employed to find the e. m. f.'s
of several cells by comparing them with that of a standard
Daniell cell (p. 159). Directions for the adjustment of the
telescope and scale are given on p. 25.
(A) Equal Resistance Method. Make R such that the
standard Daniell cell gives a deflection of about 10 cm. on
a scale about i m. from the mirror. Make a reading of the
zero; i. e., when no current passes through the galvanometer.
Send the current through the galvanometer and read the
division now on the cross-hair. In this way make at least
six readings on one side and, reversing, make six on the
I Q2 ELECTRICITY AND MAGNETISM.
other. Read the zero often, as it is liable to change. In
reading, use, if necessary, the method of vibration (see p.
23). If the vibrations are irregular on account of trolley
currents or other disturbances, estimate the position of equi-
librium from the vibrations without actually making read-
ings. With some galvanometers the damping is so great
that the system comes to rest instead of vibrating about the
position of equilibrium. In this case the true reading can
be made at once. Always, if possible, estimate tenths of
the smallest divisions. When you have thus found the
mean deflection for the standard, find similarly the deflec-
tion for as many different types of cells as time allows.
With the other cells, three readings on a side will be suffi-
cient. The internal resistance of the different batteries
varies, but the differences are negligible compared with the
total resistance of the circuit. Express in volts your final
values of the e. m. f.'s of the cells tested.
(B) Equal Deflection Method. With a resistance which
gives a deflection of about 10 cm., make at least six careful
readings of the deflection on each side given by the standard
Daniell cell. Replace the standard by one of the cells
to be tested and vary the resistance of the circuit until the
deflection is the same as you found it on this side for the
standard. Similarly find the resistance which will make
the deflection on the other side the same as that given by
the standard on that side. We can neglect, in comparison
with the resistances in the boxes, the resistance of the bat-
tery and connecting wires, but not the resistance of the gal-
vanometer. Take the mean of the two resistances deter-
mined above, plus the resistance of the galvanometer, as the
resistance required to give the same deflection as the stand-
ard cell gave through the box-resistance used with it, plus
the galvanometer resistance. The resistance of the galvan-
ometer, G, must be determined as in Exp. XLV (last
paragraph) .
In determining the possible error of your results, estimate
the possible error of resistances from the least change in
COMPARISON OF E. M. F.'S BY CONDENSER METHOD. 193
resistance which will have an appreciable effect, and the
possible error of deflections from the mean deviation from
the mean in your readings.
Questions.
1. What are the advantages and disadvantages of the type of
galvanometer used in this experiment compared with other types
used in the laboratory?
2 . Which of the two methods do you consider the better ? Why ?
3. How could this method be used for finding the internal resist-
ance of a cell?
4. Are the deflections of a galvanometer strictly proportional to
the currents? Why?
LVII. COMPARISON OF E. M. F.'S AND MEASUREMENT
OF BATTERY RESISTANCE BY CONDENSER
METHOD.
Text-book of Physics (Duff), pp. 530-534, 562; Watson's Physics, pp.
634; Watson's Practical Physics, p. 526; Henderson's Electricity
and Magnetism, pp. 185-187.
When a condenser of capacity C is connected to a battery
of e. m. f. E it receives a charge Q = CE. If it be then con-
nected to a ballistic galvanometer, the
throw, d, will be proportional to Q, or Q =
K . d, where K is a constant. We shall apply
this to (A) compare e. m. f. 's and (B) measure
the resistance of cells. We shall describe
these separately, but in practice they may
be combined.
(A) Suppose the condenser is first charged
by a battery of e. m. f., E v and the deflec-
tion when connected to the ballistic galva-
,
nometer is a u and suppose that when this
same condenser has been charged by a battery of e. m. f.,
E 2 , the deflection is d 2 ; then
Kd,; Q 2 =CE 2 =Kd 2
E d d
FIG. 60.
194
ELECTRICITY AND MAGNETISM.
Use a key with an upper and a lower contact. The
condenser should be connected to the battery when the key
is down and to the galvanometer when the key is up. Be
very careful never to connect the battery directly to the
galvanometer. When a discharge is sent through a ballistic
galvanometer, the needle swings over to one side and then
swings back. Observe the reading of the scale on the verti-
cal cross-hair of the telescope when the needle stops and
turns back. Always in such work estimate tenths of the
smallest division. Before each throw bring the needle as
nearly as possible to rest. The zero is likely to change;
| AUS
P
\\ |
F
| EIN
FIG. 61.
therefore, before each throw, record the zero, and, after each
throw, record both the turning-point and the difference
between this turning-point and the zero; i. e., the amount
of the throw.
Always charge the condenser for approximately the
same length of time, for instance, five seconds. With a
standard Daniell cell (p. 159), record six throws on one side.
Reverse the battery connections and record six throws on
the other side. Let the mean of these be d r Replace the
Daniell cell by one of another type and find as before the
mean throw d 2 . If E 2 is the e. m. f. of the latter cell
Measure in this way the e. m. f. of as many cells of different
type as possible.
(B) This method of finding the resistance of a cell depends
on the fact that when the poles of a cell of resistance B are
COMPARISON OF E. M. F.'s BY CONDENSER METHOD. 195
joined by a conductor of resistance r, the difference of
potential of the poles depends on the ratio of r to B. Let
the current be i. Then the difference of potential of the
poles is ri by Ohm's Law applied to the part r of the circuit.
If in this condition the poles be joined to a condenser of
capacity C it will receive a charge Cri and if this when
discharged through a condenser causes a throw d f
Cri=K-d'
Now if E be the e. m. f. of the cell, E=(B +r)i by Ohm's
Law applied to the whole circuit. Hence, if the condenser be
charged by the cell when it is not short-circuited as above,
the charge C E also equals Ci(B +r) and if the deflection
when the condenser is discharged is d,
Dividing one equation by the other and solving for B,
d-d'
B = r ~^- ..
The connections are the same as when comparing e. m. f.'s
with the addition of a circuit containing a resistance
and a very low resistance key (e. g., a mercury key), con-
necting the poles of the cell. The battery should be short-
circuited just before the charging key is depressed, and the
short-circuiting key should be released immediately after
the other, otherwise the battery will run down. Choose
such a short-circuiting resistance that the galvanometer
throw is reduced to about half the value which it has without
the short-circuit. Do not use the plug-box resistances
for this work, on account of the danger of burning them out,
but use open wound resistances of large wire.. Find the
internal resistance of cells of several different types.
In estimating the possible error of your results, estimate
the possible error of your mean readings from the mean
deviation from the mean in the individual readings.
Additional exercises (to be performed if time permit.)
(C) Study the effect of length of time of charge by means
196 ELECTRICITY AND MAGNETISM.
of the throws obtained with the condenser charged for
different lengths of time with the same battery.
(D) Study the leakage of the condenser by comparing the
throws when the condenser has been successively charged
with the same e. m. f., and has remained charged for different
intervals of time.
(E) Study the electric absorption of the condenser by
charging for several minutes, discharging and reading the
throw and immediately insulating; after one minute, again
discharge and insulate. Continue discharging for several
minutes, the condenser being insulated during the minute
intervals.
Questions.
1. What are the peculiarities and requirements of a good ballistic
galvanometer ?
2. What is the construction of a condenser and what do absorption
and leakage mean?
3. How could you find the resistance of the galvanometer used, em-
ploying a condenser and a known resistance ?
LVIII. CALIBRATION OF A VOLTMETER.
Ames 1 General Physics, pp. 674-675; Text-book of Physics (Duff),
PP- 57~57 1; Watson's Practical Physics, pp. 493-498; Hender-
son's Electricity and Magnetism, pp. 200-208; Hadley's Electricity
and Magnetism, p. 320.
A voltmeter may be calibrated by balancing a part of the
e. m. f. applied to the terminals of the voltmeter against
the e. m. f. of one or more standard cells. To do this a
very high resistance circuit, consisting of -a high resistance-
box in series with an ordinary resistance-box, is placed in
parallel with the voltmeter. The fall of potential in part
of the low resistance-box is measured by a side-circuit con-
sisting of the standard cell, a sensitive galvanometer and a
key, the standard cell being so turned that it tends to
send a current through the galvanometer in the opposite
direction to the fall of potential in the box.
Let e be the e. m. f. of the cell, E the potential difference
at the terminals of the voltmeter, r 1 the resistance across
CALIBRATION OF A VOLTMETER.
197
which the galvanometer circuit is connected, and r^ the
remaining resistance in the high-resistance circuit. When
r l is adjusted so that there is no deflection when the key is
pressed
A special fuse-wire for very low currents should be placed
immediately adjacent to the battery to prevent the possi-
bility of injury to the resistance-boxes. The main circuit
should be closed through a spring-key only a sufficient
length of time to enable the voltmeter or the galvanometer
to be read. A high resistance should be placed in series
with the standard cell to prevent any considerable current
passing through it. In first
performing this experiment
it is well to use a simple
and inexpensive f6rm of
standard cell, and the
Daniell cell (p. 159) will be
suitable. For later and
more accurate work, either
the Clark or the Weston cell
should be used. By vary-
ing the number of cells in
the main circuit or using different resistances in the main
circuit, different voltages at the terminals of the voltmeter
may be obtained.
The above method will not apply if the voltmeter is to
be tested at voltages less than the e. m. f. of the standard
cell. In this case, an inversion of the connections may be
used. Instead of balancing a variable part of the voltage
against the e. m. f. of the cell a variable part of the e. m. f.
of the cell is balanced against the voltage. A little con-
sideration will indicate the necessary change of connection.
Instead of the above temporary arrangement of circuits,
a Potentiometer, which consists essentially of the several
198 ELECTRICITY AND MAGNETISM.
circuits with the necessary resistances and keys in perma-
nent connection, may be used. With its aid the work may
be performed more rapidly and more accurately. Its parts
and connections should be carefully traced out with the
assistance of a large diagram (which may be attached to the
wall near the instrument) and additional explanation will be
supplied by the instructor.
A calibration curve, consisting of true volts plotted
against scale readings, should be drawn.
Questions.
1. Prove the above formula by applying Kirchoff's laws.
2. Draw a diagram showing the connections when a millivoltmeter
has to be calibrated.
3 . Draw a diagram to show how the above method could be adapted
to compare the e. m. f.'s of cells.
LIX. CALIBRATION OF AMMETER.
Hadley's Electricity and Magnetism, p. 325; Watson's Practical
Physics, pp. 516517; Henderson's Electricity and Magnetism, p.
205.
A method somewhat similar to that used for the volt-
meter may be employed. The current from a storage
battery that passes through the ammeter passes also through
a conductor of large current capacity and of measured
resistance and a switch. The potential difference at the ends
of this conductor is found by a shunt circuit, consisting of
a high resistance-box in series with a box containing low
resistances. In parallel with the latter is a circuit containing
a Daniell cell, a sensitive galvanometer and a key. A
special very fine fuse-wire should be used in series with the
two boxes, and its resistance should be known and taken
account of in calculating the current. The large conductor
should be immersed in oil and its temperature kept as nearly
as possible at the temperature at which its resistance is
determined. To prevent heating, the main current should
be closed for short intervals only.
CALIBRATION OF AMMETER.
199
The galvanometer should be protected by a shunt during
the first adjustments. Notice first in which direction the
galvanometer moves when the key in its circuit is depressed.
The deflection should be reversed or reduced when in
addition the switch is closed. If this is found not to be so,
the connection of either the Daniell cell or the storage
batteries should be reversed. The resistance in the box
nearest the galvanometer and, if necessary, in the other
box also, should be varied
until there is no deflection if
the galvanometer key is de-
pressed when the switch is
closed.
When the adjustment has
been obtained as closely as
possible, the fall of potential
between the points of the
high-resistance circuit to
which the standard cell circuit
is attached equals the e. m. f.
of the cell (p. 159). From this
and the resistances of the boxes the fall of potential between
the ends of the large conductor is found and then from the
resistance of the large conductor the current through it
and the ammeter is calculated. The total resistance in
the two boxes must be kept high. A preliminary calculation
will show about how large the resistance of large capacity
should be. A number of currents distributed over the
range of the ammeter should be used and from the results a
calibration curve should be drawn.
In the above we have assumed that the voltage applied
to the conductor exceeds that of the cell. If the reverse is
the case, the arrangement must be inverted; i. e., part of
the e. m. f. of the cell must be balanced against the voltage
applied to the conductor. This method must be applied
when the current is less than the quotient of the e. m. f. of
the cell and the resistance of the conductor.
200 ELECTRICITY AND MAGNETISM.
Instead of the arrangements of circuits above, the Po-
tentiometer referred to in Exp. LVIII may be used to
measure the fall of potential in the conductor of large cur-
rent capacity.
Questions.
1. Why must the resistance in the shunt circuit be large?
2. Storage batteries giving an e. m. f. of 50 volts are available for
calibrating an ammeter whose range is from 5 to 25 amperes and
whose resistance is 0.2 ohms, (a) What is the least possible value
for the resistance of large capacity? (b) What is the greatest
value?
LX. COMPARISON OF CAPACITIES OF CONDENSERS.
Henderson 1 s
's Practical
Hadley's Electricity and Magnetism, pp. 331334; Henderson's
Electricity and Magnetism, pp. 235241; Watson' j
Physics, pp. 530-535.
Two or more condensers are to be compared by three
methods.
(A) First Method. Each condenser is charged in turn
by the same battery and then discharged through a ballistic
galvanometer. Let the capacities of the two condensers be
C\ and C 2 . The charges which they receive when connected
to a battery of e. m. f. E, are Q 1 =C 1 E, and Q 2 .=C 2 E. Let
the throws of the galvanometer when the condensers are
discharged through it be d l and d 2 , respectively. Then
(Exp. LVII). The connections are the same as in Exp. LVII,
with the addition of one or more keys to charge alter-
nately two or more condensers. If either deflection be too
small, additional cells should be added. Storage cells may
be used if connections are made through special very fine
fuse-wires to protect the resistances.
If either deflection be too large the galvanometer should be shunted
by a known resistance, S. Let G be the resistance of the galvan-
ometer determined as in Exp. XLV (last paragraph), d' the
throw obtained with the galvanometer shunted, d the throw which
COMPARISON OF CAPACITIES OF CONDENSERS.
201
would have been obtained without the shunt, q r the quantity of
electricity passing through the galvanometer, q" the quantity passing
through the shunt. Then
9"_5
q' S
for charges of electricity, like steady direct currents, divide inversely
as the resistances (p. 158). Hence
Since
_*
q' d"
(B) Bridge Method. The two condensers to be com-
pared, C\ and C 2 , form two arms of a Wheatstone's Bridge,
two high non-inductive
resistances, R 1 and R 2 (see
figure), preferably several
thousand ohms, forming
the other two arms. These
two resistances are ad-
justed until on closing the
battery circuit at a the
galvanometer is not dis-
turbed. Then during both r~^
charge and discharge the
farther poles of the con-
denser (A and B) must
remain at the same potential as well as the nearer poles
(joined at D). Hence the charges Q^ and Q 2 in the con-
densers must have the ratio, Q t : Q 2 ' : C\ : C 2 . But the quanti-
ties which have flowed into the condensers will be inversely
proportional to the resistances through which the charges
have flowed, that is, Q^\ Q 2 : : R 2 : R^ Hence
FIG. 65.
C 2 R,'
The battery key has an upper and lower contact; the
upper contact (b), against which the lever ordinarily rests,
2O2
ELECTRICITY AND MAGNETISM.
short-circuits the battery terminals of the bridge, thus keep-
ing the condenser uncharged. The sensitiveness may be
increased by increasing the number of cells in the battery,
and also by using a double commutator (see p. 1 6 1 j . Instead
of a galvanometer a telephone may be used in this method,
the battery being replaced by a small induction coil.
(C) Thomson's Method of Mixtures. The connections
are as shown in the figure. K l is a Pohl's commutator, K 2
an ordinary single contact switch. When the swinging
arm of the commutator is in the position aa', the two con-
densers are charged, C l to the difference of potential at the
extremities of R lf C 2 to the difference of potential at the
extremities of R 2 . The swinging arm of the commutator is
now placed in the position b'b' and the two charges are
allowed to mix. If they are exactly equal, being of opposite
sign, the galvanometer will not be affected when K 2 is
depressed. R l and R 2 (which are large resistances, pref-
erably several thousand ohms), are adjusted until this is
secured. The charges being equal, C 1 V l = C 2 V 2 , and since
ABSOLUTE MEASUREMENT OF CAPACITY. 203
Questions.
1. What is the composite capacity of three microfarad condensers
in parallel? In series?
2. Which of these three methods do you consider best?
3. State briefly in words (without formulae) why charges divide
like steady currents; i. e., inversely as the ohmic resistances.
4. Why cannot series resistance be used in (A) to reduce the
sensitiveness of the galvanometer.
LXI. ABSOLUTE MEASUREMENT OF CAPACITY.
The magnitude of a capacity can also be found without
the use of a known capacity with which to compare it. This
can be done in different ways. The following is one of the
simplest.
Let Q be the charge received by the condenser of unknown
capacity C when connected to a cell of known e. m. f., E.
Then
To find Q discharge the condenser through a ballistic galvan-
ometer the constant of which has been found by the method
of Exp. LXIV. If the constant be K and the deflection
D t Q = KD.
LXII. COEFFICIENTS OF SELF-INDUCTION AND OF
MUTUAL INDUCTION.
Text-book of Physics (Duff), pp. 609-611; Ames' General Physics,
PP- 743-745; Watson's Practical Physics, pp. 543-548; Parr's
Practical Electrical Testing, p. 207; Hadley's Electricity and
Magnetism, pp. 417-422.
The coefficient of self-induction of a circuit is the number
of magnetic lines of force which link with the current when
the circuit is traversed by unit current. Owing to the
difficulty of calculating this important quantity from the
dimensions of the circuit, experimental methods of deter-
mination have much value.
204
ELECTRICITY AND MAGNETISM.
(A) Probably the best method (using direct currents)
is Anderson's Modification of Maxwell's Method. The
connections are shown in the figure. The coil of self-in-
duction L and resistance Q is made one arm of a Wheat-
stone's Bridge (preferably Post-office Box form). Ob-
tain a balance for steady currents by proper
variation of 5, so that when K l is closed
and then K 2 , the galvanometer is not dis-
turbed. For delicacy of adjustment it is
well to either have a resistance which can
be varied continuously form a part of 5,
or make the ratio arms P and R such that
5" is large. Vary r, the resistance in the
battery circuit, and if necessary, vary the
capacity of the condenser C until there is
a balance for transient currents; i. e.,
until the galvanometer is not disturbed
FIG. 67.
when K 2 is depressed and K l depressed afterward. Then
if C is the capacity of the condenser.
For at time t\et x = current in branch AB, y = current in AD = cur-
rent in D E, = current in BE .'. current in r = y + z. q = charge in
condenser, e = potential difference of its poles = Rz + r (y + z) .
Since there is a balance for transient currents we may equate the
e. m. f. in AD to that in AB. Hence
The current in the branch containing the condenser is (x z) ; but
it can also be expressed as
dq r de
dJ mC di
Hence
Now since there is a balance for steady currents RQ = PS and
since Rz = Sy, it readily follows that
= Qy + C[r
COEFFICIENTS OF INDUCTION.
If the resistances are expressed in ohms and the capac-
ity in farads, the results will be in henries.
Measure the self-induction of a coil whose length is
great compared with its diameter and compare the result
with that calculated. To calculate the coefficient of self-
induction it is necessary to know the number of lines of
force passing through the coil. This number multiplied by
the number of turns will give the number which link with
the current; i. e., the self-induction. If A be the area of
the cross section of a solenoid of practically infinite length,
with n Q turns per cm. of
length, the number of lines
is 4xn A for unit current.
The number of turns in a
length d is n Q d; hence the
coefficient of self-induction of
this length is 47iAn Q 2 d in C.
G. S. units. Reduce to
henries by dividing by io 9
(p. 162). '
To secure greater sensitive-
ness in making the balance FlG
for transient currents, replace
the battery by the secondary of a small induction coil and
the galvanometer by a telephone, or use a double commu-
tator (see p. 161).
(B) Comparison of Two Coefficients of Self-induction.
The two coils of self-induction L lf L 2 , and resistances R v
R 2 , are placed in two arms of a Wheatstone's Bridge, a
variable resistance, r, being included in one arm. By vary-
ing r, and, if possible, by varying one of the self-inductances,
if not, by varying r, P, and Q, find a balance for both steady
and transient currents.
Then for steady currents
206 ELECTRICITY AND MAGNETISM.
and for transient currents
P
where <o = 271 X frequency. Hence
The above adjustment is obtained by first securing a
balance for steady currents. A balance for transient cur-
rents is then sought by varying L l or L 2 . If this cannot
be secured, r and P must be both increased or decreased
and a balance for steady currents again obtained and then
one for transient currents found by
varying L t or L 2 . If neither L l
nor L 2 can be varied, a balance can
only be obtained by a series of trials
as above, the ratio of P and (7^ +r)
being kept constant, so that the
steady current balance may not be
disturbed. To increase the sensi-
tiveness with transient currents, a
double commutator (p. 161) should
be used or the battery may be re-
placed by the secondary of an induction coil and the galva-
nometer by a telephone.
(C) The coefficient of mutual induction of two coils is the
number of lines of force which link with the turns of the other
when the first is traversed by unit current. Pirani's
method is perhaps the most satisfactory for the experimental
determination of coefficients of mutual induction. The
connections are shown in figure 69. If M be the required
coefficient, C the capacity of the condenser, and r^ and r 2
the values of the variable resistances for which the galvan-
ometer is not disturbed,
FIG. 69.
STRENGTH OF A MAGNETIC FIELD. 207
For let the steady current in the battery circuit = i. The potential
difference at the terminals of r lt = ir^ The charge of the condenser is
Cir r If t time required to establish or destroy the battery current
the average current in the condenser branch during this time =
Cir l /t and the potential difference at the terminals of r. 2 = Cir l r. J /t.
Opposing this e. m. f. in the galvanometer circuit is that due to M,
the average value of which = Mi/t. If the galvanometer is not dis-
turbed on making or breaking the battery circuit, Cir^r^t^Mi/t
.'. M = Cr l r. 2 .
The secondary of M is acted on by two e. m. f.'s one due
to its connection with the main circuit in which there is an
e. m. f., the other due to mutual induction between the
primary and secondary. If these two do not oppose one
another a balance cannot be found. If such is found
to be the case the connections of the secondary to the
galvanometer circuit must be reversed. To increase the
sensitiveness of the method a double commutator may be
used or, better still, the battery and galvanometer may be
replaced by a small induction coil and telephone.
When an approximate adjustment has been found C and
r l should be altered until the sensitiveness is a maximum,
and then r l and r 2 treated in the same way.
Find the coefficient of mutual induction of two coils, one
wound upon the other, and one of which is long compared
with its diameter, and compare the result with that calculated
from the definition.
QUESTIONS.
1. Coils with iron cores do not have definite induction coefficients.
Explain.
2. How are resistance coils in boxes wound so as to be free from
self-induction ?
LXIII. STRENGTH OF A MAGNETIC FIELD BY A
BISMUTH SPIRAL.
Ames' General Physics, p. 758; Hadley' s Electricity and Magnetism,
p. 296.
The electrical resistance of a bismuth wire is changed
when it is placed transverse to a magnetic field and the
magnitude of the change depends on the strength of the field.
208 ELECTRICITY AND MAGNETISM.
When a curve representing the resistance of a flat spiral
of bismuth as a function of the strength of the magnetic
field has been obtained the spiral may, in connection with a
Wheatstone's Bridge, be used to measure the strength of any
magnetic field within the range of the calibration. For
instance, it may be used to study the magnetic field of an
electromagnet. The following three points may be examined :
(A) Find how the magnetic field between the poles varies
when the strength of the current actuating the electro-
magnetic is varied by means of a rheostat.
(B) Find how the strength of the field midway between
the pole-pieces changes when the distance apart of the pole-
pieces is varied, the current being kept constant.
(C) Find how the strength of the field in an equatorial
plane varies with the distance from the axis of the pole-pieces.
In each case represent the results by means of a curve.
LXIV. CONSTANT AND RESISTANCE OF A BALLISTIC
GALVANOMETER.
Text-book of Physics (Duff), pp. 556, 562; Ames' General Physics, p.
674, 712; Watson's Practical Physics, pp. 518-524; Pierce, Am.
Acad. Arts and Sc. Vol. 42, pp. 159160.
When quantities of electricity are discharged through a
ballistic galvanometer (p. 156) the throws are proportional
to the quantities or,
Q=K.d,
where K is the constant of the ballistic galvanometer. To
determine the value of K a known quantity must be dis-
charged through the galvanometer and the throw noted.
This known quantity might be obtained from a condenser
of known capacity, charged to a known potential, or by
turning an earth inductor (Exp. XLIII) in a field of
known strength. Both of these methods, however, re-
quire that other constants (capacity and e. m. f. in the first,
strength of field in the second) be determined.
BALLISTIC GALVANOMETER. 2OQ
A simpler method is to use a so-called calibra ting-coil ;
i. e., an induction coil of known winding without a magnetic
core. The primary is a long, straight helix, so long that
there is no appreciable leakage near the center. Over the
center there is wound a secondary. If the primary be of n
turns per cm. and the secondary be of n' total turns, then
the magnetizing force produced by a current of i amperes
in the primary is
10
and the quantity induced in the secondary by making or
breaking i is
han'
where a is the area of cross section of the cylinder on which
the primary is wound, r is the total resistance of the second-
ary circuit and the factor io 8 is required when q is in
coulombs and r in ohms (p. 162).
The number of lines of force that pass through the secondary is
a h. Hence when h is increasing the induced e. m. f. is in absolute
units
,d(ah)
The quantity induced is I i dt and i equals e/r. Hence
Jn' d(ah) n'ak
7~dT '
From the above expression for q and the throw d, K can
be calculated. If the throw is small it may be doubled by
reversing i and the half of the double throw taken for d.
Several currents should be tried. The value of K thus
found is in coulombs per scale division. If the distance of
the scale from the galvanometer be changed in any pro-
portion K will be increased in the same proportion. Hence
the distance should be noted unless it is not liable to change.
It is often necessary to change the sensitiveness of a ballistic
galvanometer by shunting it or putting resistance in series
14
210 ELECTRICITY AND MAGNETISM.
with it. To allow for this we must know the resistance of
the galvanometer.
The resistance of a ballistic galvanometer when used
ballistically on a closed circuit is different from its resistance
when used as an ordinary galvanometer. This is due to the
fact that the galvanometer coil moves in a magnetic field,
and thus an induced e. m. f. is produced, which opposes the
applied e. m. f. and has the same effect as an added resist-
ance. To find the effective resistance of a ballistic gal-
vanometer we may use the same apparatus and connections
as in finding the constant of the galvanometer. If a
current be reversed in the primary of the calibrating coil,
the quantity of electricity that will flow through the second-
ary will vary inversely as the total secondary resistance.
Hence, by observing the throw with a certain primary
current, and then increasing the secondary resistance by the
insertion of a box-resistance and repeating the reversal of
the primary, we can, by proportion, find the resistance of
the galvanometer when used ballistically.
The resistance of the galvanometer should also be found
by the method of Exp. XLVI or that of Exp. XLV and
compared with the above.
LXV. MAGNETIC PERMEABILITY.
Text-book of Physics (Duff), pp. 500503; Ames' General Physics, pp.
609, 615; Watson's Physics, pp. 712-714; Hartley's Electricity
and Magnetism, pp. 384-392; Watson's Practical Physics, pp.
563-568; Henderson's Electricity and Magnetism, pp. 282-284;
Swing's Magnetism in Iron, Chapter III.
A current in a long solenoid of wire will produce near
the center of the solenoid a magnetic force H, which may
be specified by the number of lines of force per unit of area
at right angles to the lines. If a long iron rod be now
thrust into the solenoid, the number of lines of force (now
called lines of induction) will be much greater, say B per
unit of area. The permeability of the iron is defined as
u=B+H.
MAGNETIC PERMEABILITY. 211
If this experiment were performed with comparatively
short iron rods, it would be found that B would be less the
shorter the rod. One consistent way of explaining this is
to consider the free poles developed at the ends of the rod
when magnetized. A little consideration will show that
they of themselves would produce a magnetic force in the
space occupied by the iron, this magnetic force being opposed
to the original magnetizing force, and so we may say that
the effective magnetic force, H, is the original magnetic
force diminished by the demagnetizing force of the poles.
It is this effective magnetic force that we should divide into
the induction to get the permeability. The calculation of
the demagnetizing force is usually difficult and uncertain,
and so it is better to take some method of eliminating it.
One such way is that implied in the statement at the out-
set, to use a long rod, for that will diminish the magnitude
of the demagnetizing force at the center. But the necessary
length makes it inconvenient to test specimens in this way.
Another method is to join the ends of the rod by a heavy
yoke of iron, for opposite poles developed in the yoke neu-
tralize the effect of the poles in the rod. (This is one way of
stating the case. Another way is to say that the yoke carries
around the lines of force. A third way is to say that the
yoke diminishes the magnetic resistance of the circuit.)
The difficulty with a yolk method is in getting a satisfactory
contact between yoke and rod. A very small gap will re-
sult in the neutralization being not quite complete (or in
leakage of lines of force or in magnetic resistance) .
A more satisfactory method is to take an endless speci-
men; i. e., a ring. Then there are no free poles and no de-
magnetizing force. On the ring a magnetizing coil of N
turns per cm. is wound. When a current of / amperes passes
through it, the magnetizing force produced is
IO
For finding the value of. B a, secondary coil is wound on the
212
ELECTRICITY AND MAGNETISM.
ring and put in series with a ballistic galvanometer. Sup-
pose the iron initially free from magnetism. The setting up
of the field B produces a discharge, Q, of electricity through
the secondary. If A be the area of cross section of the ring
and N' the total number of turns,
N'AB
R being the total (ohmic) resistance of the secondary circuit.
(The factor io 8 is not necessary if Q and R are in absolute
units. It must be used when Q is in coulombs and R in
ohms (p. 162). If the throw of the galvanometer is D
Q=K-D
when K is the ballistic constant (Exp. LXIV). If K is
not known, a calibrating coil for determining it should be
included in the arrangement of the apparatus. From the
above formulae and the data, H, B and fi can be calculated.
VVVVVVVS.'Vv'V'S
FIG. 70.
In making the connection for the practice of the method,
it is much better to have a clear understanding of the plan
and purpose of each part and to proceed systematically
than to copy the connection from a diagram. In the first
MAGNETIC PERMEABILITY. 213
place, the secondaries of both coil and ring should be kept
permanently in series with the galvanometer. Then a switch
is to be so arranged that the current can be passed through
either the primary of the calibrating coil or that of the ring.
A suitable rheostat and ammeter are needed in the primary
circuit. If, as the primary current is increased, the deflec-
tions of the galvanometer become too great to be read, a
resistance must be put in series or in parallel with the gal-
vanometer. The former is preferable. In choosing this
added series resistance, it is well to so choose it that the
whole new secondary resistance is made a simple multiple
of the former resistance. If this is done the throw will be
reduced in the proportion in which the resistance is increased,
and all throws may be reduced to what they would have been
with the original resistance by multiplying the actual throw
by the proportion in which the secondary resistance was
increased. For methods of bringing the galvanometer to
rest, see p. 156.
Before readings are begun the ring should be demagnetized
as thoroughly as possible. This can be done by passing an
alternating current through the primary and reducing it
from a large value to zero by means of a rheostat, or, by
rapidly commutating and at the same time reducing a
direct current. Also at each new value of the magnetizing
current, before readings are taken, the commutator should be
reversed several times, so that the iron may come to a steady
cyclical state. Instead of attempting to get the throw on
making the primary current, the double throw on reversing
the current is taken with both calibrating coil and ring
and divided by 2.
At least three throws that agree well should be read for
each strength of the primary current. The magnetizing
current should be increased at first by small steps to bring
out the characteristic features of the curve of magnetization,
afterward by larger steps. The work need not be continued
after the readings begin to differ in a much smaller propor-
tion than the successive magnetizing currents, for this shows
214 ELECTRICITY AND MAGNETISM.
approaching saturation. The throw at break of current
should also be carefully noted as a means of estimating the
permanent magnetism; for from the throw at break the
diminution of B, and, therefore, the residual value of B,
can be calculated as above.
In the report the various values of /, H, Q, B, and /*
should be tabulated and a curve drawn with B as ordinates
and H as abscissae (B-H curve or curve of magnetization).
On the same sheet a B-fi curve should also be drawn and a
third curve showing the permanent magnetism as deduced
from the throws at break of the current.
Questions.
1. Why is only the ohmic and not the self -inductive resistance of
the secondary considered ?
2. What is the effect of the windings being closer together on the
inside of the ring than on the outside?
3. What is meant by intensity of magnetization? Susceptibility?
Calculate a few values from your results.
LXVI. MAGNETIC HYSTERESIS.
Text-book of Physics (Duff), pp. 503-504; Watson's Physics, pp. 716
722; Hartley's Magnetism and Electricity, pp. 393-395. Watson's
Practical Physics, p. 561; Henderson's Electricity and Mag-
netism, p. 294; Ewing's Magnetism in Iron, Chapter V.
Let a magnetizing force applied to a specimen of iron as in
the preceding experiment be increased step by step and let
the resulting increases of magnetization be observed. At
some stage let the process be stopped and then the mag-
netizing force decreased by the same steps. It will be found
that the steps of decrease of magnetization are less than
those by which it at first increased, or the magnetization
lags behind the magnetizing force. This is called hysteresis.
For a complete view of the process a cycle must be com-
pleted, i. e., the magnetizing force must be decreased step
by step to zero, then increased to a negative value equal
(numerically) to the positive value at which the decreases
were begun, then decreased again to zero, and finally
MAGNETIC HYSTERESIS.
215
increased again to the highest positive value. Thus a
hysteresis loop will be obtained.
With a ring specimen, over which primary and secondary
coils are wound, there are two methods of procedure.
(A) Step by Step Method. This method follows closely
the general description given above. The successive steps
are indicated in figure 71. The increases or decreases of
7 must be made without break of the current. The steps
must not be too large or the points on the curve will be
too far apart, and they must not be too small or the work
-I -I,
FIG. 71.
will become tedious. To satisfy these conditions, place in
the primary circuit a special rheostat consisting of suitable
resistances in parallel, each of which can be short-circuited
by a knife-edge switch. Such a rheostat may be made up
with resistances permanently connected in position, but a
better plan is to use removable resistances. In the latter
case a considerable collection of units should be supplied,
and from these, by a preliminary trial, units that will
produce suitable changes of I (e. g. , from 4 to o . 5 amp. by steps
of o. 5 amp.) should be chosen and placed in position in the
rheostat.
It is not necessary to start from zero magnetization.
2l6 ELECTRICITY AND MAGNETISM.
Beginning with the highest current to be used, reverse
several times to produce a cyclical state and then find the
throw on reversal. From this the maximum value of B can
be calculated as in Exp. LXV. Then dimmish the current
by steps and note the throw in each step. After the step
that reduces the current to zero, the current must be
reversed and the resistances increased step by step. The
rest of the process needs not be described. From each throw
the corresponding change of induction, AJ5, is calculated
as in Exp. LXV. When the cycle has been completed the
algebraic sum of the throws should be zero. It should not
be necessary to change the sensitiveness of the galvanometer;
it will give the smaller throws with less accuracy, but they
are less important. This "step by step" method of meas-
uring hysteresis is the most instructive and is not difficult
after some initial practice. It has, however, the disad-
vantage that an error in one reading of the galvanometer
vitiates the whole.
(B) The Ewing-Classen Method. The last-mentioned dis-
advantage is avoided in the method by starting each
step from the maximum value of B. As before, we first find
by reversals the value of B corresponding to the maximum
value of /. We then diminish I (without breaking the cur-
rent) and from the throw we calculate the diminution of
B. This gives us a second point on the curve. We then
return to the maximum current and, after several reversals,
to re-establish the cyclical state, we again decrease /, but by
a larger amount than before. From the throw we again
calculate the diminution of B and thus get another point
on the curve. Proceeding in this way, we reach the stage
at which I is decreased from its maximum to zero. This
gives us the point at which the curve crosses the axis on
which B is plotted.
A simple method of producing the above changes of 7 is to
connect the rheostat described under (A) in parallel with one
of the cross-bars of the Pohl's commutator used for reversing
/ (Fig. 72). If this cross-bar be suddenly removed, the
MAGNETIC HYSTERESIS.
217
resistance in the rheostat will be thrown into the circuit
without breaking the current.
By the above process, we have obtained that part of the
descending branch of the hysteresis loop, which lies to the
right of the B axis. To obtain the remainder of the branch,
we again proceed by steps from the positive maximum value
of B, but, since each change of / will carry it from its positive
maximum to a smaller negative value, we must simultane-
ously diminish and reverse the current. To be able to do this,
remove the cross-bar of the commutator which is in parallel
with the rheostat and turn the commutator so that the
r
FIG. 73.
current flows to the ring, but does not pass through the
rheostat (Fig 73). If the commutator be now reversed, the
current will be reversed and will be diminished by passing
through the rheostat. Thus we get another point on the curve
and, by a series of such steps with decreasing resistances in
the rheostat, the descending branch of the loop is completed.
To trace the other branch we might proceed as above, begin-
ning each step from the negative maximum of B. This,
however, is unnecessary, since we would evidently be merely
repeating the previous readings. The loop is symmetrical
about the origin, and the co-ordinates of the ascending branch
2l8 ELECTRICITY AND MAGNETISM.
are equal to those of the descending branch but with signs
reversed.
It can be shown that the energy expended in such a cyclical
change of magnetization is
C
xJ
fidB
4xJ
ergs per c.c. of the iron. The integral also represents the
area of the loop, due allowance being made for the scale on
which it is plotted. Hence if the area be found by means
of a planimeter (the use of which will be explained by an
instructor), the energy loss per c.c. per cycle can be cal-
culated.
The total number of lines of induction through each turn of the
magnetizing coil is AB. Since the total number of turns is IN ,
when B is being increased there is induced in the magnetizing coil an
e. m. f.
_, d(lNAB} jlB r n c .,
E = -- - JT - = VN-jT C. G. S. units.
at at
V being the volume of the core ( = IA). The work done by the
battery in time dt in overcoming this opposing e. m. f. is
dW=IEdt=INVdB ergs
Now the area of the hysteresis loop is the integral of HdB and
,W - I HdB
Questions.
1. What rise of temperature would 1000 cycles produce in the iron
if no heat were lost ?
2. How much less would the energy loss be if the maximum
magnetization were half as great as in your cycle?
THE MECHANICAL EQUIVALENT OF HEAT. 2IQ
LXVII. (A) THE MECHANICAL EQUIVALENT OF HEAT.
(B) THE HORIZONTAL INTENSITY OF THE
EARTH'S MAGNETISM.
Text-book of Physics (Duff), pp. 557, 587-588; Ames' General Physics,
pp. 664-665, 688; Watson's Physics, pp. 692-693, 674-677, 775-
776; Crew's General Physics, 289, 319; Hadley's Electricity
and Magnetism, pp. 336340, 458; Watson's Practical Physics,
pp. 508-512.
If Q calories of heat be produced in a conductor by the
passage of a current i for time t, and if no other work,
chemical or mechanical, be performed, then
JQ=?Rt t
] being the mechanical equivalent. If i be expressed in am-
peres, R in ohms and Q in calories, i 2 Rt will be in joules
(one joule being io 7 ergs), and J will be obtained as the
number of joules in a calorie.
Q can be measured by immersing the conductor of resistance
R in a known mass of water contained in a vessel of known
water equivalent. The mass of water may be obtained
with sufficient accuracy by measuring it from a burette.
To eliminate the effects of radiation, conduction and con-
vection, the water should be at the beginning of the passage
of the current as much below the temperature of the room
as it finally rises above it, for the current is kept steady and
the temperature of the water therefore rises steadily, so
that it is as long above the room temperature as below.
The resistance R may be measured against a standard
ohm coil by Wheatstone's Bridge, and, since it will be
found necessary to use a wire of comparatively small re-
sistance, R should be measured with great care. Leads of
large size and small length should be employed for con-
necting the wire to the bridge. While being measured it
should be immersed in the calorimeter in water at the
temperature of the room, so that the mean resistance
throughout the experiment is obtained. To reduce to
absolute units the resistance in ohms is multiplied by io 9 .
220
ELECTRICITY AND MAGNETISM.
The current, i, may be obtained from its chemical effect
in another part of the circuit. Careful measurements have
shown that unit current (C. G. S.) flowing through a solution
of copper sulphate of a certain strength between copper
electrodes deposits 0.00326 gms. of copper per second on
the cathode.
The form of copper voltameter employed consists of a
glass vessel containing a solution of copper sulphate into
which dip three plates. The two outer are of heavy copper
Calon'meter
Rheostat
Voltameter
FIG. 74.
and are both joined, directly or indirectly, to the positive
pole of the battery, forming the anode. The intermediate
plate is thin and light, and is connected to the negative
pole of the battery, forming the cathode. A satisfactory solu-
tion consists of 15 grams of copper sulphate dissolved in
100 grams of water, to which are added 5 grams of sulphuric
acid and 5 grams of alcohol. (The alcohol is easily oxidized,
thus preventing the oxidization of the deposit on the
cathode and the formation of polarizing compounds at the
anode.)
THE MECHANICAL EQUIVALENT OF HEAT. 221
Clean the two anode plates with sand-paper and fasten
them in the two outside binding posts of the top of the
voltameter. Clean with sand-paper a cathode plate, wash
with tap water and then with alcohol. When dry, weigh
on one of the chemical balances, weighing to milligrams
with the rider. Wrap in paper and set aside. Be very
careful not to touch with the fingers any part of the plate
which will be in the solution, after it has been cleaned.
Clean with sand-paper a trial cathode and mount it on the
middle binding post. Before putting the voltameter in the
circuit, dip the two wires which are to be connected to the
voltameter in the solution of the voltameter. Decrease the
variable resistance until you have a moderate current, but
do not entirely cut it out. Notice on which wire copper is de-
posited as a brown powder. Connect this wire to the cathode
of the voltmeter and the other wire to the anode plates.
It is important that the current be kept constant. It is
true that even if the current vary, the deposit will give the
true mean value of the current. But what is needed is the
mean value of i 2 , and this is not necessarily the same as the
square of the mean value of i. If a storage battery in good
condition be used as the source of current, the current will
not vary much; nevertheless, a tangent galvanometer or an
ammeter should be included in the circuit to test the con-
stancy of the current. There is also another reason for
including a current meter of some form. The difference
of potential at the terminals of the heating coil, or iR,
must not be as great as the e. m. f. (1.6 V) that will elec-
trolyze water, otherwise some part of the energy of the
current will be spent in chemical work. Knowing R, one can
choose a safe value for i. If the constant of the galvan-
ometer be not known, it can be calculated roughly from
the dimensions of the coils and the approximate value (say
o.i 8) for the horizontal component of the earth's field (see
Exp. XLII), and so the deflection corresponding to a safe
value of i deduced. An ammeter, if available, affords a
still simpler means.
222 ELECTRICITY AND MAGNETISM.
If a tangent galvanometer be used a fairly reliable value
for the horizontal component' of the earth's field may be
deduced from the results of the experiment. For this pur-
pose the dimensions of the galvanometer should be care-
fully measured and the current through it frequently re-
versed and carefully read. The Helmholtz form of tangent
galvanometer may be used. This consists of two coils sepa-
rated a distance equal to their common radius, with the
needle on their common axis midway between them. This
arrangement of two coils produces a very uniform field
over quite an area where the needle is located, allowing the
use of a longer needle. The formula for the galvanometer
(the proof for which will be found in text-books on physics) is
r / # 2 \f
i =H 1 i H I tan .
(If a simpler type of tangent galvanometer, with but
one coil, is used, x is the distance from the plane of this coil
to the suspension of the needle.)
Equating this expression, which involves the dimensions
of the galvanometer and the deflection, to the current as de-
termined by the voltameter, the horizontal component is
deduced.
When the adjustments have been completed, open the
switch, remove the trial cathode and put in place the other
cathode, which has been kept wrapped in paper. Take
care that there is no metallic connection between the
cathode and the anode plates. Remove all iron from the
neighborhood of the tangent galvanometer and from your
pockets. All wires must be close together to avoid stray
induction and the galvanometer had best be at some distance
from the other apparatus.
After reading the temperature of the calorimeter every
minute for five minutes note the exact second on an ordinary
watch, and close the switch. As soon as possible, read both
ends of the needle. Reverse the current, making the reversal
quickly, and again read both ends of the needle. Always
THERMOELECTRIC CURRENTS. 223
estimate tenths. Reading both ends of the needle eliminates
error due to the axis about which the pointer turns, not
coinciding with the center of the graduated circle, and
reversing eliminates uncertainty about the reading for the
zero position. Keep the current constant with the variable
resistance. At intervals of three minutes (approximately)
read both ends of the needle and, reversing, again read
both ends. Read the temperature every minute to tenths
of the smallest division. Allow the current to flow until
the temperature has risen to the extent desired. Note the
exact second of breaking the circuit. Continue to observe
the temperature at minute intervals for five minutes. Re-
move the cathode, being very careful not to touch the
copper deposit. Wash it gently with tap water and then
with alcohol, allowing the liquid to simply flow over the
surface. When it is dry, weigh as before. Measure very
carefully the diameter of the coils in a number of directions,
and from the mean determine r. Count n, the total number
of turns in both coils, and measure 2 x, the distance between
the centers of the two coils. Weigh the inner calorimeter
vessel and note of what metal it consists. Plot the
temperature readings and correct for radiation (p. 63).
Questions.
1 . Calculate what the exact voltage at the terminals of the heating
coil was.
2. What sources of error remain uneliminated ?
3. Calculate the mean activity of the current in the coil during the
experiment.
4. What are the peculiarities and advantages of the tangent
galvanometer ?
5. What chemical actions take place in the voltameter? To what
is the deposition of any metal proportional?
6. Why is it advantageous to have the deflection about 45?
LXVIII. THERMOELECTRIC CURRENTS.
Text-book of Physics (Duff), pp. 593-598; Ames' General Physics, pp.
679-683; Watson's Physics, pp. 696-705; Hartley's Electricity
and Magnetism, pp. 359-381.
To the ends of wires of iron, nickel, silver etc., cop-
per wires are soldered and brought to binding posts on a
224 ELECTRICITY AND MAGNETISM.
board. Below the ends of the board are vessels contain-
ing sand or oil in which two test-tubes are supported. The
junctions are placed in these test-tubes as indicated in
figure 75. The binding posts are connected, by copper
wires, to a key of as many parts as there are wires to be
tested, so that each circuit may be completed through a
sensitive galvanometer. Thermometers are placed in the
test-tubes to note the temperatures as one vessel is being
heated by a burner.
It is especially important that the temperature should
be ascertained accurately. Hence heat should be applied
Cu.
Cu.
FIG. 75.
cautiously, especially at first, and, when observations are
to be made, the source of heat should be removed, and time
should be allowed for the temperatures to become fairly
constant.
The galvanometer reading should be noted with the great-
est care and the zero should be frequently tested. After
each reading of the galvanometer, the temperature should be
noted.
If a high resistance galvanometer of sufficient sensitive-
ness is available, the other resistances may be neglected
and the various e. m. f.'s will then be proportional to the
deflections. Or a sensitive low resistance galvanometer, with
a constant high resistance permanently in series, may be used
with similar simplicity. The constant of the galvanometer,
considered as a voltmeter, may be found by applying to it
a fraction of the e. m. f. of a standard cell (pp. 161, 179).
With this arrangement (which will be readily under-
stood from the figure) the thermo-electric force of each
circuit, consisting of copper and another* wire, may be de-
ELEMENTARY STUDY OF ALTERNATING CURRENTS. 225
termined. Curves representing the results should be plot-
ted with the differences of temperature of the junctions as
abscissae and the e. m. fs as ordinates.
If a low resistance galvanometer of low sensitiveness is
used, it will be necessary to consider it as an ammeter.
In this case, the resistances of the various circuits and of
the galvanometer must be found and the constant of the
galvanometer, in amperes per unit deflection, must be ob-
tained by connecting it in series with a standard cell and a
sufficient known resistance. Thus, the currents and the
resistances being known, the thermo-electric forces can be
calculated.
Questions.
1. State what would be observed if the temperature of the hot
junction were increased steadily beyond the highest temperature
used in this experiment.
2. Is the effect observed here due solely to differences of potential
produced at the contacts ?
LXIX. ELEMENTARY STUDY OF RESISTANCE, SELF-
INDUCTION AND CAPACITY.
Text-book of Physics (Duff}, pp. 623624; Watson's Physics, p. 758;
Hadley's Electricity and Magnetism, pp. 441450.
In the following exercises, which are intended for students
who have not made a study of the theory of alternating
currents, some of the properties of such currents are studied
and compared with those of direct currents.
Ohm's Law for steady currents states that
E
=a constant, R,
i
where R is called the resistance of the conductor.
(A) Apply various e. m. f.'s to anon-inductive conductor.
Measure the current by an ammeter, and the voltage by a
voltmeter (of any type) and calculate R for each value of
the e. m. f. The latter may be varied by means of a series
rheostat.
226 ELECTRICITY AND MAGNETISM.
(B) Apply the same method to (i) a large coil, (2) the
large coil and the non-inductive resistance, (a) in series, and,
(b) in parallel. Compare the results of (a) and (b) with the
calculated values.
(C) Repeat (A) and (B), using alternating currents and
an electrostatic voltmeter. Corresponding to Ohm's Law we
have
E /
- =constant = V R
i
where the constant is called the impedance and L is the
coefficient of self-induction and n the frequency. Find the
value of the constant for different e. m. f.'s and currents,
and, from the mean and the values of R and n, calculate L.
Contrast the results in series and
parallel combinations with the
N values calculated by treating im-
- 1 pedance in the same way as re-
Jtheo I i . ., .
A I sistance in direct currents.
Tabulate all results so that they
may be readily compared.
(D) When an alternating cur-
F rent is applied to a condenser, it
is charged, discharged, charged
oppositely and discharged during each alternation. Evi-
dently the total quantity that traverses the leads in each
unit of time is proportional to the frequency and to the
product of the capacity and the voltage (since q = CV) and
it can be shown that the current is given by
i = 2nnCV.
Measure i for various values of V and calculate C. Do
this for several condensers (i) separately, (2) in parallel,
(3) in series. Compare the results of (2) and (3) with the
calculated values.
Questions.
i. What is meant by the effective value of an alternating current
and what ratio does it bear to the maximum value?
INDUCTION AND CAPACITY, ALTERNATING CURRENTS. 227
2. How, by means of a diagram, would you find the impedance
when given the ohmic resistance R and the inductance L ?
3. Supposing the alternating e. m. f. resolved graphically into
two parts, one to overcome the ohmic resistance and the other to
overcome the inductance, what relation between the phases of these
two parts does question (2) suggest?
LXX. SELF-INDUCTION, MUTUAL INDUCTION AND
CAPACITY, ALTERNATING CURRENTS.
See references to LXIX. J . J. Thomson's Electricity and Magnetism,
233, 244-245; Jackson's Alt. Cur., pp. 90-91, 151-200; Parr's
Electrical Eng. Testing, pp. 222224, 228231, 234235. For
Electrostatic Voltmeter, see Parr, 367-371.
This exercise, which is somewhat more advanced than the
preceding, is intended, for students who have made some
study of the theory of alternating currents.
Let E be the alternating e. m. f. in a circuit of resistance
R, capacity C, and self-induction L. If i is the current,
. E
*>- i /Co) 2
We can test the above formula by calculation, after
measuring i, E, R, C, and L. An inductance with a magnetic
core has a variable value of L, the magnitude of which
depends on the strength of the current. Hence, for this
experiment, an inductance consisting of a very large coil
containing no iron is used.
(A) Measurement of C. If, in the general formula, L
be zero, C can be deduced from the values of i, E, and R,
assuming that <o, which equals 2 it times the frequency n,
is known. E, the e. m. f. across the terminals of the con-
denser, is measured by an electrostatic voltmeter, i by an
alternating current ammeter. Initially a high resistance
of large current capacity must be included. This may later
be cut out. A fuse of lower capacity than the range of the
ammeter must be permanently in circuit.
(B) Measurement of L. The value of L is found by
observing the values of i and in a circuit containing the
self-inductance coil and then applying the general formula.
228 ELECTRICITY AND MAGNETISM.
Sufficient additional resistance must be placed in the cir-
cuit, but the value of E required is that across the terminals
of the inductance coil. R, which in this case is the resistance
of the coil, is best found by Wheatstone's Bridge.
(C) Test of General Formula. Connect the condenser
and self-induction in series. Measure the current and the
total e. m. f. ; also the e. m. f. across each part. Connect the
condenser and inductance coil in parallel. Measure the
common e. m. f., the total current and the current in each
branch.
Calculate i in the series arrangement from the above
formula and compare with the experimental value. If you
are familiar with the method of complex quantities and
graphical methods, apply these also to calculate the
currents in both series and parallel arrangements.
(D) Measurement of Mutual Inductance. Measure the
mutual inductance, M, of the two coils of a transformer
(with iron core) by observing the e. m. f., E, across one coil
when a measured current, i, is applied to the other.
E=Mi<o.
Vary i several times and find how M varies.
Questions.
1 . Why is an electrostatic voltmeter necessary ?
2. Does the self-induction depend upon the frequency? Why
does the latter enter into the equation ?
LXXI. DIELECTRIC CONSTANTS OF LIQUIDS.
Text-book of Physics (Duff), pp. 533, 535; Ames' General Physics, pp.
641, 66 1 ; Watson's Physics, p. 637; Hartley's Electricity and
Magnetism, Chapter X.
The dielectric constant of a liquid, or the ratio of the
capacity of a condenser with that liquid as dielectric to its
capacity when its dielectric is air, can be determined by
a comparison of capacities by the Bridge Method of Exp.
LX. For this purpose it is convenient to use a condenser
consisting of two parallel plates, the distance between which
DIELECTRIC CONSTANTS OF LIQUIDS.
229
is adjustable, as shown in figure 7 7 . The distance between
the plates can be measured by means of a scale, B, attached
to the movable plate A, and a vernier attached to the
framework. The plates hang in a vessel for holding the
dielectric. Two methods can be used. In one the distance
between the plates is not varied; in the other it is varied.
A A'
B'
p'
FIG. 77.
The first method consists in comparing the capacity of the
above condenser with that of a Ley den jar (i) when the
dielectric is the liquid to be tested; (2) when it is air. From
these comparisons the ratio of the capacities of the condenser
in the two conditions is deduced and this equals the dielectric
constant of the liquid. Instead of a battery and galvan-
ometer, an induction coil and a sensitive telephone are used.
The second method assumes the (approximate) formula
Ae
C
for the capacity of such a plate condenser (in electrostatic
units, see p. 162), where A is the area of each plate, e is the
dielectric constant of the surrounding medium, and d is
230 ELECTRICITY AND MAGNETISM.
their distance apart. If C be the same with two dielectrics,
but with different values of d
e 2 d 2
Having obtained a balance for air as dielectric, leave
the resistances in the bridge unchanged and again obtain
a balance, after filling the jar with the liquid to be tested,
by adjusting the distance between the plates. This second
method is less accurate, since the formula assumed is only
approximate and the distances cannot be determined as
accurately as the resistances. An accurate formula will be
found in Kohlrausch's Physical Measurements, p. 379.
The above methods should be applied to two highly
insulating liquids, such as kerosene and benzol.
Questions.
1. Calculate the capacity of the two plates when separated by
(a) air; (b) liquid, the distance apart being the same as in the first
part of the experiment.
2. Calculate the charge for each case if (a) 100 electrostatic units
of potential are applied; (b) 100 volts (p. 162).
LXIII. ELECTRIC WAVES ON WIRES.
Dielectric Constants of Liquids.
Ames' General Physics, pp. 752754; Watson's Physics, pp. 856-858,
870871; Text-book of Physics (Duff}, pp. 635-639; Hadley's
Magnetism and Electricity, pp. 541-548, 585-586; J. J. Thom-
son's Electricity and Magnetism, 243; Kohlrausch's Physical
Measurements (on capacity of a plate condenser), p. 379. Drude,
An. der Phys., Vol. 8, p. 336.
In this experiment electric waves on a wire, AD, are
excited by electric oscillations in a neighboring circuit or
"exciter," E, which contains an inductance, L, and a
capacity, C. The period of such oscillations is T=2x\/LC.
The inductance is that of two thick semicircular wires.
The ends e and e' of these wires carry small spheres and
the wires are so bent that the spheres are beneath the
ELECTRIC WAVES ON WIRES.
231
surface of kerosene in a small cup and form a spark-
The length of this spark-gap can
be adjusted by means of a mi-
crometer screw attached to one
of the ebonite posts, H H', on
which the semicircular wires are
supported. The condenser is of
the variable form shown in figure
77 and is connected between the
other two ends of the semicircular
wires. The impulses that start
the oscillations in the exciter are
produced by a Tesla coil, the
secondary of which is connected
across the spark-gap ee' t while the
primary of the Tesla coil is con-
nected through another spark-gap,
Z, to the secondary of an induc-
tion coil /.
The wire, AD, on which the
waves are formed is bent to a
U-form and lies in a horizontal
plane above the plane of the
exciter. The oscillations in the
exciter act by induction on the
part, A, of the wire and produce
waves that move along the wire
toward D; between any two cor-
responding points, such as d and
d', there is an oscillating differ-
ence of potential and the trans-
mission of this oscillation consti-
tutes the wave-motion. At the
free end, D, these waves are
reflected and interfere with the
direct waves. If the wire is of
proper length, this interference
gap.
B
B
D
FIG. 78.
232 ELECTRICITY AND MAGNETISM.
produces stationary waves; that is, the wire "resonates"
to the exciter. If the wire be also sufficiently long, there
will be one or more nodes on the wire, that is, places of no
potential difference, with intervening antinodes, or places of
maximum oscillating potential difference. If a small wire
"bridge," B, be placed across the wire at a node it will not
interfere with the stationary waves ; but it will destroy them
if it is placed at any other point. (It will be instructive to
compare the above with the formation of stationary sound
waves in a resonance tube such as that of Exp. XXX, the
tuning-fork being the exciter.) When the bridge has been
placed at a node the part of the wire between it and the free
end, D, could be changed in length or removed without
appreciably diminishing the oscillations between A and B,
(just as the lower part of a violin string that is touched by
the finger does not interfere with the vibrations of the upper
part). The part A of the wire is (approximately) a node,
although it is the part where the oscillations are excited.
(Compare with this the fact that when a tuning-fork,
connected to a long thread as in Melde's experiment, throws
the latter into stationary vibrations, the point of connection
is a node).
Various means have been used for detecting such station-
ary vibrations. The simplest is a "vacuum" tube, V,
which contains helium at a very low pressure. An alter-
nating potential difference between the ends of such a tube
will cause oscillating discharges accompanied by a glow.
If placed across the wire when there are no stationary
oscillations the tube will not glow; but, if stationary oscil-
lations exist, it will glow brightly at an antinode, less
brightly between an antinode and a node, not at all at a node.
By the aid of the tube the bridge can be adjusted to each
node. If the wire be long enough to permit of more than
one node, twice the distance between two adjacent nodes
will equal the wave length.
Since the wire is in resonance with the exciter, the period
of oscillation of the wire equals that of the exciter. The
ELECTRIC WAVES ON WIRES. 233
latter, and therefore the former, can be changed by changing
C. Hence
The electric waves, while directed by the wire, are really
waves of oscillation of electric force in the medium between
the two branches of the wire. Such waves travel with a
velocity that is independent of the wave length, and, if
A! be the wave-length when the period is T lt ^ that when
the period is T 2 , v = ). l /T l =A 2 /T 2 . Hence
By determining the wave-length with air as the dielectric
in C and then with a liquid as dielectric, we can evidently
find the dielectric constant of the liquid.
In the practice of the method most trouble is likely to be
due to the spark-gap, ee'. It must be adjusted until the
spark occurs under the kerosene. To avoid danger to the
tube by accidental dropping, it may be attached loosely to
the wire by a loop of thread. Each node should be deter-
mined several times. The distance between the plates of C
should be varied four or five times with air as dielectric
and a curve drawn with d as abscissa and X as ordinate.
From this curve and the value of X for each liquid, the value
of the dielectric constant for that liquid is readily deduced.
For a complete proof of the formula works on electricity and
magnetism must be consulted (e. g., J. J. Thomson's Electricity and
Magnetism, 243). The following considerations suggest the
formula. When an alternating e. m. f., E, is applied to a circuit
containing capacity, self-inductance, and ohmic resistance in series,
The potentials across the self-inductance (including the resistance)
and the capacity, respectively, are
L = /-\/r 2 + a> 2 L 2 and E c = ~^
234 ELECTRICITY AND MAGNETISM.
Evidently the potentials across the parts of the system may be greater
than the total e. m. f. which is the resultant obtained by geometrical
(i. e.,' vector) addition of the parts. This constitutes resonance.
It is complete when
Substituting for w its value 2x/T, we get T = 2n'\/LC. This, then,
is the period of the applied e. m. f. when resonance results. It is
also, therefore, the period of the free natural vibrations of the system.
The approximate formula for the capacity of a plate condenser is
given in Exp. LXII, a more exact one in Kohlrausch, p. 379.
Questions.
1. Assuming the velocity of the waves to be that of light, calculate
the frequency of the oscillations for one value of ^.
2. From this and the approximate formula for C calculate L.
3. If a sufficient amount of liquid were available, how could its
dielectric constant be found by immersing the wire AD in it?
4. What would be observed if AD were contained in a vacuum
tube?
TABLES
236
TABLES.
TABLE I.
Logarithms of Numbers from i to 1000.
No.
o
i
2
3 4
567
8
9
10
oooo
0043
0086
0128
0170
0212
0253
0294
0334
0374
ii
0414
4 C 3
0492
Q53 1
0569
0607
0645
0682
0719
0755
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1 106
13
"39
"73
I2O6
1239
1271
J 33
J 335
1367
*399
143
14
1461
1492
1523
*553
1584
1614
1644
1673
!73
1732
IS
1761
1790
1818
1847
.1875
1903
I93 1
1959
1987
2014
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
17
2304
233
2355
2380
2405
2430
2455
2480
2504
2529
18
2553
2577
2601
262 5
2648
2672
2695
2718
2742
2765
19
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
20
3010
3032
354
3075
3096
3118
3139
3160
3181
3201
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
22
3424
3444
3464
3483
3502
3522
3541
356o
3579
3598
23
3617
3636
3 6 55
3674
3692
37"
3729
3747
3766
3784
24
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
25
3979
3997
4014
4031
4048
4065
4082
4099
4116
4133
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
27
43*4
4330
4346
4362
4378
4393
4409
4425
4440
44 5 6
28
4472
4487
4502
4518
4533
4548
4564
4579
4594
4609
29
4624
4639
4654
4669
4683
4698
47 J 3
4728
4742
4757
30
477 1
4786
4800
4814
4829
4843
4857
4871
4886
4900
31
4914
4928
4942
4955
4969
4983
4997
5011
5024
5038
32
505 1
5065
5079
5092
5io5
5"9
5132
5M5
5159
5172
33
5i85
5198
5211
5224
5237
5250
5263
5276
5289
5302
34
5315
5328
5340
5353
5366
5378
539i
5403
54i6
5428
35
544i
5453
5465
5478
549
5502
55*5
5527
5539
555 1
36
55 6 3
5575
5587
5599
5611
5623
5635
5 6 47
5658
5670
37
5682
5694
575
5717
5729
574
5752
5763
5775
5786
38
5798
5809
5821
5832
5843
5855
5866
5877
5888
5899
39
59 11
5922
5933
5944
5955
5966
5977
5988
5999
6010
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
4i
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
43
6335
6345
6355
6365
6375
6385
6395
6405
6415
6425
44
6435
6444
6454
6464
6474
6484
6493
6503
6513
6522
45
6532
6542
6551
6561
657i
6580
6590
6599
6609
6618
46
6628
6637
6646
6656
6665
6675
6684
6693
6702
6712
47
6721
6730
6 739
6749
6758
6767
6776
6785
6794
6803
48
6812
6821
6830
6839
6848
6857
6866
6875
6884
6893
49
6902
691 1
6920
6928
6937
6946
6955
6964
6972
6981
50
6990
6998
7007
7016
7024
733
7042
7050
759
7067
51
7076
7084
793
7101
7110
7118 7126
7135
7 J 43
7152
52
7160
7168
7177
7185
7 J 93
7202
7210
7218
7226
7 2 35
53
7 2 43
7251
7259
7267
7275
7284
7292
7300
7308
73 J 6
54
7324
7332
7340
7348
7356
7364 ', 7372 7380
7388
7396
No.
I 5
TABLES.
237
TABLE I. Continued.
Logarithms of Numbers from i to 1000.
No.
1 1 * \ 3
4
5 | 6
7
8
9
55
7404
7412
7419
7427
7435
7443
745i
7459
7466 7474
56
7482
7490
7497
7505
75*3
7520
7528
7536
7543 755 1
57
7559
7566
7574
7582
7589
7597
7604
7612
7619 7627
58
7^34
7642
7649
7657
7664
7672
7679
7686
7694 7701
59
7709
7716
7723
7731
7738
7745
7752
7760
7767 7774
60
7782
7789
7796
7803
7810
7818
7825
7832
7839 ! 7846
61
7853
7860
7868
7875
7882
7889 7896
7903
7910 ! 7917
62
7924
793 1
7938
7945
7952
7959 7966
7973
7980 7987
63
7993
8000
8007
8014
8021
8028
8035
8041
8048 8055
64
8062
8069
8075
8082
8089
8096
8102
8109
8116 8122
65 8129
8136
8142
8149
8156
8162
8169
8176
8182 8189
66 8195 8202
8209
8215
8222
8228 8235 8241
8248 8254
67 8261
8267
8274
8280
8287
8293 8299
8306
8312 8319
68 8325
8331
8338
8344
8351
8357
8363
8370
8376 8382
69
8388
8395
8401
8407
8414
8420
8426
8432
8439 | 8445
70
8451
8457
8463
8470
8476
8482
8488
8494
8500 ! 8506
7i 8513
8519
8525
8531
8537
8543
8549
8555
8561 ; 8567
72 8573
8579
8585
859i
8597
8603
8609
8615
8621 8627
73 1 8633
8639
8645
8651
8657
8663
8669
8675
8681 ; 8686
74 8692
8698
8704
8710
8716
8722
8727
8733
8739 8745
75
875i
8756
8762
8768
8774
8779
8785
8791
8797
8802
76 j 8808
8814
8820
8825
8831
8837
8842
8848
8854
8859
77
8865
8871
8876
8882
8887
8893
8899
8904
8910
8915
78
8921
8927
8932
8938
8943
8949
8954
8960
8965
8971
79
8976
8982
8987
8993
8998
9004
9009
9oi5
9020
9025
80
9031
9036
9042
9047
9053
9058
9063 9069
9074
9079
81
9085
9090
9096
9101 9106
9112
9117 i 9122
9128
9*33
82
9138
9M3
9149
9154 9159
9165
9170 9175
9180
9186
f 3
9191
9196
9201
9206 9212
9217 9222 i 9227
9232
9238
84
9243
9248
9253
9258
9263
9269
9274
9279
9284
9289
85
9294
9299
934
9309
9315
9320
9325
9330
9335
9340
86
9345
935
9355
9360
93 6 5
9370
9375 I 938o
9385
9390
87
9395
9400
9405
9410
9415
9420
9425 i 943 9435
9440
88
9445
9450
9455
9460
9465
9469 i 9474
9479 9484
9489
89
9494
9499
9504
9509
95 J 3
9518 ! 9523
9528 9533
9538
90
9542
9547
9552
9557
9562
9566
957i
9576 9581
9586
9i
959
9595
9600
9605 i 9609
9614
9619
9624 9628
9633
92
9638
9643 ! 9647
9652 9657
9661 9666
9671 1 9675
9680
93
9685
9689 i 9694
9699 9703
9708 ! 9713
9717
9722 9727
94
973i
9736
974i
9745
975
9754
9759
9763
9768
9773
95
9777
9782
9786
9791
9795
9800
9805
9809
9814
9818
96
9823
9827
9832
9836
9841
9845
9850
9854
9859
9863
97
9868
9872
9877
9881
9886
9890
9894
9899
9903
9908
98
9912
9917
9921
9926
9930
9934
9939
9943
9948
9952
99
9956
9961
9965
9969
9974
997 s
9983
9987
9991
9996
No.
o
z
2
3
4 I 5
6
7
8
9
238
TABLES.
TABLE II.
Natural Sines and Cosines.
Sine
Cosine
4,
o.oooo
90
I.OOOO
I
0.0175
175
89
0.9998
02
2
0.0349
1 74
88
0.9994
04
3
0.0523
174
87
0.9986
08
4
0.0698
175
86
0.9976
10
5
0.0872
174
85
0.9962
14
6
0.1045
84
0.9945
7
0.1219
174
83
0.9925
20
8
0.1392
173
82
0.9903
22
9
0.1564
172
81
0.9877
26
10
0.1736
172
80
0.9848
29
ii
0.1908
172
79
0.9816
32
12
0.2079
171
78
0.9781
35
13
0.2250
171
77
0.9744
37
14
0.2419
169
76
0.9703
15
0.2588
169
75
0.9659
44
16
0.2756
1 68
74
0.9613
46
J 7
0.2924
168
73
0.9563
50
18
0.3090
166
7 2
0.9511
52
19
0.3256
166
7 1
0.9455
56
20
0.3420
164
70
0-9397
58
21
0.3584
164
69
0.9336
61
22
0.3746
162
68
0.9272
64
2 3
0.3907
161
67
0.9205
67
24
0.4067
1 60
66
-9 I 35
70
25
0.4226
159
65
0.9063
72
26
0.4384
158
64
0.8988
75
27
0.4540
156
63
0.8910
78
28
0.4695
155
62
0.8829
81
29
0.4848
153
61
0.8746
83
30
0.5000
I 5 2
60
0.8660
86
3 1
0.5150
I 5
59
0.8572
88
32
0.5299
149
58
0.8480
92
33
0.5446
147
57
0.8387
93
34
0.5592
146
56
0.8290
97
35
0-5736
144
55
0.8192
98
36
0.5878
142
54
0.8090
102
37
0.6018
140
53
0.7986
104
38
0.6157
139
52
0.7880
106
39
0.6293
136
0.7771
109
40
0.6428
X 35
5
0.7660
in
41
9-6561
J 33
49
0-7547
XI 3
42
0.6691
130
48
Q-743 1
116
43
0.6820
129
47
0.7314
117
44
0.6947
127
46
0-7*93
121
45
0.7071
124
45 *
0.7071
122
Ctfsine
Sine
TABLES.
239
TABLE III.
For Reduction of Time of Oscillation to an Infinitely Small Arc.
4 4 64 4
If t = observed time and
T = true reduced time
T-t-kt.
a
fc
a
k
a
&
o.ooooo
7
o .00023
14
O.OOOQ3
i
ooo
8
030
J 5
107
2
002
9
39
16
122
3
004
10
048
J 7
138
4
008
1 1
058
18
154
5
012
12
069
*9
172
6
Oi;
!3
080
20
I9O
7
023
14
093
TABLE IV.
Reduction of Barometer Readings to o.
(The corrections below are in mm. and are to be subtracted. The
uncorrected height is in cm.)
Temp.
Brass Scale
Glass Scale
72
73
74
75
76
77
78
74
75
76
77
78
15
r -75
1.77 1.81 1.83
1.86
1.88
1.91
1.92
1.94
1.97
2.0O
2. 02
1 6
1.87
1.89 1.93
1.96
1.98
2.01
2.03
2.05
2.17
2.07
2. 2O
2.10
2.1 3
2.16
2 29
i7
1.98
2.01
2.05
2.08
2.IO
2.13
2.16
2.23
2.26
18
2.IO
2-13
2.17
2. 2O
2.23
2.26 2.29
2.30
2-43
2-33
2.36
2-39
2-43
2.56
"9
2.22
2.25
2.29
2.32
2-35
2.38 2.41
2.46
2-49
2.53
20
2 -33
2.37 2.41
2-44
2.47
2-5 1
2-54
2.56
-59
2.62
2.66
2.69
2 I
2-45
2.48 2.53
2.56
2.69
2.6O
2.63
2.76
2.67
2.68
2.81
2.72
2. 7 6
2.89
3.02
3.15
3.28
2.79
2.8 3
2.96
22
2-57
2.60 2.65
2.72
2.79
2.8 5
2.92
2 3
2.68
2.72
2-77
2.81
2.8 4
2 88
2.92
2.94
2.98
3- 11
3.23
3.06
3- J 9
3- J o
3-23
3-36
24
2.80
2.84 2.89
2.93
3.05
2.97
3-9
3.01
3-!3
3-05
3-J7
3.06
3- J 9
i
25
2.92
2.96
3.01
3-32
240
TABLES.
TABLE V.
Density and Volume of One Gram of Water at Different
Temperatures.
Temp.
Density
Vol. of i. gr.
Temp.
Density
Vol. of i . gr.
0.999878
I.OOOI22
J
21
0.998065
.001939
i
Q-999933
.000067
22
0.997849
.0021 56
2
0.999972
.000028
2 3
0.997623
.002383
3
0.999993
.000007
24
0.997386
.002621
4
I.OOOOOO
.000000
25
0.997140
.002868
5
0.999992
.000008
30
0-99577
.00425
6
0.999969
.000031
35
0.99417
.00586
7
Q-999933
.000067
40
0.99236
.00770
8
0.999882
.000118
45
0-99035
.00974
9
0.999819
.000181
50
0.98817
.01197
10
0-999739
.000261
55
0.98584
.01436
1 1
o 999650
.000350
60
0-98334
.01694
12
0.999544
.000456
65
0.98071
.01967
*3
0.999430
.000570
7
0.97789
.02261
U
0.999297
.000703
75
0-97493
.02570
i5
0.999154
.000847
80
0.97190
.02891
16
0.999004
.000997
85
0.96876
-03225
17
0.998839
.001162
90
0.96549
03574
18
0.998663
.001339
95
0.96208
.03941
19
0.998475
.001527
IOO
0.95856
04323
20
0.998272
.001731
TABLE VI.
Density of Gases (o, 76 cm.). 1
Hydrogen 00008987
Oxygen 0014290
Nitrogen .0012507
Air 0012928
Chlorine 003167
Carbon monoxide 0012504
Carbon dioxide 0019768
Ethane 001341
Ethylene 001252
Steam (at 100) 00060315
Largely from Guye, J. Ch, Phys., 1907, p. 203.
TABLES.
2AI
TABLE VII.
Density (o), Specific Heat (o), and Coefficient of Linear Expansion
Element
Density
Specific
Heat
Coef. of Lin. Exp.
Multiplied by io 6
\luminum
2 60
2 2
231
Antimony
6.62
O4Q
^ j. x
Bismuth
9.8
.OT, I
Cadmium
8.61
O ? ?
-2Q.7
Carbon, diamond
Carbon, graphite
Carbon gas carbon
3-52
2.25
i .00
.10
15
3 r.?8
7.8
5 A
Cobalt
8.8
. I 06
124
Copper
Copper sulphate (crys )
8.92
S-S8
.094
16.8
Gold
IQ.3
.032
M-4
Iron
7.8
. I I
I 2.1
Lead
n.^6
O2 O
202
Magnesium .
1.74.
Mercury
I "? . ^06
QT.T.T.
181 (cub. exp )
Nickel
.IO8
12.8
Phosphorus yellow .
1.83
.20
Phosphorus, red
2.IQ
I 7
Phosphorus, metallic.
2.^4
Platinum
214.
O 3 3
90
Potassium chloride
I 08
Silver
I O ^ 3
.0 ^6
I Q. 2
Sodium chloride
2.1 C
Sodium sulphate. . . .
2.6 t?
Tin
7.3
.0 ^6
22.3
Zinc
7-2
OQ4
20. 2
Zinc sulphate (anhy )
3 A Q
4V
16
242
TABLES.
TABLE VIII.
Density, Specific Heat, and Coefficient of Expansion of
Miscellaneous Substances (o).
Substance
Density
Specific
heat
Coef. of Lin. Exp.
(Xio 6 )
Castor oil
.060
Glass, green
Glass crown
2.6
2 . 7
.19
. 19
8.9
8.8
Glass crystal
2
. 18
7 . 7
Glass flint
3. I ^ 3 O
. 10
7 3
Hard rubber
Marble
Paraffin
I-I5
2-75
.89
7-7
11.7
Quartz, crystal II ....
Quartz, crystal - 1 -
Quartz fused
2-653
2-653
2 2O
.19
7.2
13.2
54
Alcohol (ethyl)
Benzol
.81
.800
54
.^8
i .048 1
I.I76 1
Carbon bisulphide. . . .
Chloroform
1.293
I *\3
.24
2 3
I.I4 1
i n 1
Ether (ethyl)
Glycerine
74
i 26
S3
. s8
i-Si 1
TABLE IX.
Average Value of Elastic Moduli.
Shear Modulus.
Coefficient of cubical expansion Xio 3 .
Young's Modulus.
Brass
V7Xio u
10. 4X10"
Iron
7.7X10"
I9-6XI0 11
Steel
8.2X10"
22 Xio 11
TABLES. 243
TABLE X.
Surface Tension T (15), Temperature Coefficient of Surface
Tension c', and Angle of Contact a.
T
c'
Ethyl ether
10
. i j
1 6
Ethyl alcohol
2 ^
087
Benzol
31
13
Water
7 6
'5
small
Mercury
527
--38
i35
TABLE XL
Coefficient of Viscosity (20). *
Water oioo
Mercury o 1 59
Acetic acid 0122
Methyl alcohol 00591
Ethyl alcohol .0119
Ethyl ether 00234
Benzol 00649
1 "\Vinkelmann, 1908, I, 2, p. 1397.
244
TABLES.
TABLE XII.
Specific Heats of Gases.'
Temp.
s v
Argon... 20 .1205
Helium 20 1.25
Mercury 27 5~3 56 .0246
Hydrogen o-2oo 3.406
Nitrogen -3o-20o .244
Oxygen o-2oo .217
Air o-2oo .2375
Chlorine i9-343 - 1 J 5
Iodine 2oo--377 -0336
Bromine 85-228 .0555
Water i 3 o -25o .480
Hydrogen sulphide... io-2oo .245
Carbon dioxide i 1 00 .217
Ammonia j 2o-2io .512
Chloroform ! 28-n8 .144
Ethyl alcohol i io-22o .453
Ether 7o-225 .480
Benzol n6-2i8 .375
1.66
1.64
1.66
1.396
1.405
1.40
1-405
1.32
1.29
1.29
1.287
1.28
1.28
1.14
1.07
1.187
1 Jiiptner, Phys. Chem. I, pp. 71-73-
TABLES. 245
TABLE XIII.
Pressure of Saturated Water Vapor (Regnault).
(mm.)
Temp.
Pressure Temp.
Pressure
|
Ice Water 29
29.782
30
3I-548
10
1.999 2.078 31
33-405
3 2
35 359
8
2-379 2.456 33
37-4io
34
39 565
6
2.821 2.890 35
41.827
40
54.906
4
3-334 3-387 45
7J-39 1
50
91.982
2
3-925 3-955 55
117.479
60
148.791
65
186.945
O
4.600
70
233-093
+ I
4.940
75
288.517
2
5-302
80
354-643
3
5.687
85
433-41
4
6.097
90
525-45
5
6-534
9 1
545-78
6
6.998
92
566.76
7
7.492
93
588.41
8
8.017
94
610.74
9
8-574
95
633-78
10
9.165
96
657-54
ii
9.792
97
682.03
12
10.457
98
707.26
13
11.062
98-5
720.15
14
11.906
99-o
733-91
15
12.699
99-5
746.50
16
!3- 6 35
IOO.O
760.00
17
14.421
100.5
773-7 1
18
15-357
IOI.O
787-63
19
16.346
IO2.O
816.17
20
i7-39i
IO4.O
875.69
21
18.495
105
906.41
22
19.659
no
1075.4
23
20.888
I2O
i49!-3
24
22.184
I 3
2030.3
25
23-550
X 5
358i-2
26
24.998
1 7S
6717
27
26.505
200
1 1690
28
28.101 225 19097
246
TABLES.
TABLE XIV.
Boiling Point of Water, t, at Barometric Pressure, p,(ui m.) .
p>
t.
P>
t.
P-
t.
740
99.26
750
99-63
760
100.00
41
.29
5i
.67
61
.04
42
33
52
.70
62
.07
43
37
53
74
63
.1 1
44
.41
54
.78
64
15
45
.44
55
.82
65
.18
46
.48
56
85
66
.22
47
52
57
.89
67
.26
48
56
58
93
68
.29
49
59
59
.96
69
33
75
99-63
760
100.00
770
100.36
TABLES.
247
TABLE XV.
Wet and Dry Bulb Hygrometer.
(Actual vapor pressures (mm.) for different temperatures of dry
thermometer and various differences of temperature between the two
thermometers.
The first vertical column gives the temperature of the dry-bulb
thermometer. The first horizontal line gives the difference between
the two thermometers. Since the difference is zero if the air is satu-
rated, the second vertical column gives the saturated vapor pressure
for the corresponding temperatures in the first column.)
tc.
o
I
2
3
4
5
6
7
8
9
10
I I
o
4.6
3-7
2-9
2.1
J -3
i
4-9
4.0
3-2
2.4
1.6
0.8
2
5-3
4-4
3-4
2.7
1.9
I.O
3
5-7
4-7
3-7
2.8
2.2
1 '3
4
6.1
5- 1
4.1
3-2
2-4
1.6
0.8
5
6-5
5-5
4-5
3-5
2.6
1.8
I.O
6
7.0
5-9
4-9
3-9
2. 9
2.0
i.i
7
7-5
6.4
5-3
4-3
3-3
2-3
1.4
0.4
8
8.0
6.9
5-8
4-7
3-7
2.7
*-7
0.8
9
8.6
7-4
6-3
S- 2
4.1
3- 1
2.1
i.i
O.2
10
9.2
8.0
6.8
5-7
4.6
3-5
2-5
i-5
o-5
ii
9.8
8.6
7-4
6.2
5- 1
4.0
2.9
1.9
0.9
12
10.5
9.2
8.0
6.8
5-6
4-5
3-4
2-3
i-3
*3
II. 2
9.8
8.6
7-3
6.2
5-o
3-9
2.8
i-7
14
II.9
10.6
9.2
8.0
6.7
56
4-4
3-3
2.2
i.i
15
12.7
n-3
9-9
8.6
7-4
6.1
S-o
3-8
2.7
1.6
o-5
16
J 3-5
12. 1
10.7
9-3
8.0
6.8
5-5
4-3
3-2
2.1
I.O
I 7
14.4
13.0
n-5
IO.I
8-7
7-4
6.2
4-9
3-7
2.6
!-5
0.4
18
i5-4
I 3 .8
12.3
10.9
9-5
8.1
6.8
5-5
4-3
3- 1
2.O
0.9
19
16.4
14.7
13-2
11.7
10.3
8.9
7-5
6.2
4-9
3-7
2-5
1.4
20
17.4
15-7
14-1
12.6
ii. i
9-7
8-3
6.9
5-6
4-3
3-i
1.9
21
18.5
16.8
i5-i
13-5
12.
10.5
9.0
7-6 6.3
5-
3-7
2-5
22
19.7
17.9
16.2
14.5
I2. 9
11.4
9-9
8. 4 | 7.0
5-7
4-4
3-i
2 3
20.9
19.0
J 7-3
15-6
13-9
12.3
10.8
9-2
7-8
6.4
5- 1
3-8
24
22.2
20.3
18.4
16.6
14.9
13-3
11.7
IO.I
8-7
7-2
5-8
4-5
25
23.6
21.6
19.7
17.8
16.0
14-3
12.7.
II. I
9-5
8.0
6.6
5-2
26
25.0
22.9
21.
19.0
17.2
i5-4
J 3-7
12. 1
10.5
8.9
7-4
6.0
27
26.5
24.9
22.3
20.3
18.4
16.6
14.8
I3- 1
11.4
9.8
8-3
6.8
28
28.1
25-9
23-7
21.7
19.7
17.6
16.0
14.2
12.5
10.8
9-2 : 7-7
29
29.8
27-5
25-3
23.1
21. 1
19.1
17.2
15-3
13.6 11.9
IO.2
8.6
3
3 1.6
29.2
26.9 24.6
22.5 20-5
18.5
16.6
14.7
13.0
I 1.2
9.6
248
TABLES.
TABLE XVI.
Vapor Pressure of Mercury (mm.).
Temp.
Pres.
Temp.
Pres.
o
O.O2
170
8.091
+ 20
O.O4
1 80
I I.OOO
40
O.o8
190
14.84
60
0.16
200
19.90
80
o-35
210
26.35
IOO
0.746
22O
34 7
no
1.073
230
45-35
I2O
*-534
240
58.82
I 3
2-175
250
75-75
140
3-059
260
96.73
ISO
4.266
270
123.01
I 60
5.900
280
I55-I7
TABLE XVII.
Melting Point of Metals. (Holborn and Day and
Waidner and Burgess. 1 )
Tin . . 232
Cadmium 321
Lead
Zinc
Antimony
Aluminum
Silver
Gold
Copper ....
327
419
63 1
657
961
1063
1084
Platinum 1 770
Rev., 1909, p. 467; Compt. Rend., 1909, cxlviii, p.
1177.
TABLES.
249
TABLE XVIII.
Wave Lengths in Angstrom Units (io- 8 cm.).
Line
Element
Wave Length
Color
C H a . . ..
Hydrogen.
6 ^63 o ?4
Red
Dr
Sodium
c8o6 i ? ^
Yellow
D,
Sodium
5890 182
Yellow
F H^
Hydrogen
4861. <C27
Blue
G' H v
Hydrogen
4. -i 4.0. 6^4.
Violet
H T " .
Calcium
^068 62 s
Violet
Helium
7O6 ^.2
Red
Helium
6678.1
Red
Helium
c8? ?.6
Yellow
Helium
t;oi <.7
Green
Helium
402 i.o
Blue
Helium
471 ^.2
Blue
Helium
4-47 I. ^
Violet
Mercury
623 2 O
Red
Mercury
C7QO.7
Yellow
Mercury
^760-6
Yellow
Mercury
^460.7
Green
Mercury
40 ^0.7
Green-Blue
Mercury
4916.4
Blue
Mercury
4358.3
Blue
Mercury
4078.1
Violet
Mercury
4046.8
Violet
K~
Potassium
7699.3
Red
Ko
Potassium
=:8^2.2
Yellow
^p
fr ;
La
Li R
Potassium
Lithium
Lithium
4047.4
6708.2
6103.8
Violet
Red
Orange
Cadmium
6438.5
Red
Cadmium
5085.8
Green
Cadmium
4799.9
Blue
2sO
TABLES.
TABLE XIX.
Refractive Indices.
[Yellow light, (D lines) 20.]
Glass, light crown (density = 2 . 50)
Glass, heavy crown (density = 3 .00)
Glass, light flint (density ==2 .87)
Glass, heavy flint (density =4.22)
Quartz, crystal, -*- , ord
Quartz, crystal, - 1 - , ext
Alcohol, ethyl
Benzol,
Carbon bisulphide
Chloroform
Ether, ethyl
Glycerine
Water
Air, o, 7 6 cm i
.5280
.5604
.5410
7102
5442
5533
.3614
.5014
.6277
.4490
356o
.4729
33 2 9
.000293
TABLE XX.
Specific Rotatory Power (20). Yellow Sodium Light (D).*
Active Substance
Concentration
(=c) (gr.iniooc.c.;
Cane-sugar R
/ 3-28
66.639 .02o8<7
Invert sugar L
\ 10-86
I 14.
66.453 .000124(7
2O 07 O4l
Glucose (dextrose). R
(crystallized)
Fructose (levulose) L . .
Milk-sugar, R
Tartaric acid R ...
0-100%
0-40
f ".'
47-73 +.oisX%
100.3 -f.ioSc
52-53
15.06 .131(7
Quartz R or L
1 22-63
i3.436-.ii9C
21 70 (for i mm thickness)
Landolt and Bornstein.
TABLES.
TABLE XXI.
Photometric Table.
251
-. . x 1 ui at,
^<-xj-pcn \> s. n
VJbVJlllClilll/ U<
( 3 oo-w) 2 '
n
I
2
3
A
50
0.0400
O.42O
0.0440
0.0460
0.0482
60
.0625
-6 5 I
.0678
.0706
735
70
.0926
.0961
.0997
1034
. 1072
80
.1322
.1368
.1414
.1643
.1512
90
1837
.1896
!957
.2018
. 2082
IOO
25OO
.2576
2653
2734
.2815
I IO
335 2
3449
3549
3652
3756
I2O
4445
457
.4698
.4829
.4964
130
.5848
. 6009
6i73
6343
.6516
I4O
.7656
.7864
.8078
.8296
.8521
J 5
i .0000
i .027
1-055
1.083
1.113
1 60
1.306
1.342
1-379
i .416
!-454
170
i . 700
i-757
i. 806
1.856
1.907
1 80
2.250
2-313
2-379
2.446
2.516
190
2.983
3.070
3 .160
3-253
3-35
200
4.000
4.122
4-285
4-380
4.516
210
5-444
5.621
5-803
5-994
6.192
220
7-563
7.826
8.100
8.387
8.687
230
10.80
I I .21
11.64
12.09
I2 -57
240
16.00
16.68
17.41
18.17
18.98
250
25.00
n
5
6
7
8
9
5
0.0504
0.0527
0.0550
0.0574
0.0599
60
.0765
.0796
.0827
.0859
.0892
70
.mi
.1151
. 1 193
1235
.1278
80
i5 6 3
.1615
.1668
!7 2 3
.1779
90
.2148
.2215
.2283
2354
.2428
IOO
.2899
.2985
3074
.3164
3256
no
.3864
3974
.4088
.4204
4323
120
.5012
5244
5389
5538
.5691
I 3
.6694
.6877
.7064
7257
7454
140
.8752
.8988
.9231
.9481
9737
I 5
1-143
1.174
i .205
1-238
1.272
1 60
1.494
i-535
i-577
i .620
i .664
170
i .960
2 .OI4
2 .071
2.129
2.188
180
2.588
2.662
2-739
2.817
2.889
190
3-449
3-552
3-658
3.768
3.882
200
4.656
4.803
4-954
5.111
5-275
2IO
6.398
6.122
6-835
7.068
7.310
22O
9.000
9-327
9.670
10.03
10.40
2 3
13.07
13.60
14-15
14.74
15-35
24O
19.84
20.75
21.72
22.75
23.84
2 5
2 5 2
TABLES.
TABLE XXII.
Specific Resistance at o C. and Temperature Coefficient.
Specific
Resistance
Temperature
Coefficient
Bismuth
Copper (
Copper (
German
Iron. . .
(hard)
132.6X10-
i .590X10-
1.622X10-
20.24X10-
10.43X10-
19.85X10-
94.07 X io-
8.957X10-
1.521X10-
1.652X10-
9.565X10-
0.0054
0.0043
annealed)
hard drawn)
silver (4Cu + 2 Ni + i Zn)
0.00027
0.007
0.0039
0.00089
0.0034
0.00377
Lead (pi
Mercury
Platinun
Silver (a
Silver (h
Tin
ressed)
i
nnealed)
ard drawn)
0.004
TABLE XXIII.
Specific Resistance and Temperature Coefficient of Solutions (18).*
Sp. Res.
! Temp.
Coef.
Sp. Res.
Temp.
Coef.
wHCl
3-3 2
i o 0165
nNaCl
I -7 A f
O O2 2 6
o.inHCl
o.oinHCl
28.5
271.
o.mNaCl....
o.oiwNaCl . .
108.1
974.
wHNO 3 .
o.iwHN0 3 ...
o.oiwHNO 3 . .
n iH 2 SO 4
3-23
28.6
272.
c o ^
0.163
o . o 1 64 1
wKCl ..
o.inKCl
o.oiwKCl . . .
nAgNO 3
10.18
89.5
817.
14. . 7 c
0.0217
o . 02 1 6
o.iniH 2 SO 4 ..
o.oiw$H 2 SO 4 .
wC 2 H 4 O 2
44-4
325-
758
1
o. in AgNO 3 .. .
o.oiwAgNO 3 .
w^Pb(NO 3 ) 2
105-7
922 .
21. 8
o.mC 2 H 4 O 2 ..
o.oiwC 2 H 4 O 2 .
2170.
6990.
o.miPb(N0 3 ) 2
o.omiPb(NO 3 ) 2
129.4
967.
nNaOH .
o.iwNaOH ..
o.oiwNaOH.
6-25
54-7
500.
0.019
w^ZnSO 4
o.iZnSO 4 ..
o.omiZnSO 4
37-6
217.
1362.
0.02 5
wNH 4 OH ..
o.mNH 4 OH .
o.oiNH 4 OH
1125.
33-
10420.
niCuSO 4 .
o.iw JCuSO 4 ..
o.oiw ^CuSO 4 .
38.8
223.
1385-
0.0225
* A normal solution (designated by the subscript w), contains in one liter a
number of grams equal to the chemical equivalent (atomic or molecular weight
divided by the valency). A solution with the subscript o. in has one-tenth this
concentration, etc. For exmaple, o. iwHCl has 3 .65 grs. of HC1 (gas) in one
trile of solution, or that proportion.
TABLES. 253
TABLE XXIV.
Dielectric Constants.
I
II
Hydrocyanic acid
Water . .
.... 96
80
Ether . .
4(T
Xylol
Benzol
2.26
2 2
Methyl alcohol
Ethyl alcohol
Ammonia (liquid)
Acetone
Sulphur dioxide
Pyridene
:::: %
.... 22
.... 17
14
.... 12
Toluol
2 2
Petroleum
2.O7
INDEX,
Aberration, 138
Absorption, electric, 196
Acceleration of gravity, 36
Air, density of, 33
thermometer, 76
Alloys, melting-point, 109
Alternating current measurements,
225, 227
Ammeter, calibration of, 198
Anderson's method (self induction),
204
Angle of prism, 131
Angular field of view, 143
Apparent expansion of gas, 76
of liquid, 74
Arc of vibration, correction, 239
Balance, 21-25
correction for air buoyancy, 24
method of oscillations, 22-24
ratio of arms, 24
Ballistic galvanometer, 156, 208
Barometer, 21
table of corrections, 239
Battery, electromotive force, 191, 193
resistance, 185, 195
Beckman thermometer, 65
Biquartz, 151
Bismuth spiral, 207
Boiling-point of water (table), 246
Bridge, Wheatstone's, 153
Bunsen photometer, 126
Cadmium cell, 160
Calibrating coil, 209, 212
Calibration of ammeter, 198
of galvanometer, 179
of resistances, 183
of scale, 26
of thermometer, 67
of voltmeter, 196
Callender's equation (platinum ther-
mometer), 117
Calorimeter, for gases, 113
for liquids, 114
for solids, no, in
simple, 89
Candle-power, measurement of, 1 26
Capacity, absolute measurement, 203
divided charge method, 178
measurement (alternating cur-
rents), 225, 227
Capacities, comparison of, 200, 228,
230
different types, 194, 229
Carey Foster bridge, 183
Cathetometer, 19
Chemical hygrometer, 84
Chromatic aberration, 138
Clark cell, 160
Clement and Desermes' method
(specific heat of gases), 91
Coefficient of apparent expension, 74,
76
of expansion, 71, 241, 242
of friction, 41-45
of increase of pressure, 76
of mutual induction, 206
of self induction, 203
of viscosity, 56, 243
Coincidence method, 38
Commutator, double, 161
Comparator, 15
Condenser, see capacity.
Conductivity, thermal, 102
of electrolyte, 188
Copper voltameter, 220
Curves, plotting of, 1 1
Daniell cell, 159
Demagnetization of iron, 213
Density, of gases, 33, 240
of liquids, 29
of powders, 32
of solids, 28, 241, 242
of water, 240
255
256
INDEX.
Dew-point, 83
Dielectric constant, 228, 230
(table), 253
Diffraction grating, 147
Dip circle, 167
Dividing engine, ry
Dolezalek electrometer, 176
Double bridge, 178
Double commutator, 161
Drude's apparatus (electric waves),
230
Earth inductor, 169
Elastic constants, 242
Electric absorption, 196
Electrical resonance, 232
units, 162
waves, 230
Electrolytes, resistance of, 188
Electrometer, quadrant, 176
Electromotive force, device for small,
161
measurement of, 191, 193
of various cells, 160
Equivalent, chemical, 31
Errors, 2-10
of weights, 27
Expansion, apparent, 74, 76
coefficient of, 71, 241, 242
Eutectic alloy, 109
Focal length of lenses, 137, 140
of mirrors, 134
Frequency of tuning fork, 44, 1 20
Friction, coefficient of kinetic, 42
coefficient of static, 41
"G," determination of, 36
Galvanometer, bringing to rest, 156
calibration of, 179, 208
damping, 157
different types, 155
resistance of, 171, 173
shunt, 157
study of ballistic, 158, 208
tangent, 222
Gas, coefficient of increase of pressure,
76
density of, 240
Grating, diffraction, 147
Heat, conductivity for, 102
Heat value of gas, 113
of liquid, 114
of solid, no
Hempel calorimeter, no
Hooke's law, 46
Horizontal component of earth's field,
163, 219
Hygrometry, 83
Hypsometer, 70
Hysteresis, 214
Incandescent lamp, study of, 128
Inclination, magnetic, 167
Index of refraction, measurement of,
132
table of, 250
Inertia, measurement of moment of, 5 2
Insulation resistance, 176
Interferometer, 149
Iron, permeability of, 210
Junker calorimeter, 113
Kundt's method (velocity of sound)'
122
Latent heat of fusion, 94
of vaporization, 96, 99
Lenses, combinations, 140
focal length, 137
rule of signs, 125
Light, filters, 124
monochromatic, 124
Logarithmic decrement, 157
tables, 236
Low resistance, measurement of, 178-
183
Lummer-Brodhun photometer, 126,
127
Magnetic field, measurement of, 207
of earth, dip, 167
of earth, horizontal component,
163, 219
hysteresis, 214
permeability, 210
Magnetometer, 165
Magnification, 141
Magnifying power of telescope, 142
Mance's method (battery resistance),
185
Mechanical equivalent of heat, elec-
trical method, 219
by friction, 105
Melting-point of alloy, 109
of metals (table), 248
INDEX.
257
Mercury, vapor pressure of (table),
248
Michelson's interferometer, 149
Micrometer caliper, 14
microscope, 15
Minimum deviation, 132
Mirror and scale, adjustment of, 25
Mirrors, spherical, measurement of
focal length, 134
rule of signs, 125
Moduli, law of, 31
Mohr-Westphal balance, 29
Moment of inertia, 52
Monochromatic light, 124
Mutual induction, 206, 227
Optical lever, 48, 73
pyrometer, 116
Passages, method of, 54
Pendulum, correction for arc (table),
239
physical, 37
simple, 36
Permeability, 210
Photometric table, 251
Photometry, 126
Pirani's method (mutual induction),
200
Pitch of tuning fork, 44, 120
Planimeter, 218
Platinum thermometer, 116
Pohl commutator, 202, 216
Polarization, rotation of plane of, 150
Possible error, 4-9
Post-office box bridge, 154
Potentiometer, 197
Pressure, coefficient of increase of, 76
of mercury vapor, 248
of water vapor (measurement), 80
(table), 245
Primus burner, 114
Prism, angle of, 131
minimum deviation, 132
Probable error, 9
Pyknometer, 33
Pyrometry, 115
Quadrant electrometer, 176
Radiation correction, 63
pyrometer, 116
Radius of curvature of mirror, 134
Ratio of specific heats, measurement
of, 91, 122, 123
table, 244
Refractive index, measurement of, 132
of lenses, 136
table, 250
Regnault's apparatus, hygrometry, 83
vapor pressure, 81
Reports, 2
Resistance, boxes, 153
electrolytic, 188
high, 175-178
low, 178-183
measurement of, 169
of ballistic galvanometer, 208
of battery, 185, 195
of galvanometer, 171, 173
temperature coefficient of, 186
Resistances, comparison of, 183
Resolving power, of eye, 146
of telescope, 145
Rigidity of metals, 51
Rosenhain calorimeter, in
Rotation of plane of polarization,
measurement of, 150
table, 250
Rubber grease, 34
Saccharimetry, 150
Scale, calibration of, 26
construction of, 26
Self induction, alternating current
method, 225, 227
Anderson's method, 204
inductions, comparison of, 205
Shunts, galvanometer, 157, 171
Shear modulus, 51
Signs (mirrors and lenses), 125
Slide wire bridge, 154
Sound velocity of, 119, 122
Specific gravity bottle, 33
heat, of gases, 91
(table), 244
of metals, 85
(table), 241
of miscellaneous substances
(table), 241, 242
inductive capacity, see dielectric
constant,
resistance of electrolytes, 188, 252
of metals, 171, 252
rotatory power (table), 250
Spectrometer, 130
Spherical aberration, 138
Spherometer, 16
258
INDEX.
Standard cells, 159.
Stroboscopic disk, 44
Surface tension, measurement of, 60
table, 243
Tangent galvanometer, 222
Telescope, adjustment of, 25
magnifying power of, 142
resolving power of, 145
Temperature coefficient of expansion,
7*> 74
of expansion (tables), 240-242
of resistance, 186
(table), 252
Thermal conductivity, 102
Thermocouple, 91, 116, 223
Thermometer, air, 76
Beckman, 65
calibration of, 67-7 1
fixed points, 69
platinum, 116
Thomson's double bridge, 181
method (galvanometer resistance),
method of mixtures, 202
Time of vibration, method of coin-
cidences, 38
method of passages, 54
reduction to infinitely small arc,
239
signals, 25
Torsion, modulus of, 51
Trigonometrical functions (table), 238
Tuning fork, pitch of, 44, 1 20
Units, electrical, 162
Vacuum, reduction of weighing to, 24
Valson's law of moduli, 31
Vapor pressure of mercury (table), 248
of water (measurement), 80
(table), 245
Velocity of sound, Kundt's method,
122
resonance method, 119
Vernier, 13
caliper, 14
Virtual image, 136, 139
Viscosity, measurement of coefficient
of, 56
table, 243
Voltameter, copper, 220
Voltmeter, calibration, 196
Volumenometer, 32
Water, boiling-point (table), 246
density (table), 240
equivalent, 89
vapor pressure, 80, 245
Wave length, of electric waves, 230
of light waves (measurement), 147
(table), 249
of sound waves, 119, 122
Weighing, by oscillations, 22
double, 24
reduction to vacuum, 24
Weight thermometer, 76
Weights, calibration of, 27
Weston (cadmium) cell, 160
Wet and dry bulb hygrometer, 84, 247
Wheatstone's bridge, 153, 169
Young's modulus, by bending, 47
by stretching, 46
table, 242
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
RENEWED BOOKS ARE SUBJECT TO IMMEDIATE
RECALL
LIBRARY, UNIVERSITY OF CALIFORNIA, DAVIS
Book Slip-Series 458
167U18
Duff, A.W.
Physical measure-
ments .
Call Number:
QC37
D8
1910
QC37
167418