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757.]

SELF-INDUCTION.

357

a third method. Beginning with an arrangement in which the transient current due to self-induction is slightly in excess of that due to mutual induction, we may get rid of the inequality by inserting a conductor whose resistance is W between A and Z. The condition of no permanent current through the galvanometer is not affected by the introduction of W. We may therefore get rid of the transient current by adjusting the resistance of Walone. When this is done the value of L is

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Comparison of the Coefficients of Self-induction of Two Coils.

757.] Insert the coils in two adjacent branches of Wheatstone's Bridge. Let L and N be the coefficients of self-induction of the coils inserted in P and in R respectively, then the condition of no galvanometer current is

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Hence, by a proper adjustment of the resistances, both the permanent and the transient current can be got rid of, and then the ratio of L to N can be determined by a comparison of the resistances.

CHAPTER XVIII.

ELECTROMAGNETIC UNIT OF RESISTANCE.

On the Determination of the Resistance of a Coil in Electro-
magnetic Measure.

758.] THE resistance of a conductor is defined as the ratio of the numerical value of the electromotive force to that of the current which it produces in the conductor. The determination of the value of the current in electromagnetic measure can be made by means of a standard galvanometer, when we know the value of the earth's magnetic force. The determination of the value of the electromotive force is more difficult, as the only case in which we can directly calculate its value is when it arises from the relative motion of the circuit with respect to a known magnetic system.

759.] The first determination of the resistance of a wire in electromagnetic measure was made by Kirchhoff*. He employed two coils of known form, 4, and A2, and calculated their coefficient of mutual induction from the geometrical data of their form and position. These coils were placed in circuit with a galvanometer, G, and a battery, B, and two points of the circuit, P, between the coils, and Q, between the battery and galvanometer, were joined by the

A,

P

A2

Fig. 63.

R

B

wire whose resistance, R, was to be measured.

When the current is steady it is divided between the wire and the galvanometer circuit, and produces a certain permanent deflexion of the galvanometer. If the coil 4, is now removed quickly

Bestimmung der Constanten von welcher die Intensität inducirter elektrischer Ströme abhängt.' Pogg. Ann., lxxvi (April 1849).

759.]

KIRCHHOFF'S METHOD.

359

from A, and placed in a position in which the coefficient of mutual induction between 4 and 4, is zero (Art. 538), a current of induction is produced in both circuits, and the galvanometer needle receives an impulse which produces a certain transient deflexion.

The resistance of the wire, R, is deduced from a comparison between the permanent deflexion, due to the steady current, and the transient deflexion, due to the current of induction.

Let the resistance of QGA, P be K, of PA, BQ, B, and of PQ, R. Let L, M and N be the coefficients of induction of Д, and A. Let be the current in G, and that in B, then the current from P to Qis -ỳ.

Let E be the electromotive force of the battery, then

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d

= E.

dt

Rx+(B+R) ÿ+ (Mx+Ny)

When the currents are constant, and everything at rest,

(K+R) –Rj= 0.

(1)

(2)

(3)

If M now suddenly becomes zero on account of the separation of 1⁄4 from Д, then, integrating with respect to t,

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(7)

(8)

(B+R) (K+R) — R2

Substituting the value of y in terms of a from (3), we find

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When, as in Kirchhoff's experiment, both B and K are large compared with R, this equation is reduced to

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Of these quantities, x is found from the throw of the galvanometer due to the induction current. See Art. 768. The permanent current,, is found from the permanent deflexion due to the steady current; see Art. 746. M is found either by direct calculation from the geometrical data, or by a comparison with a pair of coils, for which this calculation has been made; see Art. 755. From

these three quantities R can be determined in electromagnetic mea

sure.

These methods involve the determination of the period of vibration of the galvanometer magnet, and of the logarithmic decrement of its oscillations.

Weber's Method by Transient Currents*.

760.] A coil of considerable size is mounted on an axle, so as to be capable of revolving about a vertical diameter. The wire of this coil is connected with that of a tangent galvanometer so as to form a single circuit. Let the resistance of this circuit be R. Let the large coil be placed with its positive face perpendicular to the magnetic meridian, and let it be quickly turned round half a revolution. There will be an induced current due to the earth's magnetic force, and the total quantity of electricity in this current in electromagnetic measure will be

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where g1 is the magnetic moment of the coil for unit current, which in the case of a large coil may be determined directly, by measuring the dimensions of the coil, and calculating the sum of the areas of its windings. His the horizontal component of terrestrial magnetism, and R is the resistance of the circuit formed by the coil and galvanometer together. This current sets the magnet of the galvanometer in motion.

If the magnet is originally at rest, and if the motion of the coil occupies but a small fraction of the time of a vibration of the magnet, then, if we neglect the resistance to the motion of the magnet, we have, by Art. 748,

Q

=

Η Τ
2 sin 0,

(2)

where G is the constant of the galvanometer, T is the time of vibration of the magnet, and 0 is the observed elongation. From these equations we obtain

1

R = Gg

Tsin 0

(3)

The value of H does not appear in this result, provided it is the same at the position of the coil and at that of the galvanometer. This should not be assumed to be the case, but should be tested by comparing the time of vibration of the same magnet, first at one of these places and then at the other.

* Elekt. Maasb.; or Pogg., Ann. lxxxii, 337 (1851).

762.]

WEBER'S METHOD.

361

761.] To make a series of observations Weber began with the coil parallel to the magnetic meridian. He then turned it with its positive face north, and observed the first elongation due to the negative current. He then observed the second elongation of the freely swinging magnet, and on the return of the magnet through the point of equilibrium he turned the coil with its positive face south. This caused the magnet to recoil to the positive side. The series was continued as in Art. 750, and the result corrected for resistance. In this way the value of the resistance of the combined circuit of the coil and galvanometer was ascertained.

In all such experiments it is necessary, in order to obtain sufficiently large deflexions, to make the wire of copper, a metal which, though it is the best conductor, has the disadvantage of altering considerably in resistance with alterations of temperature. It is also very difficult to ascertain the temperature of every part of the apparatus. Hence, in order to obtain a result of permanent value from such an experiment, the resistance of the experimental circuit should be compared with that of a carefully constructed resistancecoil, both before and after each experiment.

Weber's Method by observing the Decrement of the Oscillations
of a Magnet.

762.] A magnet of considerable magnetic moment is suspended at the centre of a galvanometer coil. The period of vibration and the logarithmic decrement of the oscillations is observed, first with the circuit of the galvanometer open, and then with the circuit closed, and the conductivity of the galvanometer coil is deduced from the effect which the currents induced in it by the motion of the magnet have in resisting that motion.

If T is the observed time of a single vibration, and λ the Napierian logarithmic decrement for each single vibration, then, if we write

and

π

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the equation of motion of the magnet is of the form

$ Ce-at cos (wt+B).

(1)

(2)

(3)

This expresses the nature of the motion as determined by observaWe must compare this with the dynamical equation of

tion. motion.

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