# A Treatise on Electricity and Magnetism/Part IV/Chapter XVI

CHAPTER XVI.

ELECTROMAGNETIC OBSERVATIONS.

730.] So many of the measurements of electrical quantities depend on observations of the motion of a vibrating body that we shall devote some attention to the nature of this motion, and the best methods of observing it.

The small oscillations of a body about a position of stable equilibrium are, in general, similar to those of a point acted on by a force varying directly as the distance from a fixed point. In the case of the vibrating bodies in our experiments there is also a resistance to the motion, depending on a variety of causes, such as the viscosity of the air, and that of the suspension fibre. In many electrical instruments there is another cause of resistance, namely, the reflex action of currents induced in conducting circuits placed near vibrating magnets. These currents are induced by the motion of the magnet, and their action on the magnet is, by the law of Lenz, invariably opposed to its motion. This is in many cases the principal part of the resistance.

A metallic circuit, called a Damper, is sometimes placed near a magnet for the express purpose of damping or deadening its vibrations. We shall therefore speak of this kind of resistance as Damping.

In the case of slow vibrations, such as can be easily observed, the whole resistance, from whatever causes it may arise, appears to be proportional to the velocity. It is only when the velocity is much greater than in the ordinary vibrations of electromagnetic instruments that we have evidence of a resistance proportional to the square of the velocity.

We have therefore to investigate the motion of a body subject to an attraction varying as the distance, and to a resistance varying as the velocity.

731.] The following application, by Professor Tait[1], of the principle of the Hodograph, enables us to investigate this kind of motion in a very simple manner by means of the equiangular spiral.

Let it be required to find the acceleration of a particle which describes a logarithmic or equiangular spiral with uniform angular velocity ${\displaystyle \omega }$ about the pole.

The property of this spiral is, that the tangent ${\displaystyle PT}$ makes with the radius vector ${\displaystyle PS}$ a constant angle ${\displaystyle \alpha }$.

If ${\displaystyle v}$ is the velocity at the point ${\displaystyle P}$, then

${\displaystyle v.\sin \alpha =\omega .SP}$.

Hence, if we draw ${\displaystyle SP^{\prime }}$ parallel to ${\displaystyle PT}$ and equal to ${\displaystyle SP}$, the velocity at ${\displaystyle P}$ will be given both in magnitude and direction by

${\displaystyle v={\frac {\omega }{\sin \alpha }}SP^{\prime }}$.

Fig. 58.

Hence ${\displaystyle P^{\prime }}$ will be a point in the hodograph. But ${\displaystyle SP^{\prime }}$ is ${\displaystyle SP}$ turned through a constant angle ${\displaystyle \pi -\alpha }$, so that the hodograph described by ${\displaystyle P^{\prime }}$ is the same as the original spiral turned about its pole through an angle ${\displaystyle \pi -\alpha }$.

The acceleration of ${\displaystyle P}$ is represented in magnitude and direction by the velocity of ${\displaystyle P^{\prime }}$ multiplied by the same factor, ${\displaystyle {\frac {\omega }{\sin \alpha }}}$.

Hence, if we perform on ${\displaystyle SP^{\prime }}$ the same operation of turning it through an angle ${\displaystyle \pi -\alpha }$ into the position ${\displaystyle SP^{\prime \prime }}$, the acceleration of ${\displaystyle P}$ will be equal in magnitude and direction to

${\displaystyle {\frac {\omega ^{2}}{\sin ^{2}\alpha }}SP^{\prime \prime }}$,

where ${\displaystyle SP^{\prime \prime }}$ is equal to ${\displaystyle SP}$ turned through an angle ${\displaystyle 2\pi -2\alpha }$. If we draw ${\displaystyle PF}$ equal and parallel to ${\displaystyle SP^{\prime \prime }}$, the acceleration will be ${\displaystyle {\frac {\omega ^{2}}{\sin ^{2}\alpha }}PF}$, which we may resolve into

${\displaystyle {\frac {\omega ^{2}}{\sin ^{2}\alpha }}PS}$ and ${\displaystyle {\frac {\omega ^{2}}{\sin ^{2}\alpha }}PK}$.

The first of these components is a central force towards ${\displaystyle S}$ proportional to the distance.

The second is in a direction opposite to the velocity, and since

${\displaystyle PK=2\cos \alpha P^{\prime }S=-2{\frac {\sin \alpha \cos \alpha }{\omega }}v}$,

this force may be written

${\displaystyle -2{\frac {\omega \cos \alpha }{\sin \alpha }}v}$.

The acceleration of the particle is therefore compounded of two parts, the first of which is an attractive force ${\displaystyle \mu r}$, directed towards ${\displaystyle S}$, and proportional to the distance, and the second is ${\displaystyle -2kv}$, a resistance to the motion proportional to the velocity, where

${\displaystyle \mu ={\frac {\omega ^{2}}{\sin ^{\alpha }}}}$, and ${\displaystyle k=\omega {\frac {\cos \alpha }{\sin \alpha }}}$.

If in these expressions we make ${\displaystyle \alpha ={\frac {\pi }{2}}}$, the orbit becomes a circle, and we have ${\displaystyle \mu _{0}=\omega _{0}^{2}}$, and ${\displaystyle k=0}$.

Hence, if the law of attraction remains the same, ${\displaystyle \mu =\mu _{0}}$, and

${\displaystyle \omega =\omega _{0}\sin \alpha }$

or the angular velocity in different spirals with the same law of attraction is proportional to the sine of the angle of the spiral.

732.] If we now consider the motion of a point which is the projection of the moving point ${\displaystyle P}$ on the horizontal line ${\displaystyle XY}$, we shall find that its distance from ${\displaystyle S}$ and its velocity are the horizontal components of those of ${\displaystyle P}$. Hence the acceleration of this point is also an attraction towards ${\displaystyle S}$, equal to ${\displaystyle \mu }$ times its distance from ${\displaystyle S}$, together with a retardation equal to ${\displaystyle k}$ times its velocity.

We have therefore a complete construction for the rectilinear motion of a point, subject to an attraction proportional to the distance from a fixed point, and to a resistance proportional to the velocity. The motion of such a point is simply the horizontal part of the motion of another point which moves with uniform angular velocity in a logarithmic spiral.

733.] The equation of the spiral is

${\displaystyle r=Ce^{-\phi \cot \alpha }}$,

To determine the horizontal motion, we put

${\displaystyle \phi =\omega t}$,${\displaystyle x=a+r\sin \phi }$,

where ${\displaystyle a}$ is the value of ${\displaystyle x}$ for the point of equilibrium.

If we draw ${\displaystyle BSD}$ making an angle ${\displaystyle \alpha }$ with the vertical, then the tangents ${\displaystyle BX}$, ${\displaystyle DY}$, ${\displaystyle GZ}$, &c. will be vertical, and ${\displaystyle X}$, ${\displaystyle Y}$, ${\displaystyle Z}$, &c. will be the extremities of successive oscillations.

734.] The observations which are made on vibrating bodies are—

(1) The scale-reading at the stationary points. These are called Elongations.
(2) The time of passing a definite division of the scale in the positive or negative direction.
(3) The scale-reading at certain definite times. Observations of this kind are not often made except in the case of vibrations of long period[2].

The quantities which we have to determine are—

(1) The scale-reading at the position of equilibrium.
(2) The logarithmic decrement of the vibrations.
(3) The time of vibration.

To determine the Reading at the Position of Equilibrium from Three Consecutive Elongations.

735.] Let ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, ${\displaystyle x_{3}}$ be the observed scale-readings, corresponding to the elongations ${\displaystyle X}$, ${\displaystyle Y}$, ${\displaystyle Z}$, and let ${\displaystyle a}$ be the reading at the position of equilibrium, ${\displaystyle S}$, and let ${\displaystyle r_{1}}$ be the value of ${\displaystyle SB}$,

 ${\displaystyle x_{1}-a={}}$ ${\displaystyle r_{1}\sin \alpha }$, ${\displaystyle x_{2}-a=-{}}$ ${\displaystyle r_{1}\sin \alpha e^{-\pi \cot \alpha }}$, ${\displaystyle x_{3}-a={}}$ ${\displaystyle r_{1}\sin \alpha e^{-2\pi \cot \alpha }}$.
From these values we find
 ${\displaystyle (x_{1}-a)(x_{3}-a)=(x_{2}-a)^{2}}$, whence ⁠ ${\displaystyle a={\frac {x_{1}x_{3}-x_{2}^{2}}{x_{1}+x_{3}-2x_{2}}}}$.
When ${\displaystyle x_{3}}$ does not differ much from ${\displaystyle x_{1}}$ we may use as an approximate formula

${\displaystyle a={\frac {1}{4}}(x_{1}+2x_{2}+x_{3})}$.

To determine the Logarithmic Decrement.

736.] The logarithm of the ratio of the amplitude of a vibration to that of the next following is called the Logarithmic Decrement. If we write ${\displaystyle \rho }$ for this ratio

${\displaystyle \rho ={\frac {x_{1}-x_{2}}{x_{3}-x_{2}}}}$, ${\displaystyle L=\log _{1}0\rho }$, ${\displaystyle \lambda =\log _{\epsilon }\rho }$.

${\displaystyle L}$ is called the common logarithmic decrement, and ${\displaystyle \lambda }$ the Napierian logarithmic decrement. It is manifest that

${\displaystyle \lambda =L\log _{\epsilon }10=\pi \cot \alpha }$.

Hence

${\displaystyle \alpha =\cot ^{-1}{\frac {\lambda }{\pi }}}$,

which determines the angle of the logarithmic spiral. In making a special determination of ${\displaystyle \lambda }$ we allow the body to perform a considerable number of vibrations. If ${\displaystyle c_{1}}$ is the amplitude of the first, and ${\displaystyle c_{n}}$ that of the ${\displaystyle n}$th vibration,

${\displaystyle \lambda ={\frac {1}{n-1}}\log _{\epsilon }\left({\frac {c_{1}}{c_{n}}}\right)}$.

If we suppose the accuracy of observation to be the same for small vibrations as for large ones, then, to obtain the best value of ${\displaystyle \lambda }$, we should allow the vibrations to subside till the ratio of ${\displaystyle c_{1}}$ to ${\displaystyle c_{n}}$ becomes most nearly equal to ${\displaystyle \epsilon }$, the base of the Napierian logarithms. This gives ${\displaystyle n}$ the nearest whole number to ${\displaystyle {\frac {1}{\lambda }}+1}$.

Since, however, in most cases time is valuable, it is best to take the second set of observations before the diminution of amplitude has proceeded so far.

737.] In certain cases we may have to determine the position of equilibrium from two consecutive elongations, the logarithmic decrement being known from a special experiment. We have then

${\displaystyle a={\frac {x_{1}+e^{\lambda }x_{2}}{1+e^{\lambda }}}}$.

Time of Vibration,

738.] Having determined the scale-reading of the point of equilibrium, a conspicuous mark is placed at that point of the scale, or as near it as possible, and the times of the passage of this mark are noted for several successive vibrations.

Let us suppose that the mark is at an unknown but very small distance ${\displaystyle x}$ on the positive side of the point of equilibrium, and that ${\displaystyle t_{1}}$ is the observed time of the first transit of the mark in the positive direction, and ${\displaystyle t_{2}}$, ${\displaystyle t_{3}}$, &c. the times of the following transits.

If ${\displaystyle T}$ be the time of vibration, and ${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ${\displaystyle P_{3}}$, &c. the times of transit of the true point of equilibrium,

${\displaystyle t_{1}=P_{1}+{\frac {x}{v_{1}}}}$, ${\displaystyle t_{2}=P_{2}+{\frac {x}{v_{2}}}}$,

where ${\displaystyle v_{1}}$, ${\displaystyle v_{2}}$, &c. are the successive velocities of transit, which we may suppose uniform for the very small distance ${\displaystyle x}$. If ${\displaystyle \rho }$ is the ratio of the amplitude of a vibration to the next in succession,

${\displaystyle v_{2}=-{\frac {1}{\rho }}v_{1}}$, and${\displaystyle {\frac {x}{v_{2}}}=-\rho {\frac {x}{v_{1}}}}$.

If three transits are observed at times ${\displaystyle t_{1}}$, ${\displaystyle t_{2}}$, ${\displaystyle t_{3}}$, we find

${\displaystyle {\frac {x}{v_{1}}}={\frac {t_{1}-2t_{2}+t_{3}}{(\rho +1)^{2}}}}$.

The period of vibration is therefore

${\displaystyle T={\frac {1}{2}}(t_{3}-t_{1})-{\frac {1}{2}}{\frac {\rho -1}{\rho +1}}(t_{1}-2t_{2}+t_{3})}$.

The time of the second passage of the true point of equilibrium is

${\displaystyle P_{2}={\frac {1}{4}}(t_{1}+2t_{2}+t_{3})-{\frac {1}{4}}{\frac {(\rho -1)^{2}}{(\rho +1)^{2}}}(t_{1}-2t_{2}+t_{3})}$.

Three transits are sufficient to determine these three quantities, but any greater number may be combined by the method of least squares. Thus, for five transits,

${\displaystyle T={\frac {1}{10}}(2t_{5}+t_{4}-t_{2}-2t_{1})-{\frac {1}{10}}(t_{1}-2t_{2}+2t_{3}-2t_{4}+t_{5}){\frac {\rho -1}{\rho +1}}\left(2-{\frac {\rho }{1+\rho ^{2}}}\right)}$.

The time of the third transit is,

${\displaystyle P_{3}={\frac {1}{8}}(t_{1}+2t_{2}+2t_{3}+2t_{4}+t_{5})-{\frac {1}{8}}(t_{1}-2t_{2}+2t_{3}-2t_{4}+t_{5}){\frac {(\rho -1)^{2}}{(\rho +1)^{2}}}}$.

739.] The same method may be extended to a series of any number of vibrations. If the vibrations are so rapid that the time of every transit cannot be recorded, we may record the time of every third or every fifth transit, taking care that the directions of successive transits are opposite. If the vibrations continue regular for a long time, we need not observe during the whole time. We may begin by observing a sufficient number of transits to determine approximately the period of vibration, ${\displaystyle T}$, and the time of the middle transit, ${\displaystyle P}$, noting whether this transit is in the positive or the negative direction. We may then either go on counting the vibrations without recording the times of transit, or we may leave the apparatus unwatched. We then observe a second series of transits, and deduce the time of vibration ${\displaystyle T^{\prime }}$ and the time of middle transit ${\displaystyle P^{\prime }}$, noting the direction of this transit.

If ${\displaystyle T}$ and ${\displaystyle T^{\prime }}$ the periods of vibration as deduced from the two sets of observations, are nearly equal, we may proceed to a more accurate determination of the period by combining the two series of observations.

Dividing ${\displaystyle P^{\prime }-P}$ by ${\displaystyle T}$, the quotient ought to be very nearly an integer, even or odd according as the transits ${\displaystyle P}$ and ${\displaystyle P^{\prime }}$ are in the same or in opposite directions. If this is not the case, the series of observations is worthless, but if the result is very nearly a whole number ${\displaystyle n}$, we divide ${\displaystyle P^{\prime }-P}$ by ${\displaystyle n}$, and thus find the mean value of ${\displaystyle T}$ for the whole time of swinging.

740.] The time of vibration ${\displaystyle T}$ thus found is the actual mean time of vibration, and is subject to corrections if we wish to deduce from it the time of vibration in infinitely small arcs and without damping.

To reduce the observed time to the time in infinitely small arcs, we observe that the time of a vibration of amplitude ${\displaystyle \alpha }$ is in general of the form

${\displaystyle T=T_{1}(1+\kappa c^{2})}$,

where ${\displaystyle \kappa }$ is a coefficient, which, in the case of the ordinary pendulum, is ${\displaystyle {\frac {1}{64}}}$. Now the amplitudes of the successive vibrations are ${\displaystyle c}$, ${\displaystyle c\rho ^{-1}}$, ${\displaystyle c\rho ^{-2}}$, … ${\displaystyle c\rho ^{1-n}}$, so that the whole time of ${\displaystyle n}$ vibrations is

${\displaystyle nT=T_{1}\left(n+\kappa {\frac {\rho ^{2}c_{1}^{2}-c_{n}^{2}}{\rho ^{2}-1}}\right)}$,

where ${\displaystyle T}$ is the time deduced from the observations. Hence, to find the time ${\displaystyle T}$ in infinitely small arcs, we have approximately,

${\displaystyle T_{1}=T\left\{1-{\frac {\kappa }{n}}{\frac {c_{1}^{2}\rho ^{2}-c_{n}^{2}}{\rho ^{2}-1}}\right\}}$.

To find the time ${\displaystyle T_{0}}$ when there is no damping, we have
 ${\displaystyle T_{0}}$ ${\displaystyle {}=T_{1}\sin \alpha }$ ${\displaystyle {}=T_{1}{\frac {\pi }{\sqrt {\pi ^{2}+\lambda ^{2}}}}}$.
741.] The equation of the rectilinear motion of a body, attracted to a fixed point and resisted by a force varying as the velocity, is

${\displaystyle {\frac {d^{2}x}{dt^{2}}}+2k{\frac {dx}{dt}}+\omega ^{2}(x-a)=0}$,

(1)

where ${\displaystyle x}$ is the coordinate of the body at the time ${\displaystyle t}$, and ${\displaystyle a}$ is the coordinate of the point of equilibrium.
To solve this equation, let

${\displaystyle x-a=e^{-kt}y}$;

(2)

then

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+(\omega ^{2}-k^{2})y=0}$;

(3)

the solution of which is
 ⁠ ${\displaystyle y=C\cos \left({\sqrt {\omega ^{2}-k^{2}}}t+a\right)}$, when ${\displaystyle k}$ is less than ${\displaystyle \omega }$; ⁠ (4) ${\displaystyle y=A+Bt}$, when ${\displaystyle k}$ is equal to ${\displaystyle \omega }$; (5) and ${\displaystyle y=C^{\prime }\cos h\left({\sqrt {k^{2}-\omega ^{2}}}t+a^{\prime }\right)}$, when ${\displaystyle k}$ is greater than ${\displaystyle \omega }$. (6)

The value of ${\displaystyle x}$ may be obtained from that of ${\displaystyle y}$ by equation (2). When ${\displaystyle k}$ is less than ${\displaystyle \omega }$, the motion consists of an infinite series of oscillations, of constant periodic time, but of continually decreasing amplitude. As ${\displaystyle k}$ increases, the periodic time becomes longer, and the diminution of amplitude becomes more rapid.

When ${\displaystyle k}$ (half the coefficient of resistance) becomes equal to or greater than ${\displaystyle \omega }$, (the square root of the acceleration at unit distance from the point of equilibrium,) the motion ceases to be oscillatory, and during the whole motion the body can only once pass through the point of equilibrium, after which it reaches a position of greatest elongation, and then returns towards the point of equilibrium, continually approaching, but never reaching it.

Galvanometers in which the resistance is so great that the motion is of this kind are called dead beat galvanometers. They are useful in many experiments, but especially in telegraphic signalling, in which the existence of free vibrations would quite disguise the movements which are meant to be observed.

Whatever be the values of ${\displaystyle k}$ and ${\displaystyle \omega }$, the value of ${\displaystyle a}$, the scale-reading at the point of equilibrium, may be deduced from five scale-readings, ${\displaystyle p}$, ${\displaystyle q}$, ${\displaystyle r}$, ${\displaystyle s}$, ${\displaystyle t}$, taken at equal intervals of time, by the formula

${\displaystyle a={\frac {q(rs-qt)+r(pt-r^{2})+s(qr-ps)}{(p-2q+r)(r-2s+t)-(q-2r+s)^{2}}}}$.

On the Observation of the Galvanometer.

742.] To measure a constant current with the tangent galvanometer, the instrument is adjusted with the plane of its coils parallel to the magnetic meridian, and the zero reading is taken. The current is then made to pass through the coils, and the deflexion of the magnet corresponding to its new position of equilibrium is observed. Let this be denoted by ${\displaystyle \phi }$.

Then, if ${\displaystyle H}$ is the horizontal magnetic force, ${\displaystyle G}$ the coefficient of the galvanometer, and ${\displaystyle \gamma }$ the strength of the current,

${\displaystyle \gamma ={\frac {H}{G}}\tan \phi }$.

(1)

If the coefficient of torsion of the suspension fibre is ${\displaystyle \tau MH}$ (see Art. 452), we must use the corrected formula

${\displaystyle \gamma ={\frac {H}{G}}(\tan \phi +\tau \phi \sec \phi )}$.

(2)

Best Value of the Deflexion.

743.] In some galvanometers the number of windings of the coil through which the current flows can be altered at pleasure. In others a known fraction of the current can be diverted from the galvanometer by a conductor called a Shunt. In either case the value of ${\displaystyle G}$, the effect of a unit-current on the magnet, is made to vary.

Let us determine the value of ${\displaystyle G}$, for which a given error in the observation of the deflexion corresponds to the smallest error of the deduced value of the strength of the current.

Differentiating equation (1), we find

${\displaystyle {\frac {d\gamma }{d\phi }}={\frac {H}{G}}\sec ^{2}\phi }$.

(3)

Eliminating ${\displaystyle G}$,

${\displaystyle {\frac {d\phi }{d\gamma }}={\frac {1}{2\gamma }}\sin 2\phi }$.

(4)

This is a maximum for a given value of ${\displaystyle \gamma }$ when the deflexion is 45°. The value of ${\displaystyle G}$ should therefore be adjusted till ${\displaystyle G\gamma }$ is as nearly equal to ${\displaystyle H}$ as is possible; so that for strong currents it is better not to use too sensitive a galvanometer.

On the Best Method of applying the Current.

744.] When the observer is able, by means of a key, to make or break the connexions of the circuit at any instant, it is advisable to operate with the key in such a way as to make the magnet arrive at its position of equilibrium with the least possible velocity. The following method was devised by Gauss for this purpose.

Suppose that the magnet is in its position of equilibrium, and that there is no current. The observer now makes contact for a short time, so that the magnet is set in motion towards its new position of equilibrium. He then breaks contact. The force is now towards the original position of equilibrium, and the motion is retarded. If this is so managed that the magnet comes to rest exactly at the new position of equilibrium, and if the observer again makes contact at that instant and maintains the contact, the magnet will remain at rest in its new position.

If we neglect the effect of the resistances and also the inequality of the total force acting in the new and the old positions, then, since we wish the new force to generate as much kinetic energy during the time of its first action as the original force destroys while the circuit is broken, we must prolong the first action of the current till the magnet has moved over half the distance from the first position to the second. Then if the original force acts while the magnet moves over the other half of its course, it will exactly stop it. Now the time required to pass from a point of greatest elongation to a point half way to the position of equilibrium is one-sixth of a complete period, or one-third of a single vibration.

The operator, therefore, having previously ascertained the time of a single vibration, makes contact for one-third of that time, breaks contact for another third of the same time, and then makes contact again during the continuance of the experiment. The magnet is then either at rest, or its vibrations are so small that observations may be taken at once, without waiting for the motion to die away. For this purpose a metronome may be adjusted so as to beat three times for each single vibration of the magnet.

The rule is somewhat more complicated when the resistance is of sufficient magnitude to be taken into account, but in this case the vibrations die away so fast that it is unnecessary to apply any corrections to the rule.

When the magnet is to be restored to its original position, the circuit is broken for one-third of a vibration, made again for an equal time, and finally broken. This leaves the magnet at rest in its former position.

If the reversed reading is to be taken immediately after the direct one, the circuit is broken for the time of a single vibration and then reversed. This brings the magnet to rest in the reversed position.

Measurement by the First Swing.

745.] When there is no time to make more than one observation, the current may be measured by the extreme elongation observed in the first swing of the magnet. If there is no resistance, the permanent deflexion ${\displaystyle \phi }$ is half the extreme elongation. If the resistance is such that the ratio of one vibration to the next is ${\displaystyle \rho }$, and if ${\displaystyle \theta _{0}}$ is the zero reading, and ${\displaystyle \theta _{1}}$ the extreme elongation in the first swing, the deflexion, ${\displaystyle \phi }$, corresponding to the point of equilibrium is

${\displaystyle \phi ={\frac {\theta _{0}+\rho \theta _{1}}{1+\rho }}}$.

In this way the deflexion may be calculated without waiting for the magnet to come to rest in its position of equilibrium.

To make a Series of Observations.

746.] The best way of making a considerable number of measures of a constant current is by observing three elongations while the current is in the positive direction, then breaking contact for about the time of a single vibration, so as to let the magnet swing into the position of negative deflexion, then reversing the current and observing three successive elongations on the negative side, then breaking contact for the time of a single vibration and repeating the observations on the positive side, and so on till a sufficient number of observations have been obtained. In this way the errors which may arise from a change in the direction of the earth's magnetic force during the time of observation are eliminated. The operator, by carefully timing the making and breaking of contact, can easily regulate the extent of the vibrations, so as to make them sufficiently small without being indistinct. The motion of the magnet is graphically represented in Fig. 59, where the abscissa represents the time, and the ordinate the deflexion of the magnet. If ${\displaystyle \theta _{1}}$${\displaystyle \theta _{6}}$ be the observed elongations, the deflexion is given by the equation

${\displaystyle 8\phi =\theta _{1}+2\theta _{2}+\theta _{3}-\theta _{4}-2\theta _{5}-\theta _{6}}$.

Fig. 59.

Method of Multiplication.

747.] In certain cases, in which the deflexion of the galvanometer magnet is very small, it may be advisable to increase the visible effect by reversing the current at proper intervals, so as to set up a swinging motion of the magnet. For this purpose, after ascertaining the time, ${\displaystyle T}$, of a single vibration of the magnet, the current is sent in the positive direction for a time ${\displaystyle T}$, then in the reversed direction for an equal time, and so on. When the motion of the magnet has become visible, we may make the reversal of the current at the observed times of greatest elongation.

Let the magnet be at the positive elongation ${\displaystyle \theta _{0}}$, and let the current be sent through the coil in the negative direction. The point of equilibrium is then ${\displaystyle -\phi }$, and the magnet will swing to a negative elongation ${\displaystyle \theta }$, such that
 ⁠ ${\displaystyle -\rho (\phi +\theta _{1})=(\theta _{0}+\phi )}$, or ${\displaystyle -\rho \theta _{1}=\theta _{0}+(\rho +1)\phi }$.\
Similarly, if the current is now made positive while the magnet swings to ${\displaystyle \theta _{2}}$,
 ⁠ ${\displaystyle \rho \theta _{2}}$ ${\displaystyle {}=-\theta _{1}+(\rho +1)\phi }$, or ${\displaystyle \rho ^{2}\theta _{2}}$ ${\displaystyle {}=\theta _{0}+(\rho +1)^{2}\phi }$;
and if the current is reversed ${\displaystyle n}$ times in succession, we find

${\displaystyle (-1)^{n}\theta _{n}=\rho ^{n}\theta _{0}+{\frac {\rho +1}{\rho -1}}(1-\rho ^{-n})\phi }$,

whence we may find ${\displaystyle \phi }$ in the form

${\displaystyle \phi =(\theta _{n}-\rho ^{-n}\theta _{0}){\frac {\rho -1}{\rho +1}}{\frac {1}{1-\rho ^{-n}}}}$.

If ${\displaystyle n}$ is a number so great that ${\displaystyle \rho ^{-n}}$ may be neglected, the expression becomes

${\displaystyle \phi =\theta _{n}{\frac {\rho -1}{\rho +1}}}$.

The application of this method to exact measurement requires an accurate knowledge of ${\displaystyle \rho }$, the ratio of one vibration of the magnet to the next under the influence of the resistances which it experiences. The uncertainties arising from the difficulty of avoiding irregularities in the value of ${\displaystyle \rho }$ generally outweigh the advantages of the large angular elongation. It is only where we wish to establish the existence of a very small current by causing it to produce a visible movement of the needle that this method is really valuable.

On the Measurement of Transient Currents.

748.] When a current lasts only during a very small fraction of the time of vibration of the galvanometer-magnet, the whole quantity of electricity transmitted by the current may be measured by the angular velocity communicated to the magnet during the passage of the current, and this may be determined from the elongation of the first vibration of the magnet.

If we neglect the resistance which damps the vibrations of the magnet, the investigation becomes very simple.

Let ${\displaystyle \gamma }$ be the intensity of the current at any instant, and ${\displaystyle Q}$ the quantity of electricity which it transmits, then

${\displaystyle Q=\int \gamma \;dt}$.

(1)

Let ${\displaystyle M}$ be the magnetic moment, and ${\displaystyle A}$ the moment of inertia of the magnet and suspended apparatus,

${\displaystyle A{\frac {d^{2}\theta }{dt^{2}}}+MH\sin \theta =MG\gamma \cos \theta }$.

(2)

If the time of the passage of the current is very small, we may integrate with respect to ${\displaystyle t}$ during this short time without regarding the change of ${\displaystyle \theta }$, and we find

${\displaystyle A{\frac {d\theta }{dt}}=MG\cos \theta _{0}\int \gamma \;dt+C=MGQ\cos \theta _{0}+C}$.

(3)

This shews that the passage of the quantity ${\displaystyle Q}$ produces an angular momentum ${\displaystyle MGQ\cos \theta _{0}}$ in the magnet, where ${\displaystyle \theta _{0}}$ is the value of ${\displaystyle \theta }$ at the instant of passage of the current. If the magnet is initially in equilibrium, we may make ${\displaystyle \theta _{0}=0}$.

The magnet then swings freely and reaches an elongation ${\displaystyle \theta _{1}}$. If there is no resistance, the work done against the magnetic force during this swing is ${\displaystyle MH(1-\cos \theta _{1})}$.

The energy communicated to the magnet by the current is

${\displaystyle {\frac {1}{2}}A\left.{\overline {\frac {d\theta }{dt}}}\right|^{2}}$.

Equating these quantities, we find
 ⁠ ⁠ ${\displaystyle \left.{\overline {\frac {d\theta }{dt}}}\right|^{2}}$ ${\displaystyle {}=2{\frac {MH}{A}}(1-\cos \theta _{1})}$, ⁠ ⁠ (4) whence ${\displaystyle {\frac {d\theta }{dt}}}$ ${\displaystyle {}=2{\sqrt {\frac {MH}{A}}}\sin {\frac {1}{2}}\theta _{1}}$ ${\displaystyle {}={\frac {MG}{A}}Q}$ by (3). (5)

But if ${\displaystyle T}$ be the time of a single vibration of the magnet,

 ⁠ ⁠ ${\displaystyle T=\pi {\sqrt {\frac {A}{MH}}}}$, ⁠ ⁠ (6) and we find ${\displaystyle Q={\frac {H}{G}}{\frac {T}{\pi }}2\sin {\frac {1}{2}}\theta _{1}}$, (7)

where ${\displaystyle H}$ is the horizontal magnetic force, ${\displaystyle G}$ the coefficient of the galvanometer, ${\displaystyle T}$ the time of a single vibration, and ${\displaystyle \theta _{1}}$ the first elongation of the magnet.

749.] In many actual experiments the elongation is a small angle, and it is then easy to take into account the effect of resistance, for we may treat the equation of motion as a linear equation.

Let the magnet be at rest at its position of equilibrium, let an angular velocity ${\displaystyle \nu }$ be communicated to it instantaneously, and let its first elongation be ${\displaystyle \theta _{1}}$.

The equation of motion is

 ⁠ ⁠ ${\displaystyle \theta }$ ${\displaystyle {}=Ce^{-\omega _{1}\tan \beta }\sin \omega _{1}t}$, ⁠ ⁠ (8) ${\displaystyle {\frac {d\theta }{dt}}}$ ${\displaystyle {}=C\omega _{1}\sec \beta e^{-\omega _{1}t\tan \beta }\cos(\omega _{1}t+\beta )}$. (9)

When ${\displaystyle t=0}$, ${\displaystyle \theta =0}$, and ${\displaystyle {\frac {d\theta }{dt}}=C\omega _{1}=\nu }$.

When ${\displaystyle \omega _{1}t+\beta ={\frac {\pi }{2}}}$,

 ⁠ ⁠ ${\displaystyle \theta }$ ${\displaystyle {}=Ce^{-\left({\frac {\pi }{2}}-\beta \right)\tan \beta }\cos \beta =\theta _{1}}$. ⁠ ⁠ (10) Hence ${\displaystyle \theta _{1}}$ ${\displaystyle {}={\frac {\nu }{\omega _{1}}}e^{-\left({\frac {\pi }{2}}-\beta \right)\tan \beta }\cos \beta }$. (11)
 Now ⁠ ⁠ ${\displaystyle {\frac {MH}{A}}}$ ${\displaystyle {}=\omega ^{2}=\omega _{1}^{2}\sec ^{2}\beta }$, ⁠ ⁠ (12) ${\displaystyle \tan \beta }$ ${\displaystyle {}={\frac {\lambda }{\pi }}}$, ⁠ ${\displaystyle \omega _{1}={\frac {\pi }{T_{1}}}}$, (13)

${\displaystyle \nu ={\frac {MG}{A}}Q}$.

(14)

 Hence ⁠ ⁠ ${\displaystyle \theta _{1}}$ ${\displaystyle ={\frac {QG}{H}}{\frac {\sqrt {\pi ^{2}+\lambda ^{2}}}{T_{1}}}e^{-{\frac {\lambda }{\pi }}\tan ^{-1}{\frac {\pi }{\lambda }}}}$, ⁠ ⁠ (15) and ${\displaystyle Q}$ ${\displaystyle {}={\frac {H}{G}}{\frac {T_{1}}{\sqrt {\pi ^{2}+\lambda ^{2}}}}e^{{\frac {\lambda }{\pi }}\tan ^{-1}{\frac {\pi }{\lambda }}}\theta _{1}}$, (16)
which gives the first elongation in terms of the quantity of electricity in the transient current, and conversely, where ${\displaystyle T_{1}}$ is the observed time of a single vibration as affected by the actual resistance of damping. When ${\displaystyle \lambda }$ is small we may use the approximate formula

${\displaystyle Q={\frac {H}{G}}{\frac {T}{\pi }}(1+{\frac {1}{2}}\lambda )\theta _{1}}$.

(17)

Method of Recoil.

750.] The method given above supposes the magnet to be at rest in its position of equilibrium when the transient current is passed through the coil. If we wish to repeat the experiment we must wait till the magnet is again at rest. In certain cases, however, in which we are able to produce transient currents of equal intensity, and to do so at any desired instant, the following method, described by Weber[3], is the most convenient for making a continued series of observations.

Suppose that we set the magnet swinging by means of a transient current whose value is ${\displaystyle Q_{0}}$. If, for brevity, we write

${\displaystyle {\frac {G}{H}}{\frac {\sqrt {\pi ^{2}+\lambda ^{2}}}{T_{1}}}e^{-{\frac {\lambda }{\pi }}\tan ^{-1}{\frac {\pi }{\lambda }}}=K}$,

(18)

then the first elongation

${\displaystyle \theta _{1}=KQ_{0}=a_{1}}$ (say).

(19)

The velocity instantaneously communicated to the magnet at starting is

${\displaystyle v_{0}={\frac {MG}{A}}Q_{0}}$.

(20)

When it returns through the point of equilibrium in a negative direction its velocity will be

${\displaystyle v_{1}=-ve^{-\lambda }}$.

(21)

The next negative elongation will be

${\displaystyle \theta _{2}=-\theta _{1}e^{-\lambda }=b_{1}}$.

(22)

When the magnet returns to the point of equilibrium, its velocity will be

${\displaystyle v_{2}=v_{0}e^{-2\lambda }}$.

(23)

Now let an instantaneous current, whose total quantity is ${\displaystyle -Q}$, be transmitted through the coil at the instant when the magnet is at the zero point. It will change the velocity ${\displaystyle v_{2}}$ into ${\displaystyle v_{2}-v}$, where

${\displaystyle v={\frac {MG}{A}}Q}$.

(24)

If ${\displaystyle Q}$ is greater than ${\displaystyle Q_{0}e^{-2\lambda }}$, the new velocity will be negative and equal to

${\displaystyle -{\frac {MG}{A}}(Q-Q_{0}ee^{-2\lambda })}$.

The motion of the magnet will thus be reversed, and the next elongation will be negative,

${\displaystyle \theta _{3}=-K(Q-Q_{0}e^{-2\lambda })=c_{1}=-KQ+\theta _{1}e^{-2\lambda }}$.

(25)

The magnet is then allowed to come to its positive elongation

${\displaystyle \theta _{4}=-\theta _{3}e^{-\lambda }=d_{1}=e^{-\lambda }(KQ-a_{1}e^{-2\lambda })}$,

(26)

and when it again reaches the point of equilibrium a positive current whose quantity is ${\displaystyle Q}$ is transmitted. This throws the magnet back in the positive direction to the positive elongation

${\displaystyle \theta _{5}=KQ-\theta _{3}e^{-2\lambda }}$;

(27)

or, calling this the first elongation of a second series of four,

${\displaystyle a_{2}=KQ(1-e^{-2\lambda })+a_{1}e^{-4\lambda }}$.

(28)

Proceeding in this way, by observing two elongations + and −, then sending a positive current and observing two elongations − and +, then sending a positive current, and so on, we obtain a series consisting of sets of four elongations, in each of which

${\displaystyle {\frac {d-b}{a-c}}=e^{-\lambda }}$,

(29)

and

${\displaystyle KQ={\frac {(a-b)e^{-2\lambda }+d-c}{1+e^{-\lambda }}}}$;

(30)

If ${\displaystyle n}$ series of elongations have been observed, then we find the logarithmic decrement from the equation

${\displaystyle {\frac {\Sigma (d)-\Sigma (b)}{\Sigma (a)-\Sigma (c)}}=e^{-\lambda }}$,

(31)

and ${\displaystyle Q}$ from the equation

${\displaystyle KQ(1+e^{-\lambda })(2n-1)}$
${\displaystyle {}=\Sigma _{n}(a-b-c+d)(1+e^{-2\lambda })-(a_{1}-b_{1})-(d_{n}-c_{n})e^{-2\lambda }}$.

(32)

Fig. 60.

The motion of the magnet in the method of recoil is graphically represented in Fig. 60, where the abscissa represents the time, and the ordinate the deflexion of the magnet at that time. See Art. 760.

Method of Multiplication.

751.] If we make the transient current pass every time that the magnet passes through the zero point, and always so as to increase the velocity of the magnet, then, if ${\displaystyle \theta _{1}}$, ${\displaystyle \theta _{2}}$, &c. are the successive elongations,

 ⁠ ⁠ ${\displaystyle \theta _{2}=-KQ-e^{-\lambda }\theta _{1}}$, ⁠ (33) ${\displaystyle \theta _{3}=-KQ-e^{-\lambda }\theta _{2}}$. (34)
The ultimate value to which the elongation tends after a great many vibrations is found by putting ${\displaystyle \theta _{n}=-\theta _{n-1}}$, whence we find

${\displaystyle \theta =\pm {\frac {1}{1-e^{-\lambda }}}KQ}$.

(35)

If ${\displaystyle \lambda }$ is small, the value of the ultimate elongation may be large, but since this involves a long continued experiment, and a careful determination of ${\displaystyle \lambda }$, and since a small error in ${\displaystyle \lambda }$ introduces a large error in the determination of ${\displaystyle Q}$, this method is rarely useful for numerical determination, and should be reserved for obtaining evidence of the existence or non-existence of currents too small to be observed directly.

In all experiments in which transient currents are made to act on the moving magnet of the galvanometer, it is essential that the whole current should pass while the distance of the magnet from the zero point remains a small fraction of the total elongation. The time of vibration should therefore be large compared with the time required to produce the current, and the operator should have his eye on the motion of the magnet, so as to regulate the instant of passage of the current by the instant of passage of the magnet through its point of equilibrium.

To estimate the error introduced by a failure of the operator to produce the current at the proper instant, we observe that the effect of a force in increasing the elongation varies as

${\displaystyle e^{\phi \tan \beta }\cos(\phi +\beta )}$,

and that this is a maximum when ${\displaystyle \phi =0}$. Hence the error arising from a mistiming of the current will always lead to an underestimation of its value, and the amount of the error may be estimated by comparing the cosine of the phase of the vibration at the time of the passage of the current with unity.

1. Proc. R. S. Edin., Dec. 16, 1867.
2. See Gauss, Resultate des Magnetischen Vereins, 1836. II.
3. Resultate des Magnetischen Vereins, 1838, p. 98.