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

CHAPTER XVII.

COMPARISON OF COILS..

Experimental Determination of the Electrical Constants of a Coil.

752.] We have seen in Art. 717 that in a sensitive galvanometer the coils should be of small radius, and should contain many windings of the wire. It would be extremely difficult to determine the electrical constants of such a coil by direct measurement of its form and dimensions, even if we could obtain access to every winding of the wire in order to measure it. But in fact the greater number of the windings are not only completely hidden by the outer windings, but we are uncertain whether the pressure of the outer windings may not have altered the form of the inner ones after the coiling of the wire.

It is better therefore to determine the electrical constants of the coil by direct electrical comparison with a standard coil whose constants are known.

Since the dimensions of the standard coil must be determined by actual measurement, it must be made of considerable size, so that the unavoidable error of measurement of its diameter or circumference may be as small as possible compared with the quantity measured. The channel in which the coil is wound should be of rectangular section, and the dimensions of the section should be small compared with the radius of the coil. This is necessary, not so much in order to diminish the correction for the size of the section, as to prevent any uncertainty about the position of those windings of the coil which are hidden by the external windings[1]. The principal constants which we wish to determine are—

(1) The magnetic force at the centre of the coil due to a unit-current. This is the quantity denoted by ${\displaystyle G_{1}}$ in Art. 700.

(2) The magnetic moment of the coil due to a unit-current. This is the quantity ${\displaystyle g_{1}}$.

753.] To determine ${\displaystyle G_{1}}$. Since the coils of the working galvanometer are much smaller than the standard coil, we place the galvanometer within the standard coil, so that their centres coincide, the planes of both coils being vertical and parallel to the earth's magnetic force. We have thus obtained a differential galvanometer one of whose coils is the standard coil, for which the value of ${\displaystyle G_{1}}$ is known, while that of the other coil is ${\displaystyle G_{1}^{\prime }}$, the value of which we have to determine.

The magnet suspended in the centre of the galvanometer coil is acted on by the currents in both coils. If the strength of the current in the standard coil is ${\displaystyle \gamma }$, and that in the galvanometer coil ${\displaystyle \gamma ^{\prime }}$, then, if these currents flowing in opposite directions produce a deflexion ${\displaystyle \delta }$ of the magnet,
 ${\displaystyle H\tan \delta =G_{1}^{\prime }\gamma ^{\prime }-G_{1}\gamma }$ (1)

where ${\displaystyle H}$ is the horizontal magnetic force of the earth.

If the currents are so arranged as to produce no deflexion, we may find ${\displaystyle G_{1}^{\prime }}$ by the equation
 ${\displaystyle G_{1}^{\prime }={\frac {\gamma }{\gamma ^{\prime }}}G_{1}}$. (2)

We may determine the ratio of ${\displaystyle \gamma }$ to ${\displaystyle \gamma ^{\prime }}$ in several ways. Since the value of ${\displaystyle G_{1}}$ is in general greater for the galvanometer than for the standard coil, we may arrange the circuit so that the whole current ${\displaystyle \gamma }$ flows through the standard coil, and is then divided so that ${\displaystyle \gamma ^{\prime }}$ flows through the galvanometer and resistance coils, the combined resistance of which is ${\displaystyle R_{1}}$, while the remainder ${\displaystyle \gamma -\gamma ^{\prime }}$ flows through another set of resistance coils whose combined resistance is ${\displaystyle R_{2}}$.

We have then, by Art. 276,]

 ⁠ ⁠ ${\displaystyle \gamma ^{\prime }R_{1}}$ ${\displaystyle {}=(\gamma -\gamma ^{\prime })R_{2}}$, ⁠ ⁠ (3) or ${\displaystyle {\frac {\gamma }{\gamma ^{\prime }}}}$ ${\displaystyle {}={\frac {R_{1}+R_{2}}{R_{2}}}}$, (4) and ${\displaystyle G_{1}^{\prime }}$ ${\displaystyle {}={\frac {R_{1}+R_{2}}{R_{2}}}G_{1}}$. (5)

If there is any uncertainty about the actual resistance of the galvanometer coil (on account, say, of an uncertainty as to its temperature) we may add resistance coils to it, so that the resistance of the galvanometer itself forms but a small part of ${\displaystyle R_{1}}$, and thus introduces but little uncertainty into the final result.

754.] To determine ${\displaystyle g_{1}}$, the magnetic moment of a small coil due to a unit-current flowing through it, the magnet is still suspended at the centre of the standard coil, but the small coil is moved parallel to itself along the common axis of both coils, till the same current, flowing in opposite directions round the coils, no longer deflects the magnet. If the distance between the centres of the coils is ${\displaystyle r}$, we have now
 ${\displaystyle G_{1}=2{\frac {g_{1}}{r^{3}}}+3{\frac {g_{2}}{r^{4}}}+4{\frac {g_{3}}{r^{5}}}+\mathrm {\&c} }$. (6)

By repeating the experiment with the small coil on the opposite side of the standard coil, and measuring the distance between the positions of the small coil, we eliminate the uncertain error in the determination of the position of the centres of the magnet and of the small coil, and we get rid of the terms in ${\displaystyle g_{2}}$, ${\displaystyle g_{4}}$, &c.

If the standard coil is so arranged that we can send the current through half the number of windings, so as to give a different value to ${\displaystyle G_{1}}$, we may determine a new value of ${\displaystyle r}$, and thus, as in Art. 454, we may eliminate the term involving ${\displaystyle g_{3}}$.

It is often possible, however, to determine ${\displaystyle g_{3}}$ by direct measurement of the small coil with sufficient accuracy to make it available in calculating the value of the correction to be applied to ${\displaystyle g_{1}}$ in the equation
 ${\displaystyle g_{1}={\frac {1}{2}}G_{1}r^{3}-2{\frac {g_{3}}{r^{2}}}}$. (7)
 where ${\displaystyle g_{3}=-{\frac {1}{8}}\pi a^{2}(6a^{2}+3\xi ^{2}-2\eta ^{2})}$, by Art. 700.

Comparison of Coefficients of Induction.

755.] It is only in a small number of cases that the direct calculation of the coefficients of induction from the form and position of the circuits can be easily performed. In order to attain a sufficient degree of accuracy, it is necessary that the distance between the circuits should be capable of exact measurement. But when the distance between the circuits is sufficient to prevent errors of measurement from introducing large errors into the result, the coefficient of induction itself is necessarily very much reduced in magnitude. Now for many experiments it is necessary to make the coefficient of induction large, and we can only do so by bringing the circuits close together, so that the method of direct measurement becomes impossible, and, in order to determine the coefficient of induction, we must compare it with that of a pair of coils arranged so that their coefficient may be obtained by direct measurement and calculation.

This may be done as follows:

Fig. 61.
Let ${\displaystyle A}$ and ${\displaystyle a}$ be the standard pair of coils, ${\displaystyle B}$ and ${\displaystyle b}$ the coils to be compared with them. Connect ${\displaystyle A}$ and ${\displaystyle B}$ in one circuit, and place the electrodes of the galvanometer, ${\displaystyle G}$, at ${\displaystyle P}$ and ${\displaystyle Q}$, so that the resistance of ${\displaystyle PAQ}$ is ${\displaystyle R}$, and that of ${\displaystyle QBP}$ is ${\displaystyle S}$, ${\displaystyle K}$ being the resistance of the galvanometer. Connect ${\displaystyle a}$ and ${\displaystyle b}$ in one circuit with the battery.

Let the current in ${\displaystyle A}$ be ${\displaystyle {\dot {x}}}$, that in ${\displaystyle B}$, ${\displaystyle {\dot {y}}}$, and that in the galvanometer, ${\displaystyle {\dot {x}}-{\dot {y}}}$, that in the battery circuit being ${\displaystyle \gamma }$.

Then, if ${\displaystyle M_{1}}$ is the coefficient of induction between ${\displaystyle A}$ and ${\displaystyle a}$, and ${\displaystyle M_{2}}$ that between ${\displaystyle B}$ and ${\displaystyle b}$, the integral induction current through the galvanometer at breaking the battery circuit is
 ${\displaystyle x-y=\gamma {\frac {{\frac {M_{1}}{R}}-{\frac {M_{2}}{S}}}{1+{\frac {K}{R}}+{\frac {K}{S}}}}}$. (8)

By adjusting the resistances ${\displaystyle R}$ and ${\displaystyle S}$ till there is no current through the galvanometer at making or breaking the galvanometer circuit, the ratio of ${\displaystyle M_{2}}$ to ${\displaystyle M_{1}}$ may be determined by measuring that of ${\displaystyle S}$ to ${\displaystyle R}$.

Comparison of a Coefficient of Self-induction with a Coefficient of Mutual Induction.

Fig. 62.
756.] In the branch ${\displaystyle AF}$ of Wheatstone's Bridge let a coil be inserted, the coefficient of self-induction of which we wish to find. Let us call it ${\displaystyle L}$.

In the connecting wire between ${\displaystyle A}$ and the battery another coil is inserted. The coefficient of mutual induction between this coil and the coil in ${\displaystyle AF}$ is ${\displaystyle M}$. It may be measured by the method described in Art. 755.

If the current from ${\displaystyle A}$ to ${\displaystyle F}$ is ${\displaystyle x}$, and that from ${\displaystyle A}$ to ${\displaystyle H}$ is ${\displaystyle y}$, that from ${\displaystyle Z}$ to ${\displaystyle A}$, through ${\displaystyle B}$, will be ${\displaystyle x+y}$. The external electromotive force from ${\displaystyle A}$ to ${\displaystyle F}$ is
 ${\displaystyle A-F=Px+L{\frac {dx}{dt}}+M\left({\frac {dx}{dt}}+{\frac {dy}{dt}}\right)}$. (9)

The external electromotive force along ${\displaystyle AH}$ is
 ${\displaystyle A-H=Qy}$. (10)

If the galvanometer placed between ${\displaystyle F}$ and ${\displaystyle H}$ indicates no current, either transient or permanent, then by (9) and (10), since ${\displaystyle H-F=0}$,
 ${\displaystyle Px=Qy}$; (11)
 and ${\displaystyle L{\frac {dx}{dt}}+M\left({\frac {dx}{dt}}+{\frac {dy}{dt}}\right)=0}$, (12)
 whence ${\displaystyle L=-\left(1+{\frac {P}{Q}}\right)M}$. (13)

Since ${\displaystyle L}$ is always positive, ${\displaystyle M}$ must be negative, and therefore the current must flow in opposite directions through the coils placed in ${\displaystyle P}$ and in ${\displaystyle B}$. In making the experiment we may either begin by adjusting the resistances so that
 ${\displaystyle PS=QR}$, (14)

which is the condition that there may be no permanent current, and then adjust the distance between the coils till the galvanometer ceases to indicate a transient current on making and breaking the battery connexion; or, if this distance is not capable of adjustment, we may get rid of the transient current by altering the resistances ${\displaystyle Q}$ and ${\displaystyle S}$ in such a way that the ratio of ${\displaystyle Q}$ to ${\displaystyle S}$ remains constant.

If this double adjustment is found too troublesome, we may adopt 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 ${\displaystyle W}$ between ${\displaystyle A}$ and ${\displaystyle Z}$. The condition of no permanent current through the galvanometer is not affected by the introduction of ${\displaystyle W}$. We may therefore get rid of the transient current by adjusting the resistance of ${\displaystyle W}$ alone. When this is done the value of ${\displaystyle L}$ is
 ${\displaystyle L=-\left(1+{\frac {P}{Q}}+{\frac {P+R}{W}}\right)M}$. (15)

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

757.] Insert the coils in two adjacent branches of Wheatstone's Bridge. Let ${\displaystyle L}$ and ${\displaystyle N}$ be the coefficients of self-induction of the coils inserted in ${\displaystyle P}$ and in ${\displaystyle R}$ respectively, then the condition of no galvanometer current is
 ${\displaystyle \left(Px+L{\frac {dx}{dt}}\right)Sy=Qy\left(Rx+N{\frac {dx}{dt}}\right)}$, (16)
 whence ${\displaystyle PS=QR}$,⁠ for no permanent current, (17)
 and ${\displaystyle {\frac {L}{P}}={\frac {N}{R}}}$, for no transient current. (18)

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 ${\displaystyle L}$ to ${\displaystyle N}$ can be determined by a comparison of the resistances.

1. Large tangent galvanometers are sometimes made with a single circular conducting ring of considerable thickness, which is sufficiently stiff to maintain its form without any support. This is not a good plan for a standard instrument. The distribution of the current within the conductor depends on the relative conductivity of its various parts. Hence any concealed flaw in the continuity of the metal may cause the main stream of electricity to flow either close to the outside or close to the inside of the circular ring. Thus the true path of the current becomes uncertain. Besides this, when the current flows only once round the circle, especial care is necessary to avoid any action on the suspended magnet due to the current on its way to or from the circle, because the current in the electrodes is equal to that in the circle. In the construction of many instruments the action of this part of the current seems to have been altogether lost sight of.

The most perfect method is to make one of the electrodes in the form of a metal tube, and the other a wire covered with insulating material, and placed inside the tube and concentric with it. The external action of the electrodes when thus arranged is zero, by Art. 683.