Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/437

The first of these methods depends on the use of the differential galvanometer, an instrument in which there are two coils, the currents in which are independent of each other, so that when the currents are made to flow in opposite directions they act in opposite directions on the needle, and when the ratio of these currents is that of $m$ to $n$ they have no resultant effect on the galvanometer needle.

Let $I_{1},\,I_{2}$ be the currents through the two coils of the galvanometer, then the deflexion of the needle may be written

$\delta =mI_{1}-nI_{2}.$

Now let the battery current $I$ be divided between the coils of the galvanometer, and let resistances $A$ and $B$ be introduced into the first and second coils respectively. Let the remainder of the resistance of their coils and their connexions be $\alpha$ and $\beta$ respectively, and let the resistance of the battery and its connexions between $C$ and $D$ be $r$, and its electromotive force $E.$

Then we find, by Ohm's Law, for the difference of potentials between $C$ and $D,$

$C-D=I_{1}(A+\alpha )=I_{2}(B+\beta )=E-Ir,$

and since

$I_{1}+I_{2}=I,$

$I_{1}=E{\frac {B+\beta }{D}},\qquad I_{2}=E{\frac {A+\alpha }{D}},\qquad I=E{\frac {A+\alpha +B+\beta }{D}},$

where

$D=(A+\alpha )(B+\beta )+r(A+\alpha +B+\beta ).$

The deflexion of the galvanometer needle is therefore

$\delta ={\frac {E}{D}}\{m(B+\beta )-n(A+\alpha )\},$

and if there is no observable deflexion, then we know that the quantity enclosed in brackets cannot differ from zero by more than a certain small quantity, depending on the power of the battery, the suitableness of the arrangement, the delicacy of the galvanometer, and the accuracy of the observer.

Suppose that $B$ has been adjusted so that there is no apparent deflexion.

Now let another conductor $A'$ be substituted for $A$, and let $A'$ be adjusted till there is no apparent deflexion. Then evidently to a first approximation $A'=A.$

To ascertain the degree of accuracy of this estimate, let the altered quantities in the second observation be accented, then