Page:A Dynamical Theory of the Electromagnetic Field.pdf/18

In this expression F is the electromotive force of the battery, $z$ the current through the galvanometer when it has become steady. P, Q, R, S the resistances in the four arms. B that of the battery and electrodes, and G that of the galvanometer.

(44) If PS=QR, then $z=0$, and there will be no steady current, but a transient current through the galvanometer may be produced on making or breaking circuit on account of induction, and the indications of the galvanometer may be used to determine the coefficients of induction, provided we understand the actions which take place.

We shall suppose PS=QR, so that the current $z$ vanishes when sufficient time is allowed, and

$x(P+Q)=y(R+S)={\frac {F(P+Q)(R+S)}{(P+Q)(R+S)+B(P+Q)(R+S)}}.$

Let the induction coefficients between P, Q, R, S be given by the following table, the coefficient of induction of P on itself being $p$, between P and Q, $h$, and so on.

${\begin{array}{ccccc}&P&Q&R&S\\P&p&h&k&l\\Q&h&q&m&n\\R&k&m&r&o\\S&l&n&o&s\end{array}}$

Let $g$ be the coefficient of induction of the galvanometer on itself, and let it be out of the reach of the inductive influence of P,Q,R,S (as it must be in order to avoid direct action of P,Q,R,S on the needle). Let X,Y,Z be the integrals of $x,y,z$ with respect to $t$. At making contact $x,y,z$ are zero. After a time $z$ disappears, and $x$ and $y$ reach constant values. The equations for each conductor will therefore be

$\left.{\begin{array}{l}PX+(p+h)x+(k+l)y=\int Adt-\int Ddt\\Q(X-Z)+(h+q)x+(m+n)y=\int Ddt-\int Cdt\\RY+(k+m)x+(r+o)y=\int Adt-\int Edt\\S(Y+Z)+(l+n)x+(o+s)y=\int Edt-\int Cdt\\GZ=\int Ddt-\int Edt.\end{array}}\right\}.$

(24)

Solving these equations for $Z$, we find

$\left.{\begin{array}{l}Z\left\{{\frac {1}{P}}+{\frac {1}{Q}}+{\frac {1}{R}}+{\frac {1}{S}}+B\left({\frac {1}{P}}+{\frac {1}{R}}\right)\left({\frac {1}{Q}}+{\frac {1}{S}}\right)+G\left({\frac {1}{P}}+{\frac {1}{Q}}\right)\left({\frac {1}{R}}+{\frac {1}{S}}\right)+{\frac {BG}{PQRS}}(P+Q+R+S)\right\}\\\\\qquad =-F{\frac {1}{PS}}\left\{{\frac {p}{P}}-{\frac {q}{Q}}-{\frac {r}{R}}+{\frac {s}{S}}+h\left({\frac {1}{P}}-{\frac {1}{Q}}\right)+k\left({\frac {1}{R}}-{\frac {1}{P}}\right)+l\left({\frac {1}{R}}+{\frac {1}{Q}}\right)-m\left({\frac {1}{P}}+{\frac {1}{S}}\right)\right.\\\\\qquad \left.+n\left({\frac {1}{Q}}-{\frac {1}{S}}\right)+o\left({\frac {1}{S}}-{\frac {1}{R}}\right)\right\}\end{array}}\right\}$

(25)

(45) Now let the deflection of the galvanometer by the instantaneous current whose intensity is $Z$ be $\alpha$.

Let the permanent deflection produced by making the ratio of $PS$ to $QR$, $\rho$ instead of unity, be $\theta$.

Also let the time of vibration of the galvanometer needle from rest to rest be $T$.