Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/391

 

from $A_{2}$ and placed in a position in which the coefficient of mutual induction between $A_{1}$ and $A_{2}$ 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 $QGA_{1}P$ be $K$, of $PA_{2}BQ$, $B$, and of $PQ$, $R$.

Let $L$, $M$ and $N$ be the coefficients of induction of $A_{1}$ and $A_{2}$.

Let ${\dot {x}}$ be the current in $G$, and ${\dot {y}}$ that in $B$, then the current from $P$ to $Q$ is ${\dot {x}}-{\dot {y}}$.

Let $E$ be the electromotive force of the battery, then

$(K+R){\dot {x}}-R{\dot {y}}+{\frac {d}{dt}}(L{\dot {x}}+M{\dot {y}})=0$,

(1)

$R{\dot {x}}+(B+R){\dot {y}}+{\frac {d}{dt}}(M{\dot {x}}+N{\dot {y}})=E$.

(2)

When the currents are constant, and everything at rest,

$(K+R){\dot {x}}-R{\dot {y}}=0$.

(3)

If $M$ now suddenly becomes zero on account of the separation of $A_{1}$ from $A_{2}$, then, integrating with respect to $t$,

$(K+R)x-Ry-M{\dot {y}}=0$,

(4)

$-Rx+(B+R)y-M{\dot {x}}=\int E\,dt=0$.

(5)

whence

$x=M{\frac {(B+R){\dot {y}}+R{\dot {x}}}{(B+R)(K+R)-R^{2}}}$.

(6)

Substituting the value of ${\dot {y}}$ in terms of ${\dot {x}}$ from (3), we find

��x M

x~ R *

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

${\frac {x}{\dot {x}}}={\frac {M}{R}}$.

(9)

Of these quantities, $x$ is found from the throw of the galvanometer due to the induction current. See Art. 768. The permanent current, ${\dot {x}}$, 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