Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/182

strength of the current is the same in $A$ and $C$. Its direction in $B$ may be the same or opposite. Under these circumstances it is found that $B$ is in equilibrium under the action of $A$ and $C$, whatever be the forms and distances of the three circuits, provided they have the relations given above.

Since the actions between the complete circuits may be considered to be due to actions between the elements of the circuits, we may use the following method of determining the law of these actions.

Let $A_{1}$, $B_{1}$, $C_{1}$, Fig. 28, be corresponding elements of the three circuits, and let $A_{2}$, $B_{2}$, $C_{2}$ be also corresponding elements in another part of the circuits. Then the situation of $B_{1}$ with respect to $A_{2}$ is similar to the situation of $C_{1}$ with respect to $B_{2}$, but the

Fig. 28.

distance and dimensions of $C_{1}$ and $B_{2}$ are $n$ times the distance and dimensions of $B_{1}$ and $A_{2}$, respectively. If the law of electromagnetic action is a function of the distance, then the action, whatever be its form or quality, between $B_{1}$ and $A_{2}$, may be written

and that between $C_{1}$ and $B_{2}$

where $a$, $b$, $c$ are the strengths of the currents in $A$, $B$, $C$. But $nB_{1}=C_{1}$, $nA_{2}=B_{2}$, $n{\overline {B_{1}A_{2}}}={\overline {C_{1}B_{2}}}$, and $a=c$. Hence

and this is equal to $F$ by experiment, so that we have

or, the force varies inversely as the square of the distance.