Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/204

positions of the two coils are in such cases said to be conjugate to each other.

Let $B_{1}$ and $B_{2}$ be two of these positions. If the coil $B$ be suddenly moved from the position $B_{1}$ to the position $B_{2}$, the algebraical sum of the transient currents in the coil $B$ is exactly zero, so that the galvanometer needle is left at rest when the motion of $B$ is completed.

This is true in whatever way the coil $B$ is moved from $B_{1}$ to $B_{2}$, and also whether the current in the primary coil $A$ be continued constant, or made to vary during the motion.

Again, let $B^{\prime }$ be any other position of $B$ not conjugate to $A$, so that the making or breaking of contact in $A$ produces an induction current when $B$ is in the position $B^{\prime }$.

Let the contact be made when $B$ is in the conjugate position $B_{1}$, there will be no induction current. Move $B$ to $B^{\prime }$, there will be an induction current due to the motion, but if $B$ is moved rapidly to $B^{\prime }$, and the primary contact then broken, the induction current due to breaking contact will exactly annul the effect of that due to the motion, so that the galvanometer needle will be left at rest. Hence the current due to the motion from a conjugate position to any other position is equal and opposite to the current due to breaking contact in the latter position.

Since the effect of making contact is equal and opposite to that of breaking it, it follows that the effect of making contact when the coil $B$ is in any position $B^{\prime }$ is equal to that of bringing the coil from any conjugate position $B_{1}$ to $B^{\prime }$ while the current is flowing through $A$.

If the change of the relative position of the coils is made by moving the primary circuit instead of the secondary, the result is found to be the same.

539.] It follows from these experiments that the total induction current in $B$ during the simultaneous motion of $A$ from $A_{1}$ to $A_{2}$, and of $B$ from $B_{1}$ to $B_{2}$, while the current in $A$ changes from $\gamma _{1}$ to $\gamma _{2}$, depends only on the initial state $A_{1}$, $B_{1}$, $\gamma _{1}$, and the final state $A_{2}$, $B_{2}$, $\gamma$, and not at all on the nature of the intermediate states through which the system may pass.

Hence the value of the total induction current must be of the form

where $F$ is a function of $A$, $B$, and $\gamma$.

With respect to the form of this function, we know, by Art. 536, that when there is no motion, and therefore $A_{1}=A_{2}$ and $B_{1}=B_{2}$,