Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/340

308 of magnets or currents, is subject to the following equation Rh

To prove this, let us consider the number of lines of magnetic force cut by the circle when $$a$$ or $$b$$ is made to vary.

(1) Let $$a$$ become $$a + \delta a$$, $$b$$ remaining constant. During this variation the circle, in expanding, sweeps over an annular surface in its own plane whose breadth is $$\delta a$$.

If $$V$$ is the magnetic potential at any point, and if the axis of $$y$$ be parallel to that of the circle, then the magnetic force perpendicular to the plane of the ring is $$\frac{dV}{dy}$$.

To find the magnetic induction through the annular surface we have to integrate where $$\theta$$ is the angular position of a point on the ring.

But this quantity represents the variation of $$M$$ due to the variation of $$a$$, or $$\frac{dM}{da} \delta a$$. Hence Rh

(2) Let $$b$$ become $$b + \delta b$$, $$a$$ remaining constant. During this variation the circle sweeps over a cylindric surface of radius $$a$$ and length $$\delta b$$.

The magnetic force perpendicular to this surface at any point is $$\frac{dV}{dr}$$ where $$r$$ is the distance from the axis. Hence Rh

Differentiating equation (2) with respect to $$a$$, and (3) with respect to $$b$$, we get

Transposing the last term we obtain equation (1).