Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/305

Rh If we now express these equations in terms of $$r$$ and $$\theta$$, where Rhthey become

a = y co r 2 - /, (9)

Equation (10) is satisfied if we assume any arbitrary function $$\chi$$ of $$r$$ and $$\theta$$. and make

Substituting these values in equation (9), it becomes Rh

Dividing by $$\sigma r^2$$, and restoring the coordinates $$x$$ and $$y$$, this becomes Rh

This is the fundamental equation of the theory, and expresses the relation between the function, $$\chi$$, and the component, $$\gamma$$, of the magnetic force resolved normal to the disk.

Let< $$Q$$ be the potential, at any point on the positive side of the disk, due to imaginary matter distributed over the disk with the surface-density $$\chi$$.

At the positive surface of the disk Rh

Hence the first member of equation (14) becomes Rh

But since $$Q$$ satisfies Laplace s equation at all points external to the disk, Rhand equation (14) becomes Rh

Again, since $$Q$$ is the potential due to the distribution $$\chi$$, the potential due to the distribution $$\phi$$, or $$\frac{d\chi}{d\theta}$$, will be $$\frac{dQ}{d\theta}$$. From this we obtain for the magnetic potential due to the currents in the disk, Rh