Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/302

Rh Let the coordinates of the pole at the time $$t$$ be

The coordinates of the image of the pole formed at the time $$t - \tau$$ are

and if $$r$$ is the distance of this image from the point ($$x$$, $$y$$, $$z$$),

To obtain the potential due to the trail of images we have to calculate

If we write

the value of $$r$$ in this expression being found by making $$\tau = 0$$.

Differentiating this expression with respect to $$t$$, and putting $$t = 0$$, we obtain the magnetic potential due to the trail of images,

By differentiating this expression with respect to $$x$$ or $$z$$, we obtain the components parallel to $$x$$ or $$z$$ respectively of the magnetic force at any point, and by putting $$x = 0$$, $$z = c$$, and $$r = 2c$$ in these expressions, we obtain the following values of the components of the force acting on the moving pole itself,

665.] In these expressions we must remember that the motion is supposed to have been going on for an infinite time before the time considered. Hence we must not take $$\mathfrak{w}$$ a positive quantity, for in that case the pole must have passed through the sheet within a finite time.

If we make $$\mathfrak{u} = 0$$, and $$\mathfrak{w}$$ negative, $$X = 0$$, and or the pole as it approaches the sheet is repelled from it.

If we make $$\mathfrak{w} = 0$$, we find $$Q^2 = \mathfrak{u}^2 + R^2$$,