Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/101

60.] electric polarization, the outside would gradually become charged in such a manner as to neutralize the action of the internal electrification for all points outside the body. This external superficial charge could not be detected by any of the ordinary tests, and could not be removed by any of the ordinary methods for discharging superficial electrification. The internal polarization of the substance would therefore never be discovered unless by some means, such as change of temperature, the amount of the internal polarization could be increased or diminished. The external electrification would then be no longer capable of neutralizing the external effect of the internal polarization, and an apparent electrification would be observed, as in the case of tourmaline.

If a charge $$e$$ is uniformly distributed over the surface of a sphere, the resultant force at any point of the medium surrounding the sphere is numerically equal to the charge e divided by the square of the distance from the centre of the sphere. This resultant force, according to our theory, is accompanied by a displacement of electricity in a direction outwards from the sphere.

If we now draw a concentric spherical surface of radius $$r$$, the whole displacement, $$E$$, through this surface will be proportional to the resultant force multiplied by the area of the spherical surface. But the resultant force is directly as the charge e and inversely as the square of the radius, while the area of the surface is directly as the square of the radius.

Hence the whole displacement, $$E$$, is proportional to the charge $$e$$, and is independent of the radius.

To determine the ratio between the charge, and the quantity of electricity, $$E$$, displaced outwards through the spherical surface, let us consider the work done upon the medium in the region between two concentric spherical surfaces, while the displacement is increased from $$E$$ to $$E+\delta E$$. If $$V_1$$ and $$V_2$$ denote the potentials at the inner and the outer of these surfaces respectively, the electromotive force by which the additional displacement is produced is $$V_l-V_2$$, so that the work spent in augmenting the displacement is $$(V_1-V_2)\delta E$$.

If we now make the inner surface coincide with that of the electrified sphere, and make the radius of the other infinite, $$V_1$$ becomes $$V$$, the potential of the sphere, and $$V_2$$ becomes zero, so that the whole work done in the surrounding medium is $$V\delta E$$.

But by the ordinary theory, the work done in augmenting the charge is $$V\delta e$$, and if this is spent, as we suppose, in augmenting