Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/408

366 the spheres cause the resistance of the compound medium to be different in different directions.

Application of the Principle of Images.

315.] Let us take as an example the case of two media separated by a plane surface, and let us suppose that there is a source $$S$$ of electricity at a distance a from the plane surface in the first medium, the quantity of electricity flowing from the source in unit of time being $$S.$$

If the first medium had been infinitely extended the current at any point $$P$$ would have been in the direction $$SP,$$ and the potential at $$P$$ would have been $$\frac$$ where $$E = \frac$$ and $$ r_1= SP.$$

In the actual case the conditions may be satisfied by taking a point $$I,$$ the image of $$S$$ in the second medium, such that $$IS$$ is normal to the plane of separation and is bisected by it. Let $$r_2$$ be the distance of any point from $$I,$$ then at the surface of separation

Let the potential $$V_1$$ at any point in the first medium be that due to a quantity of electricity $$E$$ placed at $$S,$$ together with an imaginary quantity $$E_2$$ at $$I,$$ and let the potential $$V_2$$ at any point of the second medium be that due to an imaginary quantity $$E_1$$ at $$S,$$ then if the superficial condition $$ V_1 = V_2 $$ gives and the condition

The potential in the first medium is therefore the same as would be produced in air by a charge $$E$$ placed at $$S,$$ and a charge $$E_1$$ at $$I$$ on the electrostatic theory, and the potential in the second medium is the same as that which would be produced in air by a charge $$E_1$$ at $$S.$$