Page:Elementary Text-book of Physics (Anthony, 1897).djvu/305

§ 257] which it occupies. This necessitates a negative charge distributed over the vrhole cylinder. In other words, the earth and the cylinder may be considered as forming one conductor charged by induction, in which the neutral line is not within the cylinder.

If the ground connection be broken the electrical relations are not disturbed. If the cylinder be now removed to a region of lower potential against the attraction of the sphere, work will be done against electrical forces, which reappears as electrical energy. The potential of the cylinder is lowered, and, if it be again connected with the earth, work will be done by a flow of electricity to it.

In § 57 it was shown that the forces on the opposite side of a sheet, in which the surface density is $$\sigma,$$ differ by $$4\pi \sigma.$$ Now the force within an electrified conductor vanishes, so that the force at a point just outside it is given by $$4\pi \sigma.$$

The pressure outwards on the surface of an electrified conductor due to the repulsion of the various parts of the charge for one another is equal to $$2\pi \sigma^2 .$$ For the force just outside the conductor, which is equal to $$4\pi \sigma,$$ is due to that part of the conductor immediately under the point considered, which may be considered plane, and to the rest of the conductor. The force due to the plane part is (§ 57) equal to $$2\pi \sigma,$$ and that due to the rest of the conductor is therefore also $$2\pi \sigma.$$ Select any small portion of the surface of the conductor of area $$a.$$ The force on unit quantity acting outward from the conductor at a point in that area due to the charge of the rest of the conductor is $$2\pi \sigma.$$ This force acts on every unit of charge on the area. The force on the area acting outwards is then $$2\pi a \sigma^2 ,$$ or the pressure at a point in the area referred to unit of area is $$2 \pi \sigma^2 .$$ This quantity is often called the electric pressure.

257. Capacity.—The electrical capacity of a conductor is defined to be the charge which a conductor must receive to raise it from zero to unit potential, while all other conductors in the field are kept at zero potential. This charge varies for any one conductor in a way which cannot be always definitely determined, depending upon the medium in which the conductor is immersed and the