Page:Elementary Text-book of Physics (Anthony, 1897).djvu/308

294 divide numerator and denominator by $$R'$$ and write $$C = \frac{R}{1 - \frac{R}{R'}}\cdot$$ Now, if $$R'$$ be greater than $$R$$ by an infinitesimal, the fraction $$\frac{R}{R'}$$ is less than unity by an infinitesimal, and the capacity of the accumulator is infinitely great. It becomes infinitely small if $$R$$ be diminished without limit. The presence of any finite charge at a point would require an infinite potential at that point, which is of course impossible. The existence of finite charges concentrated at points, which we have assumed sometimes in order to more conveniently state certain laws, is therefore purely imaginary. If electricity is distributed in space, it is distributed like a fluid, a finite quantity of which never exists at a point.

If $$R'$$ increase without limit, $$C$$ becomes more and more nearly equal to $$R.$$ Suppose the inner sphere to be surrounded not by the outer sphere but by conductors disposed at unequal distances, the nearest of which is still at a distance $$R'$$ so great that $$\frac{R}{R'}$$ may be neglected in comparison with unity. Then if the nearest conductor were a portion of a sphere of radius $$R'$$ concentric with the inner sphere, the capacity of the inner sphere would be approximately $$R.$$ And this capacity is evidently not less than that which would be due to any arrangement of conductors at distances more remote than $$R'.$$ Therefore the capacity of a sphere removed from other conductors by distances very great in comparison with the radius of the sphere is equal to its radius $$R.$$ This value $$R$$ is often called the capacity of a freely electrified sphere. Strictly speaking, a freely electrified conductor cannot exist; the term is, however, a convenient one to represent a conductor remote from all other conductors.

A common form of condenser consists of two flat conducting disks of equal area, placed parallel and opposite one another. The capacity of such a condenser may be calculated from the capacity of the spherical condenser already discussed. Let $$d$$ represent the