Hence the potential at the surface of the sphere, and therefore the potential of the sphere, is Q/R, where R is the radius of the sphere in centimetres. The quantity of electricity which must be given to the sphere to raise it to unit potential is therefore R electrostatic units. The capacity of a conductor is defined to be the charge required to raise its potential to unity, all other charged conductors being at an infinite distance. This capacity is then a function of the geometrical dimensions of the conductor, and can be mathematically determined in certain cases. Since the potential of a small charge of electricity dQ at a distance r is equal to dQ/r, and since the potential of all parts of a conductor is the same in those cases in which the distribution of surface density of electrification is uniform or symmetrical with respect to some point or axis in the conductor, we can calculate the potential by simply summing up terms like [sigma]dS/r, where dS is an element of surface, [sigma] the surface density of electricity on it, and r the distance from the symmetrical centre. The capacity is then obtained as the quotient of the whole charge by this potential. Thus the distribution of electricity on a sphere in free space must be uniform, and all parts of the charge are at an equal distance R from the centre. Accordingly the potential _at_ the centre is Q/R. But this must be the potential _of_ the sphere, since all parts are at the same potential V. Since the capacity C is the ratio of charge to potential, the capacity of the sphere in free space is Q/V = R, or is numerically the same as its radius reckoned in centimetres. Capacity of a thin rod. We can thus easily calculate the capacity of a long thin wire like a telegraph wire far removed from the earth, as follows: Let 2r be the diameter of the wire, l its length, and [sigma] the uniform surface electric density. Then consider a thin annulus of the wire of width dx; the charge on it is equal to 2[pi]r[sigma]/dx units, and the potential V at a point on the axis at a distance x from the annulus due to this elementary charge is _l/2 / 2[pi]r[sigma] V = 2 | ---------------dx = 4[pi]r[sigma] {log_e (½l + [root][r² + ¼l²]) - log_e^ r}. _/ [root](r² + x²) 0 If, then, r is small compared with l, we have V = 4[pi]r[sigma]log_e l/r. But the charge is Q = 2[pi]r[sigma], and therefore the capacity of the thin wire is given by C = ½ log_e l/r (2). Potential of an ellipsoid.