87.] Since the electric force at any point outside a uni- formly electrified spherical surface is the same as if the electric charge of the surface had been concentrated at its centre, the potential due to the electrified surface must be, for points outside it, e ψ= , r where e is the whole charge of the surface, and r is the distance of the given point from the centre.

Let a be the radius of the spherical surface, then this expression for the potential is true as long as r is greater than a. At the surface, r is equal to a. The potential at the surface due to its own electrification is therefore ψa = e a [since there can be no discontinuity in the value of the potential between the surface and a point just outside it].

Within the surface there is no electromotive force, and the potential is therefore the same as at the surface for all points within the sphere.CAPACITY OF TWO CONCENTRIC SPHERES. 76 If the potential of the spherical surface is unity, then e = a, or the charge is numerically equal to the radius.

The electric capacity of a body in a given field is measured by the charge which raises its potential to unity. Hence the electric capacity of a conducting sphere placed in air at a considerable distance from any other conductor is numerically equal to the radius of the sphere.

If by means of an electrometer we can measure the potential of the sphere, we can ascertain its charge by multiplying this potential by the radius of the sphere. This method of measuring a quantity of electricity was employed by Weber and Kohlrausch in their determination of the ratio of the unit employed in electromagnetic to that employed in electrostatic researches. Since there is no electric force within a uniformly electrified sphere the potential within the e sphere is constant and equal to . a 88.] We are now able to complete the theory of the electrification of two concentric spherical surfaces.

Let a spherical conductor of radius a be insulated within a hollow conduct- ing vessel, the internal surface of which is a sphere of radius b concentric with the inner sphere. Let the charge on the inner sphere be e, then, as we have already seen, the charge on the interior surface of the vessel will be −e. At any point outside both spherical surfaces and distant r from the centre the e electric potential due to the inner sphere will be , and that due to the outer r −e sphere will be . Since these two quantities are numerically equal, but of r opposite sign, they destroy each other, and the potential at every point for which r is greater than b is zero. Between the two spherical surfaces, at a point distant r from the centre, e the potential due to the inner sphere is , and that due to the outer sphere is r −e 1 1 . Hence the whole potential in this intermediate space is e ( − ). b r bLEYDEN JAR. 77 At the surface of the inner sphere r = a, so that the potential of the inner 1 1 sphere is e ( − ). a b The potential at all points within the inner sphere is uniform and equal to 1 1 e ( − ). a b The capacity of the inner sphere is numerically equal to the value of e when the potential is made equal to unity. In this case e= 1

ab , b−a 1 1 − a b or, the capacity of a sphere insulated within a concentric spherical surface is a fourth proportional to the distances (b − a) between the surfaces and radii (a, b) of the surfaces.

By diminishing the interval, b−a, between the surfaces, the capacity of the system may be made very great without making use of very large spheres. This example may serve to illustrate the principle of the Leyden jar, which consists of two metallic surfaces separated by insulating material. The smaller the distance between the surfaces and the greater the area of the surfaces, the greater the capacity of the jar. Hence, if an electrical machine which can charge a body up to a given po- tential is employed to charge a Leyden jar, one surface of which is connected with the earth, it will, if worked long enough, communicate a much greater charge to the jar than it would to a very large insulated body placed at a great distance from any other conductor. The capacity of the jar, however, depends on the nature of the dielectric which is between the two metallic surfaces as well as on its thickness and area. See Art. 131 et sqq.