If [sigma] is the surface density and dS an element of surface, then [int][sigma]dS is the whole charge, and hence ½ [int] V[sigma]dS is the expression for the energy of charge of a conductor. We can deduce a remarkable expression for the energy stored up in an electric field containing electrified bodies as follows:[19] Let V denote the potential at any point in the field. Consider the integral _ _ _ _ _ 1 / / / | /dV\² /dV\² /dV\² | W = ----- | | | |(----) + (----) + (----) | dx dy dz. (21) 8[pi]_/_/_/ |_\dx/ \dy/ \dz/ _| where the integration extends throughout the whole space unoccupied by conductors. We have by partial integration _ _ _ _ _ _ _ _ / / / /dV\² / / dV / / / d²V | | |(----) dx dy dz = | | V -- dy dz - | | | V --- dx dy dz, _/_/_/ \dx/ _/_/ dx _/_/_/ dx² and two similar equations in y and z. Hence _ _ _ _ _ 1 / / / | /dV\² /dV\² /dV\² | ----- | | | | (----) + (----) + (----) | dx dy dz = 8[pi]_/_/_/ |_ \dx/ \dy/ \dz/ _| _ _ _ _ _ 1 / / dV 1 / / / ----- | | V -- dS - ----- | | | V[nabla]V dx dy dz (22) 8[pi]_/_/ dn 8[pi]_/_/_/ where dV/dn means differentiation along the normal, and [nabla] stands d² d² d² for the operator --- + --- + ---. Let E be the resultant electric force dx² dy² dz² at any point in the field. Then bearing in mind that [sigma] = (¼[pi])dV/dn, and [rho] = -(¼[pi])[nabla]V, we have finally _ _ _ _ _ _ _ _ 1 / / / 1 / / 1 / / / ----- | | | E²dv = -- | | V[sigma]dS + -- | | | V[rho]dv. 8[pi] _/_/_/ 2 _/_/ 2 _/_/_/ The first term on the right hand side expresses the energy of the surface electrification of the conductors in the field, and the second the energy of volume density (if any). Accordingly the term on the left hand side gives us the whole energy in the field. Suppose that the dielectric has a constant K, then we must multiply both sides by K and the expression for the energy per unit of volume of the field is equivalent to ½DE where D is the displacement or polarization in the dielectric. Furthermore it can be shown by the application of the calculus of variations that the condition for a minimum value of the function W, is that [nabla]V = 0. Hence that distribution of potential which is necessary to satisfy Laplace's equation is also one which makes the potential energy a minimum and therefore the energy stable. Thus the actual distribution of electricity on the conductor in the field is not merely a stable distribution, it is _the only_ possible stable distribution.