Energie geassocieerd met een ladingsverdeling

Consider a continuous charge distribution enclosed in a volume. The total charge inside an infinitesimal charge element can be written in terms of the volume charge density.

The work done or the energy stored in this configuration of continuous charge distribution is given by the integration of volume charge density and the corresponding potential.

Using Gauss's law and applying the product rule, the equation is rewritten in terms of electric field.

By applying the divergence theorem and replacing the potential gradient with the electric field, the energy stored is expressed as the sum of the surface and volume integrals.

Even if the integration volume is extended, the flux remains the same over any large distance, as the charge density in the extra volume is zero. However, the surface integral decreases inversely with distance, increasing the volume integral to conserve the total energy.

Eventually, integrating all over the space makes the surface integral zero, implying that the total energy stored can solely be calculated from the magnitude of the electric field.