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Section 16.3 Electrostatic Energy from a Continuous Source

In Section 16.2, we found that the energy stored in a collection of discrete charges is

\begin{align*} U \amp= \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n\sum_{j=i+1}^n \frac{q_i q_j}{|\rr_i - \rr_j|}\\ \amp= \frac{1}{8\pi\epsilon_0} \sum_{i=1}^n\sum_{\stackrel{j=1}{j\ne i}}^n \frac{q_i q_j}{|\rr_i - \rr_j|} . \end{align*}

The advantage of the second expression, in which each term is double-counted, is that it can be rewritten in the form

\begin{gather*} U = \frac12 \sum q_i V(\rr_i) \end{gather*}

where \(V(\rr_i)\) is the potential at the location \(\rr_i\) of the \(i\)th charge due to all the other charges. This expression in turn generalizes naturally to a continuous charge distribution,  1  for which it becomes

\begin{gather*} U= \frac12 \int \rho V \,\dV \end{gather*}

We now have an expression for the energy contained in a charge distribution, expressed in terms of the charge density \(\rho\) and the potential \(V\text{.}\)  2  But each of these quantities can be expressed in terms of the electric field, since \(\EE=-\grad V\) and \(\grad\cdot\EE=\rho/\epsilon_0\) from Gauss' Law. We can therefore rewrite the energy in terms of the electric field alone. Starting from

\begin{equation*} \rho V = (\epsilon_0 \grad\cdot\EE) V \end{equation*}

we are reminded of the product rule for the divergence (see Appendix A.2), namely

\begin{align*} \grad\cdot(V \EE) \amp= \grad V \cdot\EE + V \,\grad\cdot\EE \end{align*}

which in turn suggests integrating by parts. Doing so yields

\begin{align*} U \amp= \frac{\epsilon_0}{2} \int (\grad\cdot\EE) \> V \,\dV\\ \amp= \frac{\epsilon_0}{2} \int \left( -\grad V\cdot\EE + \grad\cdot(V\EE) \right) \,\dV\\ \amp= \frac{\epsilon_0}{2} \left( \int |\EE|^2 \,\dV + \int V\,\EE\cdot d\AA\right)\\ \amp= \frac{\epsilon_0}{2} \int |\EE|^2 \,\dV \end{align*}

where we have used the Divergence Theorem to obtain the surface integral in the second last line, which evaluates to zero assuming that \(\EE\) obeys reasonable falloff conditions at infinity.

See the discussion in Griffiths about some subtleties in this limit
Our derivation here follows Griffiths, sec 2.5, p. 96–106.