Skip to main content Contents Index Prev Up Next \(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
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\)
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, 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 \(V\text{.}\) 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.
