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Category:
Problem
A solid sphere of radius \(R\) carries a charge density \(\rho(\vec{r})\). Show that the average of the electric field inside the sphere is \[\vec{E}= - \frac{1}{4\pi\epsilon_0} \frac{\vec{p}}{R^3},\] where \(\vec{p}\) is the total dipole moment of the sphere.
Solution
For any charge distribution \(\rho\), the electric field at a point \(\vec{r}\) is given by \begin{equation} \vec{E} = \frac{1}{4\pi\epsilon_0} \iiint \frac{\rho(\vec{r}^\prime)(\vec{r}-\vec{r}^\prime)}{|\vec{r}-\vec{r}^\prime|^3} d^3 r^\prime. \end{equation} Therefore the average of the electric field over the sphere of radius \(R\) is given by
\begin{eqnarray}
\vec{E}_\text{av}
&=& \frac{3}{4\pi R^3} \iiint_{S_R} d^3r \vec{E}(\vec{r})\\
&=& \frac{1}{4\pi\epsilon_0} \frac{3}{4\pi R^3} \iiint_{S_R} d^3r \iiint
\frac{\rho(\vec{r}^\prime)(\vec{r}-\vec{r}^\prime)}{|\vec{r}-\vec{r}^\prime|^3}
d^3 r^\prime.\\
\end{eqnarray}
Exchanging the integrals over \(\vec{r}\) and \(\vec{r}^\prime\) we get
\begin{eqnarray}
\vec{E}_\text{av}
&=& \frac{1}{4\pi\epsilon_0} \frac{3}{4\pi R^3} \iiint d^3r \iiint_{S_R}
\frac{\rho(\vec{r}^\prime)(\vec{r}-\vec{r}^\prime)}{|\vec{r}-\vec{r}^\prime|^3}
d^3 r^\prime.\\
&=& \frac{1}{4\pi\epsilon_0} \frac{3}{4\pi R^3} \iiint d^3r^\prime \rho(\vec{r}^\prime)\iiint_{S_R}
\frac{(\vec{r}-\vec{r}^\prime)}{|\vec{r}-\vec{r}^\prime|^3}d^3r.\label{EQ07}
\end{eqnarray}
Suppose we have a uniformly charged sphere of radius \(R\) with some constant density \(\rho_0\), the electric field at a point \(\vec{r}_P\) inside the sphere will be
\[\frac{1}{4\pi\epsilon_0} \iiint_{S_R}
\rho_0 \frac{(\vec{r}_P-\vec{r})}{|\vec{r}_P-\vec{r}|^3}
d^3r.\]
This answer is known and is given by \(\dfrac{Q}{3\epsilon_0}. \dfrac{\vec{r}_P}{R} \), where \(Q\) is the total charge \(Q=\frac{4}{3}\pi R^3 \rho_0\).
Therefore we know the integral
\begin{equation}
\frac{1}{4\pi\epsilon_0} \iiint_{S_R} \rho_0 \frac{(\vec{r}_P-\vec{r})}{|\vec{r}_P-\vec{r}|^3}
d^3r = \frac{Q}{3\epsilon_0}. \frac{\vec{r}_P}{R^3}.
\end{equation}
This equation with obvious replacements \(\rho_0\to1, Q \to \frac{4}{3}\pi R^3,
\vec{r}_P \to \vec{r}^{\,\prime}\) in \Eqref{EQ07} gives \begin{equation} \vec{E}_\text{av}= - \frac{1}{4\pi\epsilon_0} .\frac{\vec{p}}{R^3} \end{equation} where \(\vec{p}\) is the dipole moment of the sphere. We write the dipole moment \(\vec{p}= \frac{4}{3}\pi R^3 \vec{P}\), in terms of polarization per unit volume \(\vec{P}\): \[ \vec{E}_\text{av} = - \frac{1}{3\epsilon_0}\, \vec{P}.\]
