[Solved/EM-05001] Electromagnetic Theory

Question:
A dielectric sphere contains a dipole \(\vec{p}_0\). Find the net dipole moment of the system.

Solution:
The dielectric gets polarized and the dipole and acquires a polarization \(\vec{P}\). The dipole moment of the dielectric is
\begin{eqnarray}
\vec{p} &=& \int_V \vec{P} d^3r   = \int_V \vec{D}-\epsilon_0 \vec{E} d^3 r\\      &=& \int_V\epsilon_0(\kappa-1) \vec{E} d^3r  = \epsilon_0(\kappa-1) \int_V\vec{E} d^3r \end{eqnarray}
For arbitrary charge distribution the average of electric field, \( \int_V\vec{E} d^3r\), over sphere is \(-\dfrac{\vec{p}_\text{tot}}{3\epsilon_0}\).  Hence we get

\[ \vec{p}= - \frac{(\kappa-1)}{3} \vec{p}_\text{tot}\]

using \(\vec{p}_\text{tot}= \vec{p}_0 + \vec{p}\), we get

\begin{eqnarray}\vec{p}_\text{tot} &=& \vec{p}_0 -\frac{(\kappa-1)}{3} \vec{p}_\text{tot}\\\text{or } \vec{p}_\text{tot} &=&\frac{3}{\kappa+2}  \vec{p}_0\end{eqnarray}

Zangwill