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Conductors and Dielectrics in an Electric Field

Problem 3.74

A point charge \(q\) is located at the centre of a ball made of uniform isotropic dielectric with permittivity \(\varepsilon .\) Find the polarization \(\mathbf{P}\) as a function of the radius vector \(\mathbf{r}\) relative to the centre of the system, as well as the charge \(q^{\prime}\) inside a sphere whose radius is less than the radius of the ball.

Reveal Answer
P=ε−1εq4πr3r,q′=−ε−1εq\begin{aligned} &\mathbf{P}=\frac{\varepsilon-1}{\varepsilon} \frac{q}{4 \pi r^{3}}\\ &\mathbf{r}, q^{\prime}=-\frac{\varepsilon-1}{\varepsilon} q \end{aligned}
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