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Conductors and Dielectrics in an Electric Field

Problem 3.74

A point charge qq is located at the centre of a ball made of uniform isotropic dielectric with permittivity ε.\varepsilon . Find the polarization P\mathbf{P} as a function of the radius vector r\mathbf{r} relative to the centre of the system, as well as the charge q′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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