All Problems Conductors and Dielectrics in an Electric Field
Problem 3.81
Extraneous charges are uniformly distributed with space density \(\rho>0\) over a ball of radius \(R\) made of uniform isotropic dielectric with permittivity \(\varepsilon\). Find: (a) the magnitude of the electric field strength as a function of distance \(r\) from the centre of the ball; draw the approximate plots \(E(r)\) and \(\varphi(r)\) (b) the space and surface densities of the bound charges.
3.81. (a) \(E=\left\{\begin{array}{l}\rho r / 3 \varepsilon_{0} \varepsilon \text { for } r<R \\ \rho R^{3} / 3 \varepsilon_{0} r^{2} \text { for } r>R ;\end{array}\right.\) (b) \(\rho^{\prime}=-\rho(\varepsilon-1) / \varepsilon, \sigma^{\prime}=\rho R(\varepsilon-1) / 3 \varepsilon .\) See Fig. 22 .