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Conductors and Dielectrics in an Electric Field

Problem 3.81

Extraneous charges are uniformly distributed with space density ρ>0\rho>0 over a ball of radius RR made of uniform isotropic dielectric with permittivity ε\varepsilon. Find: (a) the magnitude of the electric field strength as a function of distance rr from the centre of the ball; draw the approximate plots E(r)E(r) and φ(r)\varphi(r) (b) the space and surface densities of the bound charges.

Reveal Answer
3.81. (a) E={ρr/3ε0ε for r<RρR3/3ε0r2 for r>R;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) ρ′=−ρ(ε−1)/ε,σ′=ρR(ε−1)/3ε.\rho^{\prime}=-\rho(\varepsilon-1) / \varepsilon, \sigma^{\prime}=\rho R(\varepsilon-1) / 3 \varepsilon . See Fig. 22 .
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