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Conductors and Dielectrics in an Electric Field

Problem 3.91

A point charge \(q\) is located on the plane dividing vacuum and infinite uniform isotropic dielectric with permittivity \(\varepsilon .\) Find the moduli of the vectors \(D\) and \(E\) as well as the potential \(\varphi\) as functions of distance \(r\) from the charge \(q\).

Reveal Answer
 3.91. D={q/2π(1+ε)r2 in vacuum, εq/2π(1+ε)r2 in dielectric; E=q/2πε0(1+ε)r2φ=q/2πε0(1+ε)r} both in vacuum and in dielectric. \begin{aligned} &\text { 3.91. } D=\left\{\begin{array}{l} q / 2 \pi(1+\varepsilon) r^{2} \text { in vacuum, } \\ \varepsilon q / 2 \pi(1+\varepsilon) r^{2} \text { in dielectric; } \end{array}\right.\\ &\left.\begin{array}{l} E=q / 2 \pi \varepsilon_{0}(1+\varepsilon) r^{2} \\ \varphi=q / 2 \pi \varepsilon_{0}(1+\varepsilon) r \end{array}\right\} \text { both in vacuum and in dielectric. } \end{aligned}
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