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 $q$ .

\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}

Conductors and Dielectrics in an Electric Field