Problem 3.80

An infinite plane of uniform dielectric with permittivity 8 is uniformly charged with extraneous charge of space density \(\rho\). The thickness of the plate is equal to \(2 d\). Find: (a) the magnitude of the electric field strength and the potential as functions of distance \(l\) from the middle point of the plane (where the potential is assumed to be equal to zero); having chosen the \(x\) coordinate axis perpendicular to the plate, draw the approximate plots of the projection \(E_{x}(x)\) of the vector \(\mathbf{E}\) and the potential \(\varphi(x)\) (b) the surface and space densities of the bound charge.

\begin{aligned} &\text { 3.80. (a) } E=\left\{\begin{array}{l} p l / \varepsilon \varepsilon_{0} \text { for } l<d, \\ \rho d / \varepsilon_{0} \text { for } l>d, \end{array} \varphi=\left\{\begin{array}{l} -\rho l^{2} / 2 \varepsilon \varepsilon_{0} \text { for } l \leqslant d, \\ -(d / 2 \varepsilon+l-d) \rho d / \varepsilon_{0} \end{array}\right.\right.\\ &\text { for } l \geqslant d\\ &\text { The plots } E_{x}(x) \text { and } \varphi(x) \text { are shown in Fig. 21. (b) } \sigma^{\prime}=\rho d(\varepsilon-1) / \varepsilon\\ &\rho^{\prime}=-\rho(\varepsilon-1) / \varepsilon \end{aligned}

Conductors and Dielectrics in an Electric Field