The field potential in a certain region of space depends only on the $x$-coordinate, as $\mathrm{\phi }=-a{x}^3+b$, where $a$ and $b$ are constants. Find the distribution of the space charge $\rho \left(x\right)$.

The space between the plates of a parallel-plate capacitor is filled, consecutively, with two dielectric layers $1$ and $2$, having thicknesses $d_1$ and $d_2$ and permittivities ${\epsilon }_1$ and ${\epsilon }_2$, respectively. The area of each plate is equal to $S$. Find:

(a) the capacitance of the capacitor;

(b) the density $\sigma \text{'}$ of the bound charges on the boundary plane, if the voltage across the capacitor equals $V$, and the electric field is directed from layer $1$ to layer $2$.

The gap between the plates of a parallel-plate capacitor is filled with an isotropic dielectric, whose permittivity $\epsilon$ varies linearly from ${\epsilon }_1$ to ${\epsilon }_2 \left({\epsilon }_2>{\epsilon }_1\right)$, in the direction perpendicular to the plates. The area of each plate equals $S$, the separation between the plates is equal to $d$. Find:

(a) the capacitance of the capacitor;

(b) the space density of the bound charges as a function of $\epsilon$, if the charge of the capacitor is $q$ and the field $\overrightarrow{\mathit{E}}$ in it, is directed towards the growing $\epsilon$ values.

Find the capacitance of a spherical capacitor whose electrodes have radii $R_1$ and $R_2 (R_2>R_1)$, and which is filled with isotropic dielectric whose permittivity varies as $\epsilon =\frac{a}{r}$, where $a$ is a constant and $r$ is the distance from the centre of the capacitor.