91Ó°ÊÓ

There is a dielectric cube of side a,centered at the origin.

The cube carries a polarization $\stackrel{→}{p}=k\stackrel{→}{r}.$

Volume bound charge density

${\mathbf{p}}_{\mathbf{b}}\mathbf{=}\mathbf{-}\stackrel{\mathbf{→}}{\mathbf{∇}}\mathbf{.}\stackrel{\mathbf{→}}{\mathbf{P}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Here, $\stackrel{→}{p}$ is polarization.

Surface bound charge density

${\mathbf{σ}}_{\mathbf{b}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{P}}\mathbf{.}\hat{\mathbf{n}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$

Here, $\hat{n}$is the unit vector along the normal to the surface.

The polarization is

$$\begin{gathered} \stackrel{→}{P}=K\stackrel{→}{r} \\ =k(x\hat{x}+y\hat{y}+z\hat{z}) \end{gathered}$$

From equation (1), the volume bound charge density is

$$\begin{gathered} {\mathrm{ÒÏ}}_b=-\stackrel{→}{∇}.\left(k(x\hat{x}+y\hat{y}+z\hat{z})\right) \\ =-3k \end{gathered}$$

Total volume charge from the volume charge density is

$$\begin{gathered} Q_{\mathrm{ÒÏ}}={\mathrm{ÒÏ}}_b{a}^3 \\ =-3k{a}^3 \end{gathered}$$

Let the top surface be perpendicular to the z axis.

Thus,

$\hat{n}=\hat{z}$and $z=a/2$

Thus, from equation (2), the surface charge density on the top surface is

role="math" localid="1657775319786" $$\begin{gathered} {\mathrm{σ}}_b=k(x\hat{x}+y\hat{y}+z\hat{z}).\hat{z} \\ =kz \\ =\frac{ka}{2} \end{gathered}$$

The total charge on the top surface is

$$\begin{gathered} {\mathrm{σ}}_b={\mathrm{σ}}_b{a}^2 \\ =k{a}^3 \end{gathered}$$

The surface charges are equal on all the six surfaces. The total bound charge in the cube is then

$$\begin{gathered} Q_b=Q_{\mathrm{ÒÏ}}+6Q_{\mathrm{σ}} \\ =-3k{a}^3+6×\frac{k{a}^3}{2} \\ =0 \end{gathered}$$

Thus, the net bound charge in the cube is 0.