Question

3. [2 points] A rectangular parallelepiped with dimensions a, b, c is filled with a dielectric in which the polarization is given by kr, where k is a constant with the appropriate units, and r is the position vector of a point in the dielectric with respect to the center of the parallelepiped (i.e., the center of the parallelepiped is at the origin, and the faces are parallel to the Cartesian axes). Calculate the bound surface charge density on the top and bottom sides of the parallelepiped.

Answer 1

Dimensions of rectangular parallelepiped is a*b*c.

So the distance between the origin and top or bottom surface is c/2.

Here $\Vec{P} =k \Vec{r}$

So the surface charge density on both the surface is given by

$\sigma _{b} =\Vec{P} . \hat {n} = \frac{kc}{2} \hat{r}. \hat{r}= \frac{kc}{2}$

This is the surface charge density on both top and bottom surface each.