Physics 2420 In-Class Problems: Electric Fields in Materials

1. A thin dielectric rod of cross section A extends along the x-axis from x = 0 to x = L. The polarization of the rod is along its length, and is given by P_x = ax² + b. Find the volume density of polarization charge and the surface polarization charge on each end. Show explicitly that the total bound charge vanishes in this case.
2. A dielectric cube of side L has a radial polarization given by P = Ar, where A is a constant, and r = ix + jy + kz. The origin of the coordinates is at the centre of the cube. Find all bound charge densities, and show explicitly that the total bound charge vanishes.
3. A dielectric rod in the shape of a right circular cylinder of length L and radius R is polarized in the direction of its length. The polarization is uniform and of magnitude P. Calculate the electric field resulting from the polarization at a point on the axis of the rod outside the cylinder.
4. Two semi-infinite blocks of dielectric are placed almost in contact so that there exists a narrow gap of constant separation between them. The polarization P is constant throughout all the dielectric material and it makes an angle θ with the normal to the planes bounding the gap. Determine the electric field in the gap.
5. A metal sphere of radius a is surrounded by a thick dielectric shell of inner radius a, outer radius b, and dielectric constant ε. The metal sphere carries a free charge Q. There is no free charge on or in the dielectric. Determine the electric field everywhere. Determine the polarization of the dielectric and determine the bound charge distributions. Analyze this system thoroughly, determining the potential of the metal sphere and the distribution of bound charge.

Questions?mike.coombes@kpu.ca