Consider a dielectric sphere with radius R and dielectric constant K1, surrounded by another dielectric sphere with outer radius 2R and inner radius R, and dielectric constant K2. The inner sphere has a charge Q distributed uniformly in the radius R. i) Find the Displacement vector and electric field everwhere (). 2R D. 3 ii) Find the polarization surface charge density on the inner surface D of the dielectric. R +Q iii) Find the potential at a point located at r= R/2. iv) How would the previous results differ if the inner sphere were replaced by a conducting sphere of radius R and charge Q. K2

A long cylinder of radius R is made of the dielectric material with Er=4. a) What is the permittivity of the material, in F/m? b) What is the electric displacement inside the cylinder?

A parallel plate capacitor with a charge of +/-177nC has square plates length L=20.0cm across on each side. The plates are separated by a distance d=10.0cm. A positive +4.0microC point charge is held in place halfway between each plate on the right-most edge of the capacitor. While solving this problem ignore any distortion of the capacitor's electric field along its edges. A) What is the magnitude and direction of the capacitor's electric field at point P, which is midway between the plates and on the left-most edge of the capacitor B) What is the magnitude and direction of the electric field due to the +4.0 microC point charge at point P. C) What is the magnitude and direction of the net electric field at point P

Another special case turns out to be of considerable interest. Let's see what the electric field is like near the center of a uniformly charged disk (x small, but not zero, for the reasons just given). Suppose that we are so close to the disk that x is very small (x > x). QIA Show that near the disk E- R

1. The electric field in a region of space near the origin is given by () E(x, y, z) = Eo a (a) Evaluate the curl V x E(x, y, z) (b) Setting V(0, 0, 0) = 0, select a path from (0,0, 0) to (x, y,0) and compute V (x, y, 0).

When you polarize a neutral dielectric, charge moves a bit, but the total charge remains zero. This fact should be reflected in the bound charges σ and ρ. Prove from σb = P. n and Pb = −√. P that the total bound charge vanishes.

4. A cylindrical dielectric of permittivity &, radius R, and length h, is centered about the z-axis. Assume this Cylinder has a uniform electric polarization along its axis given by P = Poz. a) What are the polarization charge densities on its surfaces? b) What is the electric displacement vector D anywhere on its axis? c) What is the electric field vector E anywhere on its axis?

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