Griffiths: 2.31 [2 points], 2.32 [5 points], 2.34 [5 points].

E1 [8 points]: Consider an infinite coaxial cable, i.e. two concentric cylinders, which extend along the z axis. The inner surface has radius a and the outer has radius b. Both the inner and the outer surfaces have uniform surface charge/area densities, sigma _{inner} and sigma _{outer}, respectively. You can figure these out from the given information that the inner surface (radius a) has total charge per unit length along the z axis of lambda, while the outer surface (radius b) has total charge per unit length along the z axis of - lambda. (So that the entire assembly is neutral, and lambda has units of C/m.) (part a) Find the electric field in all regions. (part b) Find the scalar potential (V or phi) in all regions. (part c) Find the stored potential energy per unit length along the z axis, U/L, using charge density and the scalar potential. (part d) Find the stored potential energy per unit length along the z axis, U/L, using the volume integral of the square of the electric field.

E2 [2 points]: Same setup as the previous problem. (part a) Evaluate the actual energy, in Joules, stored in a 2m length of the coaxial cable when a = 1cm, b=2cm, and lambda is 1C per meter. (part b) How much work (again in Joules) is required to pull an electron from the inner cylinder (at radius a=1cm) to the outer one?

E3 [8 points]: You are given that the scalar potential is phi = A(r/R) cos (theta) for r less than R, and phi=A (R/r)^2 cos(theta) for r greater than R. Here A is a constant, R is a constant radius in spherical coordinates, r is the radial coordinate of spherical coordinates, and theta is the angle from the z axis. (part a) Find the electric field everywhere. (part b) Find the charge density rho everywhere and the surface density sigma everywhere. (part c) Find the total energy U stored by doing the volume integral over all space of the square of the electric field. (part d) Find the total energy U by doing the appropriate integral of rho or sigma times the potential phi.