**Problem 1** [10 points]
Consider a grounded, conducting, spherical shell of outer radius b and inner radius a.
Using the method of images, dicuss the problem of a point charge q inside the shell, i.e. at a distance r<a from the center. Find

a) the potential inside the sphere;

b) the induced surface charge density on the inner surface of the shell at r=a; what is the total induced charge?

c) the magnitude and direction of the force acting on q. Does q get pushed towards the center, or away from the center?

d) Is there any change in the solution if the sphere is kept at a fixed potential φ_{o}? If the sphere has a fixed total charge Q?

**Problem 2** [10 points]
A line charge λ is placed parallel to, and a distance R away from, the axis of a conducting cylinder of radius b that is held at fixed voltage such that the potential vanishes at infinity. Find

a) the magnitude and position of the image charge(s)

b) the potential at any point (expressed in polar coordinates with the origin at the axis of the cylinder and the direction from the origin to the line charge as the x axis)

c) the induced surface charge density, and plot it as a function of angle for R/b = 2 and 4 in units of λ/2πb

d) the force on the line charge per unit length

**Problem 3** [10 points]
Consider an infinitely long grounded metal cylinder, of radius R, placed at right angles to an applied uniform electric field **E**_{o} (for example, the axis of the cylinder is along the z direction, while **E**_{o} is in the x direction).

a) Find the potential outside the cylinder.

b) Find the surface charge density σ induced on the surface of the cylinder.

**Problem 4** [10 points]
Two concentric spherical shells of radii R_{1} and R_{2}, with R_{1}<R_{2}, are fixed with the following values of the electrostatic potential:

φ(R_{1}, θ) = φ_{1}cosθ and
φ(R_{2}, θ) = φ_{2}

where φ_{1} and φ_{2} are constants and θ is the usual azimuthal spherical angle. Find the electrostatic potential for:

a) r<R_{1}, inside the inner shell

b) r>R_{2}, outside the outer shell

c) R_{1}<r<R_{2}, between the two shells

d) Find the surface charge density σ(θ) on each of the two shells.