1. An isolated parallel-plate capacitor has charge +Q on the upper plate and -Q on the lower plate. If the gap between the plates is increased, does the potential difference between the plates increase, decrease, or stay the same? Why?
2. The potential in a charge-free region of space is given by V(x,y,z) = B(x2+y2+bz2), where B and b are constants. What must be the numerical value of b? Briefly explain your reasoning. (3 marks)
3. A sphere with radius R has a total charge q. If the potential inside the sphere is given by V(r) = q/8Π∈0R(3-r2/R2), show that the volume charge density inside the sphere must be uniform.
4. Problem 2.39 (two spherical cavities in a conductor)
5. A variable capacitor consists of two thin coaxial metal cylinders of radii a and b, with (ba) << a, free to move with respect to each other in the axial direction. The length of the cylinders is L, and the potential difference between the two cylinders is V. Initially, the inner cylinder (radius = a) is completely enclosed by the outer cylinder (radius = b). Using energy methods, find the magnitude and direction of the force on the inner cylinder when it is displaced along the axial direction by a small amount with respect to the outer one. Explain how this force arises.
6. Problem 3.10 (image charge for a charged wire above a ground plane)
7. Problem 3.39 (force on point charge between two grounded conducting planes)