Note that when we deal with potential energy, we have the choice of setting the "origin", or "reference point". It is this choice that when dealing with gravitational potential energy, we may take the potential energy at infinity to be 0. The mathematics and physics wouldn't change if we had taken the potential energy at infinity to be 8, 1230, or -13120135091 J. So, when granted with this freedom, physicists choose to define the potential energy of U in the above manner to simplify calculations, as well as to present a notational similarity with the answer to 4.1.

4.2 Determine the Electric Field and Potential at an arbitrary point due to the dipole.

5. Conductors

It is important in most electrostatic questions to note if the object in question is a conductor or insulator, because both materials have very different properties and thus implications in questions.
  1. The one most important property of a conductor is that it allows for free movement of electrons and therefore charges.
  2. If the conductor has a net charge, all the charges will move to the surface. This is a necessary consequence of nature trying to minimise the potential energy where possible.

5.1 Knowing the second property of conductor mentioned above, what is the electric field inside the conductor?

5.2 Take any two points connected by a conductor. Is the point nearer to the centre of higher, lower, or equal potential? Why?

5.3 From 5.2, are you able to determine the direction of the electric field just outside the conductor?

5.4 Consider two spherical conducting shells of similar surface charge densities. Which shell will have the lower potential on the surface? From 5.2, are you able to tell which direction the charges will flow to when the two surfaces touch?

5.5 Apply Gauss' Law to a infinitesimally small cylinder on the surface of the conductor. What relationship do you get between the surface charge and the electric field? 

5.6 A metal sphere, of radius R and cut into two along a plane whose minimum distance from the sphere's centre is h, is uniformly charged by a total electric charge Q. What force is necessary to hold the two parts of the sphere together?

6. Method of Images

Consider a case where a point charge is located at (0, 0, d) above an infinite grounded conducting plane along the x-y plane, and we are asked to determine the potential above the conducting plane. Clearly it isn't as straightforward as applying \(V\ =\ \frac{kq}{r}\), as the point charge will induce a certain charge in the grounded conductor.
Without going into the details, a combination of the corollary of the first uniqueness theorem and Poisson's Equation allows for the problem solving technique of the method of images, where all conductors are assumed to be "mirrors", and we are able to place "image charges" of a negative sign at the location where the image will be.
In the above question, this will be a charge of -q at (0, 0, -d). 

6.1 With the above information, deduce the potential function for z>0.

6.2 Based on the potential function and problem 5.5, deduce the total induced charge on the conductor surface.

Note two things with regards to the method of images:
  1. The mathematics of the derivation allows it to be applied this straightforwardly only in the case of infinite planes and spheres.
  2. In the above example, at z < 0, there is no potential. You are not allowed to put image charges where you are calculating 
Another point to be cautious about is the potential energy involved. In the two point charges case the work needed to create the system is \(-\frac{kq^2}{2d}\). For a single charge and conducting plane, however the energy required is half. There are a few ways to think about this:

6.3 Derive the result for the method of images on a grounded spherical surface.

6.4 A point charge is at rest inside a thin metallic spherical grounded shell, but is not at its centre. What is the force acting on the charge?

7. Capacitors

We define capacitance of a device as: