Charge images

      While working a problem given at IPHO 2010, which can be found at this link:http://ipho.phy.ntnu.edu.tw/problems-and-solutions/2010/IPhO_2010_Theo_Problem%201.pdf

I gained a great interest in the charge image method.

The problem asks what's the attraction force between the charge q and the sphere connected to a 0 potential. Following the relatively laborious solution, I started wondering what happened if the sphere was connected to a V potential, or at no potential at all. This is exactly what I will study in this post.

First, let's see what happens if we have two charges, q and -q at a distance d from each other, in an infinite empty space. The equipotential lines and field lines look like this :

You can observe that the equipotential ares have the shapes of " elongated spheres" with one exception (right in the middle is an infinite plane of 0 potential)
The trick is that by replacing charge q with a conductor charged with q as well that has the exact shape of any of the equipotential areas, the exterior field and potential distribution is maintained. O...the conductor must be either initially charged with q, or connected to a potential equal to the potential of the equipotential surface it replaced. q is thus called the "image charge" of the conductor.
For example I'll give you the example when I replace the infinite plane with a conductor. As I said, there are two options: either to charge the conductor with q from the beginning, either to connect it to a zero potential( which is the potential of the area it replaces). In either case, the final result is the same: the conductor is charged with q, has 0 potential and the exterior distribution of field is the same. The attraction force between the plane (or any other conductor) is equal to the attraction force between the initial charges,since the field distribution is kept constant.

So, in order to solve the case with the sphere and the charge q, we must find the charge q', that, in the presence of q, creates an equipotential sphere of 0 potential and radius R.

If you write the equations for the potential in P and solve the equations in the ZOY plane, you will obtain that the equipotential zone of 0 potential is a circle, with the equation  Z^2+ (Y-OO')^2=R^2  (that means that in XYZO space, the zone is a sphere)

You can solve the IPHO problem by finding  the equation of the circle I mentioned previously. Thus you will have the radius and OO' distance. R=-q'd/q; OO'=d q'^2/q; this distribution is possible only if q'/q<0!!!, so pay attention to the signs. In the problem , the radius R and charge q are given, so you can easily find q' and OO'. Since the field distribution is the same outside the sphere( the field is 0 inside), the attraction force between the charge and the sphere is equal to the attraction force between the 2 charges.

Now,what happens if the sphere is connected to Ve? You must find a distribution of charges that gives you desired equipotential  surface,of Ve potential. I said distribution! Nobody said that there must be just one image charge! There can be more. 

If you place another charge right in the middle of the sphere from the previous drawing, you will obtain the equipotential spherical surface of Ve potential I demanded!

the condition that must be respected is simple:


 The electrostatic force between the exterior charge and the sphere will equal the sum of the electrostatic forces between the exterior charge and the two image charges!

What if we don't connect it to a potential? Well, since the sphere cannot take charge through the contact anymore, it will remain neutral. Every conductor is an equipotential surface( electrons move freely until they find equilibrium), so the sphere remains an equipotential surface, but the value of its potential is not interesting anymore. There are also two image charges, but the condition that must be respected is Q1+Q2=0! so, Q1=-Q2. 
         I hope you understand now what's up with the image charges after all. If you understand their concept, you can play easily with electrical charges  conductors and field distributions.