I’ve seen quite a few questions where some charge is given to a conducting plate and we are to find the final charge on each face of the plate and on the faces of some nearby plates. While there have been some great answers (like this one), I don’t feel the questions were entirely useful to my needs. I would like to build a general method to go about solving such systems (using the basic laws and formulas of electrostatics, of course).
Here are some assumptions:
- There are no edge effects
- The initial charge distribution is transient
My attempt:
Let’s first assume the basic case of two parallel plates a distance $d$ apart with a charge $q$ given to one and $kq$ to the other.
Let’s assume $k,q>0$ for simplicity.
The resultant field will cause the charges to shift. Here is the first problem I have. In what configuration will the charges settle? I would imagine all the like charges would want to be as far from each other as possible. Does this mean there would be 0 charge on their inner surfaces, leaving net $q$ and $kq$ for the outer ones respectively?
Once we have determined this, we may consider some other cases.
Considering the case where the plates are given $q$ and $kq$ and they are connected to each other by a conducting wire. First of all, the potentials of both plates have been made equal. So some charge will flow through the wire and the final charges on both plates will be $(k+1)\frac q2$.
But again, the question arises: how much charge is on each surface? Per my understanding of electrostatic shielding, and due to symmetry considerations the charges must be evenly split on the inner and outer surfaces of both plates otherwise the residue charges will cause a field in between the supposedly equipotential surfaces.
And, let’s consider the case where the plate with charge $kq$ is grounded. Again, it seems rather arbitrary to me. All we know is the plate which began with $kq$ has potential $V=0$.
I suppose that the potential at the right face of the grounded plate should be $0$, and the potential due to the plate with $q$ is $qd\over A\epsilon$ so potential due to charges on the grounded plate should be the same. But I’m unsure how this may be achieved.
I believe these cases are sufficient to solve more complex combinations. However, I am a bit wary of systems where the distance between some pair of plates is different than others. But, I feel it will be worked out with the rest of our cases.