Charge Distribution on Conductor

[chomebole]

Does anyone know how to find the actual charge dustribution on a conductor? More like the alignment of all the charges on a conductor? I know that the electric field should be zero everywhere on the conductor and that on pointy edges there is more charge buildup, but I have no idea how to find the actual distribution or alignment of the charges. I know it must be very difficult, but I also think that there must be a way. I know that for a sphere the charge should be uniformly distributed but what about for a conductor in the shape of a brick? Or a triangular prism etc. There should be some mathematical approach to finding this out, no?

 

[Meir Achuz]

You have to solve for the electric field outside the conductor.
There are various ways of doing this, depending on the exact situtation.
Then sigma=E/4pi at the surface (in gaussian units).

 

[arunma]

To be honest, I've never seen this problem addressed, even in upper division E&M textbooks. I suppose it's either because it's too difficult, or because no one cares (since the electric field inside is always zero). But if you have a conducting sheet of very large size, you can find the electric field outside the conductor using something called the method of images. You can treat the conductor as a sort of mirror, and draw mirror image charges on the other side of the conducting sheet, which will be drawn according to the charge distribution on the side of the sheet where you're trying to find the electric field. Of course, the answer you get will not tell you about the field inside the conductor, so that's not really an answer to your question.

 

[Meir Achuz]

The calculation is done in many texts just as I described it.
For instance for a point charge q a distance d from an infinite plane, the E field can be found by using an image charge. Then using the E field at the surface of the plane, the surface charge is sigma=-qd/[r^2+_d^2]^{3/2}

 

[mdelisio]

Usually people attack this kind of problem using so-called "moment methods". There's a good book by R.F. Harrington on the subject.

In general, you know that the charges on the conductor are all going to reside on the surface. You also know that the potential at the conductor's surface (and inside its volume) must be constant. One application of the method of moments divides the surface up into many smaller subsurfaces, and assumes a constant charge density on each subsurface. You can then solve for the potential at, say, the center of any subsurface due to the charge at a single subsurface. Superposition also applies. You wind up with a matrix equation:

Vm = Lmn*qn (summing over the n's is implied)

Where Vm is the potential at subsurface m, qn is the charge density at subsurface n, and Lmn is the matrix connecting them. You invert this matrix to find the q's for constant V's. The accuracy depends on how finely you chop your surface up into.

It can be a very messy problem, especially for complex geometries, and requires a computer to help with the matrix inversions.

As a simple example, you might want to consider a thin square metal plate. Divide it up into, say, a 4x4 array of smaller metal squares. Solving this will show you that the charge tends to pile up in the corners.

 

[arunma]

The calculation is done in many texts just as I described it.
For instance for a point charge q a distance d from an infinite plane, the E field can be found by using an image charge. Then using the E field at the surface of the plane, the surface charge is sigma=-qd/[r^2+_d^2]^{3/2}

You're right, I completely forgot about this. I just checked my old E&M text (Griffiths), and found that he does this for the case of the infinite conducting sheet.

 

[vish_al210]

I have a small questio though., I understand that the charges on a conductor may//can vary along its length and periphery. But then when I consider a conductive wire, at what point do u think I can get a potential difference != 0, but a value... and if so for how long. By 'what point', i am referring to any segment or length of wire where a probe maybe placed.
Please a lot of my theory depends on it.

 

[Delta2]

If the conductive wire is in electric equilibrium then there is no potential difference. Each time one adds or removes charge from the conductor or applies an external E field, the conductor will take a small time to reach electric equilibrium. During this time charges are moving inside and on the surface of the conductor so that the potential will eventually become equal everywhere in and on the conductor. During this time u can measure a potential difference.

 

[vish_al210]

During this time u can measure a potential difference.

Well how infinitely small can one assume this time to be if the conductor in question is any normal conductive, copper, Al wire... or any other conductor... is that a reasonably long duration for that potential change to affect the effective potential difference of the system. Thanks..

 

[Delta2]

Well how infinitely small can one assume this time to be if the conductor in question is any normal conductive, copper, Al wire... or any other conductor... is that a reasonably long duration for that potential change to affect the effective potential difference of the system. Thanks..

This time is generally small and it depends on the size and on the conductance (reciprocal of resistivity) of the conductor. The smaller the size and the bigger the conductance, the smaller the time will be. If the conductor we talk about is part of an AC circuit you want this time to be small when it is compared to the period of the AC current/voltage.