Charge distribution in a conducting sphere

[Plantis]

Homework Statement

At time t = 0 a charge distribution \rho(r) exists within an idealized homogeneous conductor whose permitivity \epsilon and conductivity \sigma are constant. Obtain \rho(r,t) for subsequent times.

Homework Equations

Maxwell's equations = Gauss' Equation + Ohm's law in Differential form

The Attempt at a Solution

By using Gauss' law I can find how an electric field depend from radius. After it I can use Ohm's law to find current density through the surface of a sphere of radius R. Then I can find quantity of charges that leave the sphere of radius R per unit time. See attachment.

q - the number of the charge coming out of the sphere of radius r.

 

[Spinnor]

I wonder if it would help to first solve a simpler problem, say one where the charge density was uniform, rho(r) = constant inside the sphere. You can almost picture the resulting current, it will be radial and can easily figure out rho(r,t) for t large, all the charge will lie on the surface? Good luck, challenging problem!

 

[Plantis]

I wonder if it would help to first solve a simpler problem, say one where the charge density was uniform, rho(r) = constant inside the sphere. You can almost picture the resulting current, it will be radial and can easily figure out rho(r,t) for t large, all the charge will lie on the surface? Good luck, challenging problem!

So what? it is obviously. In case t large all charge will be on the surface. I understand that there will be radial current in proposed situation by you. I understand what happen inside the sphere. But, I can not solve the problem.I think that my solution is correct. However, It need to be checked.

 

[Spinnor]

So what? it is obviously. In case t large all charge will be on the surface. I understand that there will be radial current in proposed situation by you. I understand what happen inside the sphere. But, I can not solve the problem.

Could you solve the easier problem?

 

[Plantis]

Could you solve the easier problem?

I do not understand you.

 

[Spinnor]

I do not understand you.

When you are stuck and getting nowhere I thought it was good practice to step back and solve an easier problem.

Anyway does the continuity equation help to solve your problem? You have not used it.

Good luck.

 

[sciillit]

1. The problem statement...

Why should there be a flux of electrons from the surface of the sphere?
Do you mean you are creating an arbitrary Gaussian surface within the sphere and figuring out the current density through that surface?

You can treat this as a diffusion problem with spherical symmetry and the classical assumption that the charge must reside on the surface after a very long time.
A trickier consideration is that the charge will not in fact reside completely on the surface, but will have a skin depth.
How does that affect your solution?

In any event, there is a key symmetry that will make the solution obvious... and also see if working the problem in time reversal helps.