Is Charge Conservation Violated in Spherical Volumes with Zero Current Density?

[walkinginwater]

hi, guys:
Charge conservation Violated? It seems to be: The charge density inside a spherical volume changes from $$\rho$$ at t=0, after period of time $$\tau$$, the charge density decrease to 0. However, the author claimed that during the processing there is no current density in the material surrounding the initially charged spherical region. Is this possible?

I am now reading a something about Charge Relaxation in a conductor.
You can find the example in the following way:
Click on the weblink:http://web.mit.edu/6.013_book/www/book.html"
Then click chapter 7 on the left column , the whole index of chapter 7 will appear on the right;
After that, you click on the section 7.7;
At last, you go to the example 1. where you see what I want to talk about.

Note: Equation (7.15) can be found in the following weblink http://web.mit.edu/6.013_book/www/book.html" click on Chapter 7 on the left column, and then click on the section 7.1 on the right column

In the example it states that: " one or both of these migrate in the electric field caused by the net charge[in accordance with (7.15)] while exactly neutralizing each other"

Anybody agree with this? Can he/she give some explanation?

 

[Meir Achuz]

However, the author claimed that during the processing there is no current density in the material surrounding the initially charged spherical region. Is this possible?

This is correct. If a charge distribution of llimited size is placed inside a conductor, the shape of the charge distribution remains constant, but decreases in magnitude to zero with a time constant tau. The original charge slowly appears on the surface of the conductor. There is a current in the conductor, but no charge distribution other than the original one occurs.
The equation governing the decay of the charge distribution is
$$\partial_t\rho=-\sigma\nabla\cdot{\vec E} =-(4\pi\sigma/\epsilon)\rho.$$.
Solve it. The result is not diffusion of the charge, but its disappearance and then reappearance on the surface of the conductor.
A good discussion of this is in Sec. 6.93 of "Classical Electromagnetism"
by J. Franklin (AW.com).

 

[cesiumfrog]

That's interesting. Does it have any application?

 

[walkinginwater]

hi, Meir Achuz:
Thank you very much for your reply. But in fact I am confused over what you have said. In the website of MIT, it states that there is no current density in the material, i.e., there is no $$\vec J$$. But you said

There is a current in the conductor, but no charge distribution other than the original one occurs.

So it seems that your statement doesn't coincide with what from the MIT Website.
Our library doesn't have the book "Classical Electromagnetism" by J. Franklin (AW.com). Would you be so nice to explain in more detail. Or can you scan section 6.93 and email it to me?
p.s.: I make some modification about my post, now you can find the example I mentioned before!

 

[Meir Achuz]

During the decay of the charge distribution, there is an E in the conductor.
Since j=\sigma E, there is a j. That is the problem with trying to learn physics from the web. even MIT.

 

[walkinginwater]

During the decay of the charge distribution, there is an E in the conductor.
Since j=\sigma E, there is a j. That is the problem with trying to learn physics from the web. even MIT.

Thanks for your answer! In fact, I continue to read the following sections from http://web.mit.edu/6.013_book/www/book.html" and I find in Section 7.8 and 7.9, there are more detailed explanation about this question. The conclusion is that:

• 
  There are current density through the conduction material

• 
  The spatial distribution of the charge density don't change and only the magnitude of the Field decay exponentially.

• The charge will drift to the boundary of the conduction materials

 

[walkinginwater]

This Thread Has been Solved! Thanks for Meir Achuz

This Thread has been Solved! Thanks for the reply of Meir Achuz!