Work Done to Separate Sheets

1. Oct 4, 2012

NullSpace0

1. The problem statement, all variables and given/known data
I have a current problem set question about the work per unit area required to separate infinite sheets of charge with equal and opposite charge densities from a separation of d to a separation of 2d.

2. Relevant equations
U=(1/8∏)∫E2dV
W=∫F*dr
E=4∏σ

3. The attempt at a solution
I was thinking I could just find the difference in energy stored in the field before and after... so I would integrate E2 over the initial and final volumes, and then the difference must come from work I have put in to the system, which shows up as energy stored in the electric field.

If I do this, I get Ui=2∏σ2d and Uf=4∏σ2d

And so the work is just 2∏σ2d. But how am I sure that this has units work per area? It seems like work per volume because you have (esu^2)/(cm^3) for the units written out fully.

Is there a way to do this by integrating the force (or perhaps the field) rather than finding stored energy changes?

2. Oct 4, 2012

vela

Staff Emeritus
Remember the potential energy of two charges is $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$ (in SI), so energy is charge2/length.

3. Oct 4, 2012

schaefera

I believe the answer is correct.

4. Oct 4, 2012

NullSpace0

What about the force method? Or any other method that's valid?

5. Oct 5, 2012

schaefera

I actually don't know how to do any other method- ideas from other people?

6. Oct 5, 2012

TSny

I think you can argue that the force felt by a patch of area dA on one sheet is due solely to the electric field produced by the other sheet (E = 2∏σ). The force on the patch is then dq*E where dq = σdA. Since the force will be constant as you separate the plates, the work will just be F*d. I believe this will give you the same answer as the energy method.