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Work Done to Separate Sheets

  1. Oct 4, 2012 #1
    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. jcsd
  3. Oct 4, 2012 #2

    vela

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    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.
     
  4. Oct 4, 2012 #3
    I believe the answer is correct.
     
  5. Oct 4, 2012 #4
    What about the force method? Or any other method that's valid?
     
  6. Oct 5, 2012 #5
    I actually don't know how to do any other method- ideas from other people?
     
  7. Oct 5, 2012 #6

    TSny

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    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.
     
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