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Force on Three Parallel Electrical Sheets

  1. Dec 19, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider three plane charged sheets, A, B and C. The sheets are parallel with B below A and C below B. On each sheet, there is a surface charge of uniform density:
    -(4/3) * 10^-5 (Call α)
    -(7/3) * 10^-5 (Call β)
    -(3/3) * 10^-5. (Call γ)

    (The density given includes charge on both sides of the sheet). What is the magnitude of the electrical force on each sheet, in N/m^2

    Answers are 3.2pi, 1.4pi and 1.8pi.

    2. Relevant equations
    For an infinite plane, the electrical field strength is:

    σ/2ε where ε is 8.85 * 10^-12 and σ is charge density.

    Also, Force = Electric Field * Charge

    3. The attempt at a solution
    dF = EΩdA or
    dF/dA = EΩ where Ω is the charge density of the plane whose force is being calculated

    (Can someone verify if this line of reasoning is correct?):

    The electric field for any patch of area on A should be the average of the electric fields above and below. This comes out to simply be the superposition of the electric fields generated by sheets B & C because the electric field generated by A cancel out on both sides.

    Therefore, for a small patch of area on A, E:

    E = (β+γ)/2ε

    Finally:

    dF/dA = (β+γ)α/2ε = 8pi

    For Sheets B & C:
    dF/dA = (abs(-α+γ))β/2ε = 1.4pi
    dF/dA = (α+β)γ/2ε = 6.6pi

    EDIT: So I did a quick test to check something. Apparently if I subtract the two density charge contributions for the electric field for Sheets A & C, I get the right answer but this has only made sense for Sheet B because the electric fields were in opposite directions. For Sheet A, the electric fields are both pointing down towards B & C while for Sheet C, the electric fields are both pointing up towards A & B.

    Also, I'm curious as to why the answers have pi in it. All my calculations simply didn't require it. Is there another way to do this?
     
    Last edited: Dec 19, 2012
  2. jcsd
  3. Dec 21, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Your solution looks correct. I think pi is used to replace 1/ε with 4pik≈36 pi 109 which makes calculations with hand possible.

    ehild
     
  4. Dec 22, 2012 #3
    Thanks. Guess the book was wrong =)
     
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