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Cylindrical Vacuum Capacitor

  1. Mar 17, 2014 #1
    1. The problem statement, all variables and given/known data
    We want to design a cylindrical vacuum capacitor of a given radius a for the outer cylindrical shell, which will be able to store the greatest amount of electrical energy, subject to the constraint that the electric field strength at the surface of the inner sphere may not exceed Eo
    (a) What radius b should be chosen for the inner cylindriclal conductor?
    (b) How much energy can be stored per unit length


    2. Relevant equations
    E=[itex]{λ}/{2πεEo}[/itex]
    λ=2πεrEo

    3. The attempt at a solution
    Electric potential difference:
    Vb-Va=∫Edl=∫EdA=∫Edr=[itex]λ/2πε[/itex]∫1/r=([itex]λ/2πε[/itex])*ln([itex]\frac{a}{b}[/itex])
    Electrical Energy in a capacitor:
    U=[itex]\frac{1}{2}[/itex]λΔV=[itex]\frac{1}{2}[/itex](2πεrEo)*([itex]λ/2πε[/itex])*ln([itex]\frac{a}{b}[/itex])
    I would take the derivative of this with respect to b to find the radius. I am not sure that I have set this up correctly.
     
  2. jcsd
  3. Mar 17, 2014 #2

    rude man

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    Homework Helper
    Gold Member

    r = b, right? Try to take the derivative wrt b then.

    I did not check your algebra.

    Also, lambda needs to be substituted for via your 2nd 'relevant equation'. It's a function of b and Eo.
     
    Last edited: Mar 17, 2014
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