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Uniformly Charged Circular Sector problem (Electrostatics)

  1. Jul 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi, I am having problems trying to solve an electrostatics question. Basically there are two circular sectors of radius R and with angle 2β. Both of these sectors have their vertices at the same point in the xy plane (ie at the origin) but have a separation d in the z axis. Also the displaced from one another by an angle in the xy plane of α. Both of these are equally but oppositely charged with a uniform surface charge density σ.

    Given this, find the rotational component of the force between these sectors (ie if you where using cylindrical co-ordinates the θ component


    2. Relevant equations


    b7c736dbf1124dead466554152d29642.png

    53b51b40e41bc1e714223d0eda4d5773.png



    3. The attempt at a solution

    Differential charge in cylindirical co-ords is;

    dq = σ*ρ*d[itex]\varphi[/itex]*dρ

    E = ∫∫ ke**ρ*d[itex]\varphi[/itex]*dρ/z^2 + ρ^2 (sin^2([itex]\varphi[/itex]) + cos^2([itex]\varphi[/itex]) r(hat)

    After this i get stuck, I can't resolve this into the different cylindrical components
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 2, 2013 #2

    rude man

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    Homework Helper
    Gold Member

    This looks like the old-fashioned type of radio variable capacitor!

    I would approach this problem from an energy viewpoint:

    what is the E field between the section of the two sectors opposite each other? Ignore fringing effects.

    then: assume α > 0 , what is the energy content of the electric field?
    then: moving one of the sectors to α = 0, what is the change in energy content of the E field?

    Equate the change in energy to the work done to turn the sector thru the angle α. BTW I think the answer should be a torque, not a force, unless you want to assume the force is applied at radius = R.
     
    Last edited: Jul 2, 2013
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