Uniformly Charged Circular Sector problem (Electrostatics)

[Boardy222]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Differential charge in cylindirical co-ords is;

dq = σ*ρ*d$\varphi$*dρ

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

After this i get stuck, I can't resolve this into the different cylindrical components

 

[rude man]

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.