This is basically question 3.18 in Purcell. You have a plate being suspended into the middle of two other plates, and then you need to determine the force downward on the hanging plate given that the separation between it and either side of the other conductor is s; the plates have length b; the amount of the inside dangling plate between the other two is height y.
F= (1/2)Q^2 * d/dx(1/C)
The Attempt at a Solution
I'm trying to say that this thing acts as two capacitors side by side, but only for the distance y by which it is dangling down. Since U= (1/2)CV^2, the total energy of the system is 2*(1/2)*C*V^2, and C=(by)/(4*pi*s). So I get the total energy as a function of capacitance; I differentiate the energy to get the force on it, and in the end I find that:
Is this right?
Can the field between the plates be determined as 4*pi*sigma, or is it 2*pi*sigma since the center plate really has 2*pi*sigma amount of charge per area on each side?