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Homework Statement
Two infinite conducting planes are held at zero potential at z=-d and z=d. An infinite sheet with uniform charge per unit area σ is interposed between them at an arbitrary point.
a) Find the charge density induced on each grounded plane and the potential at the position of the sheet of charge.
b) Find the force per unit area which acts on the sheet of charge.
Homework Equations
C = [itex]\frac{Q}{\varphi}[/itex]
∫[itex]\rho[/itex]a[itex]\varphi[/itex]bd3r = ∫[itex]\rho[/itex]b[itex]\varphi[/itex]ad3r
σ = [itex]\frac{dQ}{dS}[/itex]
The Attempt at a Solution
There is a picture associated with the problem but all it shows is that the sheet of charge must be off-center. Because both plates are grounded [itex]\varphi[/itex]=0 for both. Charge for both plates are non-zero. The electric field is the gradient of the potential, thus E at the plates and from z>|d| must also equal zero. I'm unclear what the electric field would be inside though. If the sheet was exactly in between the two plates the charge induced on both plates would be equal and have to balance the charge on the sheet so that E remains 0 outside. Offset, like it is I'm unclear how to solve for the induced charge density.
Potential of Plate further away from sheet (P=C-1):
0 = P11Q1+P12Q2+P13Q3
Potential of Plate closest to the sheet:
0 = P21Q1+P22Q2+P23Q3
Potential of the sheet:
[itex]\varphi[/itex] = P31Q1+P32Q2+P33Q3
Capacitance:
C = [itex]\frac{Q}{\varphi}[/itex]
Where:
[itex]\varphi[/itex] = P31Q1 + P32Q2 + P33Q3 - (P11Q1 + P12Q2 + P13Q3) - (P21Q1 + P22Q2 + P23Q3)
From here I'm thinking there must be a way to solve for each individual Q in a nice manner, but I'm stuck. Any help would be much appreciated.