Understanding the Effects of a Conducting Plate on a Parallel Plate Capacitor

AI Thread Summary
The discussion focuses on analyzing the effects of inserting an isolated conducting plate into a parallel plate capacitor. The electric fields between the conductors are derived using Gauss's law, resulting in expressions for both positively and negatively charged plates. The potential difference is calculated by integrating the electric field, leading to a new capacitance formula that incorporates the thickness of the conducting plate. The energy density of the charged capacitors is also addressed, with a note on the uniform potential within the conducting slab. The conversation emphasizes the importance of understanding electrostatic principles in this context.
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Homework Statement


Consider a parallel plate capacitor of area A and separation d. The plates are isloated. One has charge +Q and the other -Q. An isolated conducting plate of area A and thickness t is inserted between the plates as shown.
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a. Find the E fields between the conductors

b. Find the potentials between the conductors

c. Using the above, find the capacitance of this new arrangement and compare with C before inserting the conducting plate.

d. Calaculate the energy density
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of the charged capacitors before and after inserting the conducting plate

Homework Equations


σ = Q / A

The Attempt at a Solution


a. From Gauss's law, the E field for parallel plates E = σ / ε = Q / Aε , for the plate that is negatively charged, E = - Q / Aε

b. From V(x) = -∫E dx (from 0 to x), The potential is V = E * d = Qd / Aε , but we want to find the potential to the slab in the middle, so substitute d with d/2 + t/2, then V = E * (d+t)/2 = (Q(d+t)/2) / Aε = Q(d+t) / 2Aε

c. The capacitance C = Q/V
C = Q / (Q(d+t) / 2Aε) = 2Aε / (d+t)
Without the slab in the middle, C = 2Aε / d

I hope my steps are correct so far, and I don't know how to find the energy density here, I believe the energy density of a field is 1/2 * ε E^2 , but not sure how the slab would affect it
 
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