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This is my first question and appreciate all your help.

An infinitely wide conductor is parallel to two infinitely wide planes with surface charge density -2𝜎 and +𝜎 respectively.

How to find the induced charge density on both sides of the conductor?

I know the electric field generated by a charged plate with surface density 𝜎 is

𝐸=𝜎/2πœ–. While for a charged conductor the electric field is

𝐸=𝜎/πœ–.

But I'm not sure how to deal with the unbalanced charges on both sides of the conductor. I think after all the induced charges should add up to zero? So the induced charge densities can NOT be +2𝜎 and -𝜎 for the up and down side of the conductor?

this question is the case when both planes have equal but opposite charges, and in this case the set-up could be seen as two capacitor in parallel.

But I still do not know how to deal with the unbalanced cases. And maybe I'm not using the right keywords because I cannot find relevant questions online. enter image description here

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You can "divide" the charges between two sides of the plates and using formula $E = \frac{\sigma}{2\epsilon_0}$ find electric field on every region. The constraint on the system is that $E$ must be zero inside the plates.

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