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Consider a charge +q placed inside a cavity of conducting body. Charges will be induced as shown in the diagram below. How do we show that the electric field at a point A inside the cavity due to the induced charges is equal to zero? I was able to prove it for a spherical shell using symmetry but how do I do it for the following shape?

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Since point A is inside the cavity the field there is not zero.

The field there is the sum of the field due to the charge on the inner surface and the charge in the cavity.

The field due to the charges on the outside surface does produce zero field inside the cavity as well as inside the conductor. If that's what you want to show then you need to use a uniqueness result.

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Consider a conductor of arbitrary shape after it achieves electrostatic equilibrium.

If an electric field did exist in the conductor material, then the electric field would exert a force on the charges that are present there. This net force would begin to accelerate and move these charges. So if this were to occur, then the original claim that the object was at electrostatic equilibrium would be a false claim.

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This is the case of isolated fields , the effect of this Field is only valid inside the cavity

Just draw a guassian surface outside it ,it proves E= 0

But draw a guassian surface inside E=Kq/r^2 ( Which might not be accurate as field inside is not radially outward if charge is not placed at geometric centre)

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