# Why does the induced charge have to have the same magnitude as the inducing charge?

Why is it that the total induced charge on a conducting, grounde,d infinite plane must be of the same magnitude as the inducing charge?

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Imagine you have a hollow cubic conductor with some charge in the cavity. This will polarize the charge in the conductor to the inner and outer surfaces. If you now define a Gaussian surface inside the conductor where the electric field must be zero, then the total electric flux out of this surface is zero which must equal the charge inside from Gauss's law. Therefore, the total charge on the inner surface is equal and opposite to that inside the cavity.

Keeping the distance of the charge from one surface fixed while expanding all the other surfaces to infinity, makes all the charge induced on this one surface equal to the charge a finite distance away from it. You can now remove the other surfaces at infinity so you're just left with one infinite conducting plane where the total induced charge on the surface equals the total inducing charge.

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If you have an infinite conducting plane at potential $0$ or $V_0$, what is the potential at any point above the plane? – Revo Apr 4 '12 at 11:36

Well, the field lines from the external charge have to terminate somewhere. If they don't terminate on the conductor, then they must diverge off to infinity. If they all terminate on the conductor, the total charge on the surface must then equal the external charge.

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