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An insulated uniformly charged sphere of radius $R$, has been smeared with charge $q$ uniformly throughout its volume. Now this sphere is surrounded with a charged conducting spherical shell of inner radii $a$ and outer radii $b$, smeared with charge $-2q$.

What is the resulting $\vec E$ for all the regions?

So the case where the conducting shell is uncharged, the inner surface will be induced with charge $-q$ and the outer surface with charge, $+q$; and one can easily find the $\vec E$ using Gauss' Law.

Now in the case of a charged spherical shell, the charge will be so distributed so that inside the conductor the electric field is zero.

Now naively if I apply Gauss's Law, then I come to the conclusion that on the inner surface with radii $a$, the induced charge is $-q$, so as to cancel the field in side the conductor and then on the outside surface the induced charge is again, $-q$, so as the total charge on the conducting sphere is $-2q$.

But as we know that if their is a charged cavity inside the conductor, it shields the cavity from external charged sources. So outside the conductor we should have seen a charge $+q$. So where is my logic failing?

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closed as off-topic by AccidentalFourierTransform, Kyle Kanos, Jon Custer, Emilio Pisanty, ZeroTheHero Jul 31 '18 at 2:37

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Nothing you do to the conducting shell will change the induced charge of $-q$ on the inside surface of the conducting sphere.
This is because that induced charge will ensure that there is no electric field inside the conducting shell.
In this case because the total change on the conducting shell is $-2q$ that will mean that a change of $-q$ on the outside of the conducting shell as you have stated.

Think of it as the superposition of three electric fields.

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The field in region $A$ and region $B$ is not influenced by what goes on outside in region $C$.
For example, if the outside of the conducting sphere was earthed nothing would change in region $A$ or region $B$.

In a similar way if you are in region $C$ all you know is that there is a charge of $-q$ on the outside of the conducting shell.
You cannot determine what is happening inside the conducting shell as it could be that insulated uniformly charged sphere has no charge on it and the conducting shell only has a charge of $-q$ residing on it.

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