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If we have a solid conducting sphere with charges around it, then the electric field inside the sphere is zero, otherwise the electrons of the sphere would not be in equilibrium as there would be a net force acting on it. However, if its a hollow sphere, then why does the electric field inside the hollow sphere have to be zero?

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Okay, draw a circle around the spherical void; done?

Then see is there any charge?

If not, there should not be any divergence of electric field, know that?

So, from Gauss Law, $$\text{div} \mathbf{E} \cdot dv= 0$$ which implies $$\mathbf{E}\cdot d\mathbf S= 0 \implies \mathbf E= 0$$ over that volume .

Edit:

Don't know what OP is trying to convey through his message but still let me elaborate a bit.

Suppose, there be an isolated conductor.

Initially, there are no net charges. So, the electric field inside & outside is null.

Then some, external charges are placed inside the conductor. They would feel force of repulsion among them & in order to minimise it, they spread over the surface in such a way that there is no net electric field inside.

Then the conductor is placed in some external electric field.

The field inside the surface of the conductor is the sum of the external electric-field & the field due to the charges residing on it. They arrange in such a way that they nullify the external electric field inside the surface.

This is indeed proved by the fact that the potential inside the surface needs to follow Laplace's equation. Now, a solution could be the potential of the surface itself which is equipotential.

But by Uniqueness theorem, we know there can be only one solution: one potential inside the surface.

This means potential inside the surface is same as that of the surface.

Now, electric field is the gradient of potential which implies it is zero inside the surface of the conductor.

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  • $\begingroup$ if we were to apply the same logic for a non conducting sphere then the electric field inside it would be zero, however this is not the case. electrstatic shielding is applicable only for conductors. $\endgroup$ – Azwar Abdulsalam Dec 1 '15 at 14:48
  • $\begingroup$ it has to do something with the fact that it is a conductor, and not gauss law as gauss law would give the same result for a condcting as well as a non conducting shell $\endgroup$ – Azwar Abdulsalam Dec 1 '15 at 14:51
  • $\begingroup$ Yes the electric field in the conducting part of the sphere is zero, but your answer doesn't make clear why the electric field in the hollow is also zero. $\endgroup$ – John McVirgo Aug 25 '16 at 21:49
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$E$ is necessarily zero inside hollow sphere because,

Inside hollow sphere $Q = 0$

from gauss’s law $$\phi=\frac{Q}{\epsilon}$$

$$E.A = \frac{Q}{\epsilon}$$

Since,

$E.A = 0$

$E = 0$ or $A = 0$ but $A\ne0$ so $E = 0$

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    $\begingroup$ if that was the case then instead of a conducting shell if we had a insulating shell the electric field would still be zero, but however that is not the case, electrostatic shielding is applicable only for a conducting sphere. $\endgroup$ – Azwar Abdulsalam Dec 1 '15 at 14:46
  • $\begingroup$ You could still have static charges on the inner surface of the hollow sphere contributing to a zero divergence electric field in the hollow space; you need to show why they can't appear there if there's no charge in the hollow. $\endgroup$ – John McVirgo Aug 25 '16 at 21:43

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