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Let us consider a conductor having a cavity (both can have an arbitrary shape) wherein a positive charge $+q\,$ is located:

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At the electrostatic equilibrium, the inner wall of the conductor will have a negative charge $-q_{1}$, and the outer wall a positive charge $q_{1}$.

The $E$-field in the region between the inner and outer wall is determined only by $+q$ and $-q_{1}$. How can one prove the total $E$-field in the layer is null?

The electrostatic equilibrium argument only tell us the potential is constant on the outer and inner surface of the conductor, it does not give any additional information about the $E$-field or the potential inside the thickness since there is no charges in it. Also, we cannot claim $q_{1}= q$, because, since we don't know the $E$-field within the layer, we cannot apply Gauss's law. How can we get out of this impasse?

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    $\begingroup$ Are you not satisfied with the argument showing that the field inside any conductor is $0$? $\endgroup$ Sep 7, 2019 at 1:39
  • $\begingroup$ @Aaron Stevens Which argument? $\endgroup$
    – Hilbert
    Sep 7, 2019 at 1:45
  • $\begingroup$ That if there was a field the charges would then move until the field is $0$. So it just has to be $0$. $\endgroup$ Sep 7, 2019 at 1:59
  • $\begingroup$ @Aaron Stevens But there are no charges inside the layer, $Q=0$, charges, in this case, are located only on the inner and outer wall of the conductor. $\endgroup$
    – Hilbert
    Sep 7, 2019 at 2:04
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    $\begingroup$ I never said absence of charge means no field. You seemed to be saying to me that the field cannot be $0$ because there is only charge on the surface. That's why I was trying to show you how you can still have a zero field when this is the case. I agree, you need to start with $0$ field for the entire argument. I have told you why it must be zero, and if you don't believe that then if you look up any resource they will say the same thing. $\endgroup$ Sep 7, 2019 at 10:07

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It is important to realize that conductors inside contain mobile electrically charged particles, so called "charge carriers", even if the conductor is macroscopically neutral inside.

Since electrostatic equilibrium is assumed, there are no electric currents anywhere in the body or on its surface. In real conductors, the weakest macroscopic electric field inside the conductor means the mobile charge carriers present in the conductor (electrons in metals, or also holes in some semiconductors) experience electric force due to that electric field. If that was the only external force, it would make the mobile charge carriers move in concerted way and that means macroscopic electric current inside would occur.

If that current does not occur, either the electric forces on all charge carriers is counterbalanced by other force (for example, magnetic force can do that when the body moves through magnetic field) or, if there is no such counterbalancing force, the only possible conclusion is that electric force must be zero, hence electric field inside must be zero.

So, for conductor at rest with no other forces pushing the mobile charge carriers, and no current inside, the electric field inside must be zero. This is accomplished by special arrangement of electric charge near the surface of the conductor, whose electric field inside cancels any possible external static field, so that total field inside is zero. This is possible unless the external field is extremely high. In case the body is in vacuum in too high external electric field, the electric charge will jump out of the conductor surface and the system will no longer be in equilibrium.

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