Shielding, and other consequences of Gauss' Law

We can immediately state the most important consequence of Gauss' Law:

The electrostatic field inside any isolated conductor is always zero.

Suppose that at a given moment, this is not the situation. Then there would be a net electric field wholly inside an isolated conductor. This means that according to Gauss' Law there must be an accumulation of net charge at some place inside the conducting body. Now every conductor has free electrons inside it, which can move rapidly due to this field and redistribute the charge. The accumulated charge, therefore, due to mutual repulsion, gets away as far as possible. It will move to the outer surface of the conductor, leaving no net accumulation inside . The electric field then becomes zero everywhere inside the conductor.

Franklin's Ice-pail Experiment:

A tall, cylindrical metal can (the 'ice pail') is put on the top of a gold-leaf electroscope. The can is initially uncharged, so the electroscope leaves show no divergence. A small metal sphere is given some charge, and suspended by an insulating thread. Using the thread, the sphere is lowered into the can, without touching its inside surface. As it is lowered, the electroscope leaves start diverging, showing the presence of charge on the can. Since the sphere does not touch the can, this is obviously due to induced charge on the inside of the can. To maintain charge neutrality of the can, an equal and opposite charge appears on its outside, causing divergence of the electroscope leaves.
If the outside of the can is now grounded for a second, the outside charges escape to the earth, and the leaves collapse. If the sphere is now made to touch the inside of the can, there is still no divergence. This shows that the induced opposite charge is equal and opposite to the inducing charge on the sphere.
Without the earth connection (grounding), if the sphere is made to touch the inside of the can, the divergence of the leaves remain the same. The reason for this is that the ball and the can now form a single conductor, and the original charge on the ball is transferred to the outside of the can. This is verified further by taking the ball out and testing it for the presence of any charge on it -- it shows no charge.

Electrostatic Shielding:

If a cavity is scooped out of a conducting body, the net charge, and thus the electric field inside this cavity is always zero, no matter how strongly the conductor is charged, or no matter in how strong a field it is placed. We can show this impressively by placing a metal and a plastic container on top of a Van de Graaff generator:

[The big "Demo" button is for a large video clip (typically 1 MB); and the small one for a version reduced in size (typically 100 kB) and quality, but suitable for download via phone lines].

The proof of this follows simply from Gauss' law:

Take a closed Gaussian surface entirely surrounding the cavity, wholly inside the metal. At each point of this surface the field is zero. Therefore the net flux over this surface is also zero. Therefore by Gauss' law, it follows that it encloses no charge. Therefore the space inside the cavity is totally shielded from any external influence.

What happens if you take a charge and enclose it completely by a metallic body? You should be able to show by the same argument as above that this induces bound charges on the inside surface of the cavity, exactly equal in magnitude but opposite in sign as the original charge.

It is interesting to note that the conducting body could be quite thin, and may even have holes in it. Faraday verified this by having a cage made from wire grid, having it connected to electrostatic machines producing high charge, while he himself was inside. Of course, he was not affected at all.

[The big "Demo" button is for a large video clip (typically 1 MB); and the small one for a version reduced in size (typically 100 kB) and quality, but suitable for download via phone lines].