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Suppose we have an ebonite rod (insulator). This rod has a negative charge, and once it touches a neutral pith ball, charges are distributed amongst the pith ball. However, why doesn't the same happen if we have neutral paper rather than a pith ball? Even though the paper is an insulator, shouldn't the charge migrate from the rod to the insulator and stay on the point of contact?

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The charges should also migrate to a piece of paper upon touching the ebonite rod. But probably the paper is not as good an insulator as the pith ball, which is possibly also hanging on an insulating thread.

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Cellulose, of which paper is made, is a very good insulator.

In terms of electrostatic experiments a sheet of writing paper that you might find in a laboratory is rather a good conductor of electricity because of the water which it has absorbed; the same being true of wood.

In the case of the experiments that you describe pith is used because it is a very low density material and again a conductor because of its moisture content.
A hanging pith ball would be insulated from the ground because the silk or plastic thread which is a good insulator.

A pith ball could be replaced with a ball of paper but the ball of paper would be much heavier.

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