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Question:

A cross-section of two conducting concentric spheres is shown in the figure. The larger sphere of radius $R$ has a charge $ Q $ on it. The smaller sphere of radius $r$ has no charge on it initially. Now, the smaller sphere is earthed. How much charge leaves the inner sphere?

Intuition:

There are initially no charges on the inner sphere. Also the electric field due to the larger sphere is zero at all points inside it because of the fact that it is a conductor. I don't see any forces that could push charges from the sphere to the earth.

Solution:

The potential at all points on the inner sphere initially is the same and is equal to $V=\frac{kQ}{R}$ (taking the potential at infinity to be zero). The potential of the earth is zero. As long as there is a difference in potential between the earth and the sphere there will also be an electric field that pushes positive charges to the region of lower potential. After the sphere is earthed if a total charge $P$ has flown out of the inner sphere when it is at equilibrium, $$\frac{kQ}{R}-\frac{kP}{r} =0 \implies P=\frac{rQ}{R}$$ $\rule{17cm}{0.4pt}$

The math tells me that there is an electric field that pushes charges from the sphere to the earth. What is creating the electric field?

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That charge Q on the outer sphere does not go away and so it must be somewhere on that outer sphere. Where ever that charge resides bringing a positive charge from infinity to the outer surface of the outer sphere will require work to be done. As the inner and outer surfaces of the outer sphere are the same the same amount of work must be done in bringing the charge from the inner sphere (which is at the same potential as infinity) to the inside surface of the inner sphere.

This tells you that there must be some charge on the inner surface of the outer sphere and the rest of the total charge Q on the outer surface of the outer sphere.

The situation can be thought of as two capacitors connected in parallel storing a total charge Q.

One capacitor has its plates as the concentric spheres and the other has its plates as the outer sphere and infinity (isolated sphere).
The charge on each capacitor (charge on inner surface of outer sphere and the charge on outer surface of outer sphere) will be such that the potential difference across each of the capacitors connected in parallel will be the same.

The charge on the other plates located at “infinity” must be of the opposite sign but of the same magnitude as those on the inner and outer surfaces of the outer sphere.

Since the inner sphere had no charge on it before the earth was connected the charge on the inner surface of the outer sphere will be the charge which must have flies from the inner sphere to earth.

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