A sphere of radius is filled with positive charge with uniform density . Then a smaller sphere of radius is carved out, as shown in the figure below, and left empty. What are the direction and magnitude of the electric field at ? At ?

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**Solution**

This problem can be solved by using the principle of *superposition*. For instance, consider a point charge at some point in space. It creates an electric field everywhere. However, if you place a negative charge also at , it exactly cancels the electric field created by the original charge . Therefore, it appears that there is no charge anywhere, even though there is in fact a and sitting atop the other one.

Using this idea, the given charge distribution can be realized as the sum of a cavity-free sphere of radius and charge density , with another sphere of radius with charge (note the negative sign).

The electric field can now be determined by a simple application of Gauss’s law, since sphere I and sphere II are symmetric objects. Then, the field due to each solid sphere is

(1)

and the net electric field at is

(2)

Similarly, for the point

(3)

which means the point experiences a field

(4)

**Bonus problem:** If a long cylindrical wire of radius had a portion of radius removed, such that its cross section looks like the figure above, what would be the magnetic field at points and ? Assume that the wire is carrying current of density out of the plane of the paper.