A nonconducting spherical shell, of inner radius a and outer radius b, has a volume charge density (within its thickness), where A is a constant and r is the distance from the centre of the shell. In addition, a point charge q is located at the centre. We have to find the value of A if the electric field in the shell () is to be uniform.

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The inner and outer radii of the nonconducting spherical shell are a and b, respectively. It is given that the shell contains a spherically symmetric charge distribution. The volume charge density within the shell is described by the function

,

where A is a constant. A point charge q is placed at the centre of the spherical shell.

We have to determine the constant A for which the electric field inside the shell, , will be uniform, that is it will not depend on r.

We will apply Gauss’ law for calculating the electric field at a distance r from the centre of the shell and within the shell. Let us consider a spherical surface of radius r such that . Because of the spherical symmetry the field can be a function of r only. The outward flux on this surface will be

The total amount of charge contained within this surface will be

Gauss’ law sates that

.

Therefore,

For the field to be uniform, the condition is

or

.