In the figure a nonconducting spherical shell of inner radius a = 2.23 cm and outer radius b= 2.70 cm has (within its thickness) a positive volume charge density p = A/r, where A is a constant and r is the distance from the center of the shell. In addition, a small ball of charge q = 46.7 fC is located at that center. What value should A have if the electric field in the shell (a srs b) is to be uniform?

The electric field within the region of the shell is said to be uniform.

Imagine a Gaussian surface having a radius r, concentric with the center of the given shell. This radius (a<r<b)

So the Gaussian surface lies within the shell of the non conducting sphere.

From Gauss' law, the electric field within this Gaussian surface of radius r is given as

$$\begin{gathered} \oint E.dS=\frac{q_{enc}}{{\epsilon }_o} \\ E is the electric field \\ dS is a surface element of the gaussian surface \\ {q}_{enc} is the total charge enclosed by the gaussian surface \end{gathered}$$