Question

The region between two concentric spheres of radii a and b (> a) contains volume charge density $\rho \left(r\right)=\frac{C}{r}$, where C is a constant and r is the radial distance, as shown in figure. A point charge q is placed at the origin, r= 0.Find the value of C for which the electric field in the region between the spheres is constant (i.e., r independent).

Answer 1

Total charge in a sphere of radius R ( <b and >a ) is
${\int }_a^R\frac{C}{r}4\pi r^{2}dr=2C\pi (R^2-a^2)$
Field at a distance R from center is then,
$F=\frac{1}{4\pi ϵ}\frac{2C\pi (R^2-a^2)}{R^2}+\frac{1}{4\pi ϵ}\frac{q}{R^2}=\frac{1}{4\pi ϵ}\frac{2C\pi R^2-2C\pi a^2+q}{R^2}$
For F to be independent of R, numerator of F should contain a multiple of $R^2$
Therefore ,
$q=2C\pi a^2\Rightarrow C=\frac{q}{2\pi a^2}$