Question

The region between two concentric spheres of radii ‘a’ and ‘b’ respectively (see figure,) has volume charge density r = A/r, where A is a constant and r is the distance from the centre. At the centre of the sphere is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant is

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

At r = a

$E_a=\frac{kQ}{a^2}$

Take a shell at r = r

($a\le r\le b)$

$dq=4\pi r^{2}dr\frac{A}{r}$
$\therefore qfromr=ator=r$

$q=4\pi A{\int }_a^{r}rdr=2\pi A[r^2-a^2]$

$\therefore$ Charge from r = a to r = b

$q=2\pi A[b^2-a^2]$

$Now,fieldatr=bis{E}_b=\frac{2\pi A[b^2-a^2]+Q}{{ϵ}_{\circ }\times 4\pi b^2}$

$Now,E_a=E_{b}gives,A=\frac{Q}{2\pi a^2}$