Constant electric field in the sphere

1. The problem statement, all variables and given/known data

The region between two concentric spheres of radii a and b(>a) contains volume charge density ρ(r) = 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).

A) q/πa2
B) q/π(a2+b2)
C) q/2πa2
D) q/2πb2

2. Relevant equations

ε∫E.dS = qfree

3. The attempt at a solution

Let us consider a a gaussian surface, a concentric sphere at radius r .

Applying Gauss law

ε∫E.dS = q + 4πC(r2-a2)

εE(4πr2) = q + 4πC(r2-a2)

E = q/(4πr2ε) + C(r2-a2)/(εr2)

Now for E to be constant ,

q-4πca2 = 0 or C=q/(4πa2) .This is not given as one of the options .

I would be grateful if somebody could help me with the problem.

Attached Images
File Type: gif sphere.GIF (3.1 KB)

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