# 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/πa^{2}

B) q/π(a^{2}+b^{2})

C) q/2πa^{2}

D) q/2πb^{2}

**2. Relevant equations**

ε∫**E**.**dS** = q_{free}

**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(r^{2}-a^{2})

εE(4πr^{2}) = q + 4πC(r^{2}-a^{2})

E = q/(4πr^{2}ε) + C(r^{2}-a^{2})/(εr^{2})

Now for E to be constant ,

q-4πca^{2} = 0 or C=q/(4πa^{2}) .This is not given as one of the options .

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

http://ift.tt/1fh5wqJ

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