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Between two concentric spheres of radii a and b (a < b) the space carries a volume charge density rho = A/r, where A is a constant and r is the distance from the centre. A point charge Q sits at the common centre. For the electric field to be constant (independent of r) throughout the region a < r < b, the value of A must be:

1. 2Q/(pi*a²)
2. Q/(2*pi*a²)
3. Q/(2*pi*(b² - a²))
4. 2Q/(pi*(a² - b²))

Correct answer: Q/(2*pi*a²)

Solution

The charge enclosed within radius r is Q + 2*pi*A*(r² - a²). The field is this divided by 4*pi*eps0*r². For E to be constant the term that does not scale as r² (namely Q - 2*pi*A*a²) must be zero, giving A = Q/(2*pi*a²).

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