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Question:

The region between two concentric spheres of radii 'a' and 'b', respectively, has volume charge density ρ=Ar, where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is :

Q2πa²

2Qπa²

Q2π(b²-a²)

2Qπ(a²-b²)

Solution:

Let us find total charge enclosed in a sphere of radius r, Q'=Q+∫₀ʳ Ar4πr²dr=Q+2πAr⁴/4 = Q+(2πAr⁴)/4
By Gauss law, E×4πr²=Q'+(2πAr⁴)/4
Given, E is independent of r
Hence, Q'+(2πAr⁴)/4=0
This gives A=-Q/2πa²