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