In the figure, the inner (shaded) region A represents a sphere of radius r A = 1 , within which the electrostatic charge density varies with the radial distance r from the center as ρ A = k r , where k is positive. In the spherical shell B of outer radius r B , the electrostatic charge density varies as ρ B = 2 k r . Assume that dimensions are taken care of. All physical quantities are in their SI units. Which of the following statement(s) is/(are) correct?
B. If r B = 3 2 , then the electric potential just outside B is k ε 0 .
Since both the densities are positive, the total charge can not be zero. Hence, option A is incorrect. For total charge, Q Total = ∫ 0 r A k r 4 π r 2 d r + ∫ r A r B 2 k r 4 π r 2 d r = 4 π k 4 r A 4 + 8 π k 2 r B 2 - r A 2 = π k + 4 π k r B 2 - r A 2 If r B = 3 2 Q Total   = π k + 4 π k 9 4 - 1 = π k + 4 π k 5 4 = 6 π k The potential just outside B will be, V = 1 4 π ε 0 Q total r B = 1 4 π ε 0 6 π k r B = 3 k 2 2 3 ε 0 = k ε 0 Hence, option B is correct. If r B = 2 Q Total   = π k + 4 π k 4 - 1 = 13 π k Hence, option C is incorrect If r B = 5 2 Q Total   = π k + 4 π k 25 4 - 1 = π k + π k 21 = 22 π k Therefore, the electric field just outside B will be, E = 1 4 π ε 0 Q Total r B 2 1 4 π ε 0 22 π k 25 × 4 = 22 k 25 ε 0 Hence, option D is incorrect.
Related: Physics — Electrostatics · All PYQ Banks