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Question:

Let a total charge 2Q be distributed in a sphere of radius R, with the charge density given by ρ(r)=kr, where r is the distance from the centre. Two charges A and B, of -Q each, are placed on diametrically opposite points, at equal distance, a from the centre. If A and B do not experience any force, then:

a=R3

a=2⁻¹/4R

a=3R²/⁴

a=8⁻¹/⁴R

Solution:

Correct option is C. a=8⁻¹/⁴R
E4πa² = ∫₀ᵃ kr 4πr²dr
ε₀E = k4πa⁴/4
74πε₀2Q = ∫₀ᴿ kr 4πr²dr
k = 2Q/πR⁴ ———(1)
QE = 1/4πε₀ Q²/ (2a)² ———(2)
sum. (1) and (2)
R = a8¹/⁴