A total charge 2Q is distributed throughout a sphere of radius R with a radial charge density rho(r) = k*r (r = distance from centre). Two point charges A and B, each of charge -Q, are placed at diametrically opposite points, each at the same distance a from the centre (inside the sphere).

Question

A total charge 2Q is distributed throughout a sphere of radius R with a radial charge density rho(r) = k*r (r = distance from centre). Two point charges A and B, each of charge -Q, are placed at diametrically opposite points, each at the same distance a from the centre (inside the sphere). If neither A nor B experiences any net force, find a.

Accepted Answer

a = 8^(-1/4) R. First find k from total charge: 2Q = integral₀^R rho 4*pi r² dr = pi k R⁴, so k = 2Q/(pi R⁴). The charge enclosed within radius a is q(a) = pi k a⁴ = 2Q (a/R)⁴. Charge A (-Q) at distance a feels: (i) attraction toward centre from the enclosed positive charge q(a), and (ii) attraction toward B from the other -Q... wait, both are negative so they repel. The two forces must cancel: the inward pull from the sphere's enclosed charge equals the outward repulsion from the identical charge B at distance 2a. Setting magnitudes equal and solving gives a = R/8^(1/4) = 8^(-1/4) R.

Solution

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