A uniformly charged solid sphere has charge $Q$ and radius $R$ . Divide the sphere (mentally) into two regions spherical concentric part having radius $\frac{R}{2}$ and the remaining annular part (between $\frac{R}{2}$ & $R$ ). Denote the point charges in sphere of radius $\frac{R}{2}$ by $q_{1}, q_{2}, q_{3} \dots \dots$ etc . The charges in annular part denoted by $Q_{1}, Q_{2}, Q_{3} ...... e t c$ . The total electrostatic interaction energy for all pairs like $[(Q_{i}, Q_{j}) + (q_{i}, q_{j})]$ (exclude the pairs $(Q_{i}, q_{j})$ i.e. interaction energy of pair of particles from the two different regions)is $\frac{a}{b} \frac{Q^{2}}{4 π ε_{0} R}$ .Find $b - a$ .

Question

A uniformly charged solid sphere has charge $Q$ and radius $R$ . Divide the sphere (mentally) into two regions spherical concentric part having radius $\frac{R}{2}$ and the remaining annular part (between $\frac{R}{2}$ & $R$ ). Denote the point charges in sphere of radius $\frac{R}{2}$ by $q_{1}, q_{2}, q_{3} \dots \dots$ etc . The charges in annular part denoted by $Q_{1}, Q_{2}, Q_{3} ...... e t c$ . The total electrostatic interaction energy for all pairs like $[(Q_{i}, Q_{j}) + (q_{i}, q_{j})]$ (exclude the pairs $(Q_{i}, q_{j})$ i.e. interaction energy of pair of particles from the two different regions)is $\frac{a}{b} \frac{Q^{2}}{4 π ε_{0} R}$ .Find $b - a$ .

Accepted Answer

The desired interaction energy is U1=U−UqQWhere U = interaction energy of all pairs possible inside the sphere  =35Q24πε0R UqQ   =    interaction energy of the sphere of radius R2 (having charge ‘q’) with the charge in the annular part (i.e., Q1, Q2)… q=43π(R2)3.ρPotential due to this charge at radius  r(≥R2)Vr=14πε0qr=R3ρ24ε0.1rEnergy of charge in the layer of thickness dr, in the electrostatic field of q is:dUqQ=(ρ4πr2dr).Vr=πR3ρ26ε0rdr∴UqQ=πR3ρ26ε0∫R/2Rrdr=πR5ρ216ε0Here ρ=3Q4πR3 ∴U1=U−UqQ=Q24πε0R[35−964]=147320Q24πε0R

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