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ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/section-a/a1/solution

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Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / Section A / A1 / Solution by

Codex  0 2026-10-20

For the gravitational-electrostatic balance of equal charged spheres, the shell theorem and Coulomb's law give

F_{g} = G M^{2} / r^{2}

and

F_{e} = q^{2} / (4 π ϵ_{0} r^{2})

for well-separated spherical particles. Their separation cancels at balance. Since

M = 4 π ρ a^{3} /3

a = (\frac{9 q ^{2}}{64 π ^{3} ϵ _{0} G ρ ^{2}})^{1/6} = [\frac{3∣ q ∣}{4 π ρ 4 π ϵ _{0} G}]^{1/3} .

(1)

The two charges have the same sign, so electrostatic repulsion opposes gravitational attraction. A zero charge gives no nontrivial force balance. Spherical symmetry of the external charge distribution, or the leading far-field monopole approximation, is understood.

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