Two unit charges are confined to a unit sphere and repel with energy

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

Two unit charges are confined to a unit sphere and repel with energy f=1/∣x1−x2∣f=1/|\mathbf{x}_1-\mathbf{x}_2|, in units of e2/4πϵ0Re^2/4\pi\epsilon_0R. Write each constraint as gi=∣xi∣2−1=0g_i=|\mathbf{x}_i|^2-1=0. ((a)) At the constrained minimum the charges are antipodal. Solve the stationarity condition ∇if=λi∇igi\nabla_if=\lambda_i\nabla_ig_i for the multiplier λ1\lambda_1, and find the size of the force the sphere exerts on each charge. ((b)) Replace the constraints by the penalty 12μ∑igi2\tfrac12\mu\sum_ig_i^2. By comparing the stationarity conditions of the penalised and the constrained problems, show that for large μ\mu the penalty minimum has gi≈−λi/μg_i\approx-\lambda_i/\mu, so each charge sits outside the sphere by δ≈1/(16μ)\delta\approx1/(16\mu). ((c)) Check this at μ=100\mu=100: write the penalised energy for antipodal charges at radius rr and find its minimum numerically. ((d)) At N=5N=5 the outward Coulomb force on each of the two polar charges of the constrained minimum is 1.3111.311 in the same units. Predict the largest radial violation at μ=100\mu=100 and compare with the chapter's 3.2×10−33.2\times10^{-3}.

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