Electroneering - Classical Model of Electron · USAPhO 2021 #A3

Physics · electromagnetism

Problem

An electron is a particle with charge −q, mass m, and magnetic moment μ. In this problem we will explore whether a classical model consistent with these properties can also explain the rest energy E₀ = mc² of the electron. Let us describe the electron as a thin spherical shell with uniformly distributed charge and radius R. Recall that the magnetic moment of a closed, planar loop of current is always equal to the product of the current and the area of the loop. For the electron, a magnetic moment can be created by making the sphere rotate around an axis passing through its center. a. If no point on the sphere's surface can travel faster than the speed of light (in the frame of the sphere's center of mass), what is the maximum magnetic moment that the sphere can have? You may use the integral: ∫0πsin⁡3θdθ=43.\int_0^{\pi} \sin^3 \theta d\theta = \frac{4}{3}. b. The electron's magnetic moment is known to be μ=qℏ/2m\mu = q\hbar/2m, where ℏ\hbar is the reduced Planck constant. In this model, what is the minimum possible radius of the electron? Express your answer in terms of m and fundamental constants. c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of E₀ = mc² and the fine structure constant, α=q24πϵ0ℏc≈1137.\alpha = \frac{q^2}{4\pi\epsilon_0\hbar c} \approx \frac{1}{137}. d. Roughly estimate the total energy stored in the magnetic field of the electron, in terms of E₀ and α. (Hint: one way to do this is to suppose the magnetic field has roughly constant magnitude inside the sphere and is negligible outside of it, then estimate the field inside the sphere.) e. How does your estimate for the total energy in the electric and magnetic fields compare to E₀? In parts (a) and (b), you can also give your answers up to a dimensionless multiplicative constant for partial credit.

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