Classical Electrodynamics · Problem 4.7

Magnetostatics, Faraday's Law, Quasi-Static Fields — John David Jackson

Physics — E&M · multipole-expansion-charge-density

Problem

A localized distribution of charge has a charge density ρ(r)=164π r2e−rsin⁡2θ\rho(\mathbf{r}) = \frac{1}{64\pi}\, r^2 e^{-r} \sin^2\theta ((a)) Make a multipole expansion of the potential due to this charge density and determine all the nonvanishing multipole moments. Write down the potential at large distances as a finite expansion in Legendre polynomials. ((b)) Determine the potential explicitly at any point in space, and show that near the origin, correct to r2r^2 inclusive, Φ(r)≈14πϵ0[14−r2120 P2(cos⁡θ)]\Phi(\mathbf{r}) \approx \frac{1}{4\pi\epsilon_0}\left[\frac{1}{4} - \frac{r^2}{120}\, P_2(\cos\theta)\right] ((c)) If there exists at the origin a nucleus with a quadrupole moment Q=10−28Q = 10^{-28} m2^2, determine the magnitude of the interaction energy, assuming that the unit of charge in ρ(r)\rho(\mathbf{r}) above is the electronic charge and the unit of length is the hydrogen Bohr radius a0=4πϵ0ℏ2/me2=0.529×10−10a_0 = 4\pi\epsilon_0 \hbar^2/me^2 = 0.529 \times 10^{-10} m. Express your answer as a frequency by dividing by Planck's constant hh. The charge density in this problem is that for the m=±1m = \pm 1 states of the 2p2p level in hydrogen, while the quadrupole interaction is of the same order as found in molecules.

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