(a) Starting from the generating function identity , expand the left side in powers…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential

The Boundary-Value Problem, Ch The multipole expansion of the potential

Problem

(a) Starting from the generating function identity (1−2tcos⁡γ+t2)−1/2=∑ℓ=0∞tℓPℓ(cos⁡γ)(1-2t\cos\gamma+t^2)^{-1/2}=\sum_{\ell=0}^\infty t^\ell P_\ell(\cos\gamma), expand the left side in powers of tt up to and including t3t^3 using the binomial series (1+u)−1/2=1−12u+38u2−516u3+⋯(1+u)^{-1/2}=1-\tfrac12u+\tfrac38u^2-\tfrac5{16}u^3+\cdots with u=−2tcos⁡γ+t2u=-2t\cos\gamma+t^2, and show that the coefficient of t3t^3 is P3(cos⁡γ)=12(5cos⁡3γ−3cos⁡γ)P_3(\cos\gamma)=\tfrac12(5\cos^3\gamma-3\cos\gamma). (b) A point dipole p=pz^\mathbf{p}=p\hat{\mathbf{z}} sits at the origin. Starting from the vector form E(r)=3(p⋅r^)r^−p4πε0r3\mathbf{E}(\mathbf{r})=\dfrac{3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{4\pi\varepsilon_0r^3}, valid for r≠0\mathbf{r}\neq0, derive the spherical components ErE_r, EθE_\theta, EϕE_\phi in terms of pp, rr and the polar angle θ\theta measured from z^\hat{\mathbf{z}}. (c) Starting from the ℓ=2\ell=2 term of the multipole expansion, V2(r)=14πε0r3∫r′2P2(cos⁡γ) ρ(r′) dτ′V_2(\mathbf{r})=\dfrac{1}{4\pi\varepsilon_0r^3}\displaystyle\int r'^2P_2(\cos\gamma)\,\rho(\mathbf{r}')\,d\tau', and the identity r′cos⁡γ=r^⋅r′r'\cos\gamma=\hat{\mathbf{r}}\cdot\mathbf{r}', show that V2=18πε0r3∑i,jQijr^ir^jV_2=\dfrac{1}{8\pi\varepsilon_0r^3}\sum_{i,j}Q_{ij}\hat r_i\hat r_j, with Qij=∫(3xi′xj′−r′2δij)ρ dτ′Q_{ij}=\int(3x_i'x_j'-r'^2\delta_{ij})\rho\,d\tau' the traceless quadrupole tensor.

Browse the problem library