(a) Expand in powers of up to and including the term in , using the binomial series…

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) Expand (1−2tcos⁡γ+t2)−1/2(1-2t\cos\gamma+t^2)^{-1/2} in powers of tt up to and including the term in t3t^3, using the binomial series (1+u)−1/2=1−12u+38u2−516u3+⋯(1+u)^{-1/2}=1-\tfrac12u+\tfrac38u^2-\tfrac{5}{16}u^3+\cdots with u=−2tcos⁡γ+t2u=-2t\cos\gamma+t^2. 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 V(r,θ)=pcos⁡θ4πε0r2V(r,\theta)=\dfrac{p\cos\theta}{4\pi\varepsilon_0r^2} for r≠0r\ne0, derive the spherical components ErE_r, EθE_\theta, EϕE_\phi of the field, and show they combine into E=3(p⋅r^)r^−p4πε0r3\mathbf{E}=\dfrac{3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{4\pi\varepsilon_0r^3}. (c) Starting from r′2P2(cos⁡γ)r'^2P_2(\cos\gamma) and the identity r′cos⁡γ=r^ixi′r'\cos\gamma=\hat r_ix_i', show in index notation that r′2P2(cos⁡γ)=12r^ir^j(3xi′xj′−r′2δij)r'^2P_2(\cos\gamma)=\tfrac12\hat r_i\hat r_j(3x_i'x_j'-r'^2\delta_{ij}), and hence that the ℓ=2\ell=2 term of the multipole expansion is V2(r)=18πε0r3∑ijQijr^ir^jV_2(\mathbf{r})=\dfrac{1}{8\pi\varepsilon_0r^3}\sum_{ij}Q_{ij}\hat r_i\hat r_j.

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