Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 2

Use separation of variables, writing $\Psi =F(r)\Theta (\theta )$. After multiplication by $r^2/(F\Theta )$, the radial and angular terms are functions of independent variables, so setWith $x=\cos \theta$, the angular equation isRegularity at both poles selects $\lambda =\ell (\ell +1)$, $\ell =0,1,\dots$, and the Legendre polynomials $P_{\ell }(x)$; the other angular branch is singular at a pole. The radial Euler-Cauchy equation is $r^2{F}^{\prime \prime }+2r{F}^{\prime }-\ell (\ell +1)F=0$. A power $r^s$ gives $(s-\ell )(s+\ell +1)=0$, hence $F=a_{\ell }{r}^{\ell }+b_{\ell }{r}^{-\ell -1}$.

Superposing these axisymmetric spherical harmonics givesThe coefficients must give convergence appropriate to the domain. In a region containing the origin, regularity removes the negative powers; in an exterior region with zero potential at infinity, it removes the growing powers and the constant. No such radial restriction is needed for a general annular region.

Resolve the angular surface charge density into Legendre polynomials:For a single degree $\ell$, write the common surface potential as $V_{\ell }{P}_{\ell }$. Regularity and decay require radial factors $(r/R{)}^{\ell }$ inside and $(R/r{)}^{\ell +1}$ outside. Their exterior-minus-interior radial derivative at $R$ is $-(2\ell +1)V_{\ell }{P}_{\ell }/R$. Thus spherical harmonic matching with a derivative jump gives $V_{\ell }=R{\sigma }_{\ell }/(2\ell +1)$.

Only degrees zero and two occur, yieldingBoth expressions have surface value $2AR/3-2AR{P}_2/15$, and their derivative difference is $-2A/3+(2A/3)P_2=-A{\sin }^2\theta$, checking both interface conditions. The interior quadratic is regular at the origin, and the exterior potential tends to zero.

The leading far-field term is the monopole term. Indeed the total charge is $Q=R^2\int \sigma d\Omega =8\pi A{R}^2/3$, and the normalization of the specified jump gives $Q/(4\pi r)=2A{R}^2/(3r)$. No extra permittivity factor is inserted into the supplied normalization. Regularity here means finite piecewise smooth potential; its normal derivative necessarily jumps at the charged surface.