Paper 3, Section II, C

Vector Calculus | Part IA, 2018

Use Maxwell's equations,

E=ρ,B=0,×E=Bt,×B=J+Et\boldsymbol{\nabla} \cdot \mathbf{E}=\rho, \quad \boldsymbol{\nabla} \cdot \mathbf{B}=0, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \quad \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}

to derive expressions for 2Et22E\frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E} and 2Bt22B\frac{\partial^{2} \mathbf{B}}{\partial t^{2}}-\nabla^{2} \mathbf{B} in terms of ρ\rho and J\mathbf{J}.

Now suppose that there exists a scalar potential ϕ\phi such that E=ϕ\mathbf{E}=-\nabla \phi, and ϕ0\phi \rightarrow 0 as rr \rightarrow \infty. If ρ=ρ(r)\rho=\rho(r) is spherically symmetric, calculate E\mathbf{E} using Gauss's flux method, i.e. by integrating a suitable equation inside a sphere centred at the origin. Use your result to find E\mathbf{E} and ϕ\phi in the case when ρ=1\rho=1 for r<ar<a and ρ=0\rho=0 otherwise.

For each integer n0n \geqslant 0, let SnS_{n} be the sphere of radius 4n4^{-n} centred at the point (14n,0,0)\left(1-4^{-n}, 0,0\right). Suppose that ρ\rho vanishes outside S0S_{0}, and has the constant value 2n2^{n} in the volume between SnS_{n} and Sn+1S_{n+1} for n0n \geqslant 0. Calculate E\mathbf{E} and ϕ\phi at the point (1,0,0)(1,0,0).

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