3.II.17H

Electromagnetism | Part IB, 2005

If E(x,t),B(x,t)\mathbf{E}(\mathbf{x}, t), \mathbf{B}(\mathbf{x}, t) are solutions of Maxwell's equations in a region without any charges or currents show that E(x,t)=cB(x,t),B(x,t)=E(x,t)/c\mathbf{E}^{\prime}(\mathbf{x}, t)=c \mathbf{B}(\mathbf{x}, t), \mathbf{B}^{\prime}(\mathbf{x}, t)=-\mathbf{E}(\mathbf{x}, t) / c are also solutions.

At the boundary of a perfect conductor with normal n\mathbf{n} briefly explain why

nB=0,n×E=0\mathbf{n} \cdot \mathbf{B}=0, \quad \mathbf{n} \times \mathbf{E}=0

Electromagnetic waves inside a perfectly conducting tube with axis along the zz-axis are given by the real parts of complex solutions of Maxwell's equations of the form

E(x,t)=e(x,y)ei(kzωt),B(x,t)=b(x,y)ei(kzωt).\mathbf{E}(\mathbf{x}, t)=\mathbf{e}(x, y) e^{i(k z-\omega t)}, \quad \mathbf{B}(\mathbf{x}, t)=\mathbf{b}(x, y) e^{i(k z-\omega t)} .

Suppose bz=0b_{z}=0. Show that we can find a solution in this case in terms of a function ψ(x,y)\psi(x, y) where

(ex,ey)=(xψ,yψ),ez=i(kω2kc2)ψ,\left(e_{x}, e_{y}\right)=\left(\frac{\partial}{\partial x} \psi, \frac{\partial}{\partial y} \psi\right), \quad e_{z}=i\left(k-\frac{\omega^{2}}{k c^{2}}\right) \psi,

so long as ψ\psi satisfies

(2x2+2y2+γ2)ψ=0\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\gamma^{2}\right) \psi=0

for suitable γ\gamma. Show that the boundary conditions are satisfied if ψ=0\psi=0 on the surface of the tube.

Obtain a similar solution with ez=0e_{z}=0 but show that the boundary conditions are now satisfied if the normal derivative ψ/n=0\partial \psi / \partial n=0 on the surface of the tube.

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