Paper 3, Section II, B

Vector Calculus | Part IA, 2021

For a given charge distribution ρ(x,t)\rho(\mathbf{x}, t) and current distribution J(x,t)\mathbf{J}(\mathbf{x}, t) in R3\mathbb{R}^{3}, the electric and magnetic fields, E(x,t)\mathbf{E}(\mathbf{x}, t) and B(x,t)\mathbf{B}(\mathbf{x}, t), satisfy Maxwell's equations, which in suitable units, read

E=ρ,×E=BtB=0,×B=J+Et\begin{aligned} \boldsymbol{\nabla} \cdot \mathbf{E}=\rho, & \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \boldsymbol{\nabla} \cdot \mathbf{B}=0, & \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t} \end{aligned}

The Poynting vector P\mathbf{P} is defined as P=E×B\mathbf{P}=\mathbf{E} \times \mathbf{B}.

(a) For a closed surface S\mathcal{S} around a volume V\mathcal{V}, show that

SPdS=VEJdVtVE2+B22dV\int_{\mathcal{S}} \mathbf{P} \cdot d \mathbf{S}=-\int_{\mathcal{V}} \mathbf{E} \cdot \mathbf{J} d V-\frac{\partial}{\partial t} \int_{\mathcal{V}} \frac{|\mathbf{E}|^{2}+|\mathbf{B}|^{2}}{2} d V

(b) Suppose J=0\mathbf{J}=\mathbf{0} and consider an electromagnetic wave

E=E0y^cos(kxωt) and B=B0z^cos(kxωt)\mathbf{E}=E_{0} \hat{\mathbf{y}} \cos (k x-\omega t) \quad \text { and } \quad \mathbf{B}=B_{0} \hat{\mathbf{z}} \cos (k x-\omega t)

where E0,B0,kE_{0}, B_{0}, k and ω\omega are positive constants. Show that these fields satisfy Maxwell's equations for appropriate E0,ωE_{0}, \omega, and ρ\rho.

Confirm the wave satisfies the integral identity ()(*) by considering its propagation through a box V\mathcal{V}, defined by 0xπ/(2k),0yL0 \leqslant x \leqslant \pi /(2 k), 0 \leqslant y \leqslant L, and 0zL0 \leqslant z \leqslant L.

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