Electromagnetism | Part IB, 2004

Starting from Maxwell's equations, derive the law of energy conservation in the form

Wt+S+JE=0\frac{\partial W}{\partial t}+\nabla \cdot \mathbf{S}+\mathbf{J} \cdot \mathbf{E}=0

where W=ϵ02E2+12μ0B2W=\frac{\epsilon_{0}}{2} E^{2}+\frac{1}{2 \mu_{0}} B^{2} and S=1μ0E×B\mathbf{S}=\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}.

Evaluate WW and S\mathbf{S} for the plane electromagnetic wave in vacuum

E=(E0cos(kzωt),0,0)B=(0,B0cos(kzωt),0),\mathbf{E}=\left(E_{0} \cos (k z-\omega t), 0,0\right) \quad \mathbf{B}=\left(0, B_{0} \cos (k z-\omega t), 0\right),

where the relationships between E0,B0,ωE_{0}, B_{0}, \omega and kk should be determined. Show that the electromagnetic energy propagates at speed c2=1/(ϵ0μ0)c^{2}=1 /\left(\epsilon_{0} \mu_{0}\right), i.e. show that S=WcS=W c.

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