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Electromagnetic Waves. Radiation

Problem 4.193

A plane electromagnetic wave E=Emcos⁡(ωt−kr)\mathbf{E}=\mathbf{E}_ {m} \cos (\omega t-\mathbf{k r}), where Em=Emey,k=kex,ex,ey\mathbf{E}_ {m}=E_{m} \mathrm{e}_ {y}, \mathbf{k}=k \mathrm{e}_ {x}, \mathrm{e}_ {x}, \mathrm{e}_ {y} are the unit vectors of the xx yy axes, propagates in vacuum. Find the vector H\mathrm{H} at the point with radius vector r=xex\mathbf{r}=x e_{x} at the moment (a) t=0t=0 (b) t=t0t=t_{0}. Consider the case when Em=160 V/m,k=0.51 m−1,x=7.7 m,E_{m}=160 \mathrm{~V} / \mathrm{m}, k=0.51 \mathrm{~m}^{-1}, x=7.7 \mathrm{~m}, and t0=t_{0}= =33 ns=33 \mathrm{~ns}

Reveal Answer
4.193. (a) H=ezEmε0/μ0cos⁡kx=−0.30ez\mathbf{H}=\mathbf{e}_{z} E_{m} \sqrt{\varepsilon_{0} / \mu_{0}} \cos k x=-0.30 \mathbf{e}_{z} (b) H=ezEmε0/μ0cos⁡(ckt0−kx)=0.18ez\mathbf{H}=\mathbf{e}_{z} E_{m} \sqrt{\varepsilon_{0} / \mu_{0}} \cos \left(c k t_{0}-k x\right)=0.18 \mathbf{e}_{z}. Here ez\mathbf{e}_{z} is the unit vector of the zz axis, HH is expressed in A/m\mathrm{A} / \mathrm{m}
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