5.6.11 Problems and solutions

The problems of this subsection use the tools of the section: the reading of a monochromatic field $u=A\cos [\omega t-\varphi (r)]$ with $v_p=\omega /|\nabla \varphi |$ and $\lambda =2\pi /\beta =v_p/f$, the phasor rule $u=\mathrm{Re}\{U{e}^{j\omega t}\}$ with $\partial /\partial t\to j\omega$, Maxwell's curl equations in phasor form,

the Helmholtz equation ${\nabla }^{2}E+k^{2}E=0$ with $k^2={\omega }^2\mu \epsilon -j\omega \mu \sigma$, $k=\beta -j\alpha$ and $\tan \delta =\sigma /\omega \epsilon$, and Faraday's law $\mathrm{emf}=-d\Phi /dt$ of Section 5.3. Some problems also ask for the power carried by the wave. Section 6.1 shows that the instantaneous power density of a wave is the Poynting vector $P=E\times H[W/{\mathrm{m}}^2]$, that its time average is the real part of the complex Poynting vector,

where the asterisk denotes the complex conjugate, and that a conducting medium turns the average power $\frac{1}{2}\sigma |E{|}^2$ per unit volume into heat. These results are used here as given. Constants: ${\epsilon }_0=10^{-9}/(36\pi )F/m$, ${\mu }_0=4\pi \times 10^{-7}H/m$, $c=3\times 10^{8}m/s$ and ${\eta }_0=120\pi \Omega$.

Problems 5.6.1–5.6.5 follow the lecture. Problems 5.6.6–5.6.14 come from the exercise lecture that follows it, nine exercises in reading time-harmonic fields with the recipe of Section 5.6.9. Table 5.6.6 shows what each of them practises.

Table 5.6.6. The exercise problems 5.6.6–5.6.14.

Problem	Given	Practised
5.6.6	modulated wave, loop	Faraday's law, a mixer
5.6.7	phasor $E(x)$, ${\epsilon }_r$	Helmholtz: $f$ and $\sigma$ from $k^2$
5.6.8	damped $E(x;t)$	phasor, $k$, ${\epsilon }_r$, $\sigma$, $H$, power
5.6.9	two-component $E(x;t)$	$\lambda$ and ${\epsilon }_r$ from the phase
5.6.10	$E$, $H$ at $t=0$	$E(y;t)$, $H(y;t)$, then $\epsilon$, $\mu$
5.6.11	phasor $E(x)$, ${\mu }_r$, ${\epsilon }_r$	$H(x;t)$ from Faraday's law
5.6.12	two snapshots (a plot)	$v_p$, $f$, ${\epsilon }_r$ and $E(x;t)$
5.6.13	phasor $E(y,z)$, $f$	phasor $\to$ instantaneous
5.6.14	instantaneous $H(x;t)$	instantaneous $\to$ phasor