Worked example 6.2.1: Microwave oven heating depth

$\tan {\delta }_c={\epsilon }_r^{\prime \prime }/{\epsilon }_r^{\prime }=10/78=0.128$ and ${\sigma }_{\mathrm{eff}}=2\pi \times 2.45\times 10^9\times \frac{10^{-9}}{36\pi }\times 10=1.36S/m$: neither low-loss nor a good conductor, so the exact formula is the safe one.

$\gamma =\mathrm{j}\frac{2\pi f}{c}\sqrt{78-\mathrm{j}10}=\mathrm{j}51.31(8.850-\mathrm{j}0.565)$:

$\beta \approx 2\pi f\sqrt{78}/c=453rad/m$ and $\alpha \approx \frac{\beta }{2}\tan {\delta }_c=29.05Np/m$: only $0.2\%$ high.

$D_p=\frac{1}{2\alpha }=1.72\mathrm{cm}$, $z_{1/2}=\frac{\ln 2}{2\alpha }=1.20\mathrm{cm}$, and $e^{-2\alpha \times 0.02}=0.31$ of the power is left at $2\mathrm{cm}$.

$\alpha \beta$ must equal $\frac{1}{2}\omega {\mu }_0{\sigma }_{\mathrm{eff}}$ by Eq. (6.2.9): $29.0\times 454=1.32\times 10^4$ and $\frac{1}{2}\times 1.539\times 10^{10}\times 1.257\times 10^{-6}\times 1.36=1.32\times 10^4$.