A plane EM wave travelling along z-direction is given by $\overrightarrow{\mathrm{E}}={\mathrm{E}}_0<mspace\; width="1em"></mspace>\sin (\mathrm{kz}-\mathrm{\omega t})\hat{\mathrm{i}}<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}={\mathrm{B}}_0<mspace\; width="1em"></mspace>\sin (\mathrm{kz}-\mathrm{\omega t})\hat{\mathrm{j}}.<mspace\; width="1em"></mspace>\mathrm{Show}<mspace\; width="1em"></mspace>\mathrm{that}:$

$\left(\mathrm{i}\right)<mspace\; width="1em"></mspace>\mathrm{Evaluate}<mspace\; width="1em"></mspace>\oint \overrightarrow{\mathrm{E}}.\mathrm{d}\overrightarrow{\mathrm{l}}<mspace\; width="1em"></mspace>\mathrm{over}<mspace\; width="1em"></mspace>\mathrm{the}<mspace\; width="1em"></mspace>\mathrm{rectangular}<mspace\; width="1em"></mspace>\mathrm{loop}<mspace\; width="1em"></mspace>1234<mspace\; width="1em"></mspace>\mathrm{shown}<mspace\; width="1em"></mspace>\mathrm{in}<mspace\; width="1em"></mspace>\mathrm{figure}.$
$\left(\mathrm{ii}\right)<mspace\; width="1em"></mspace>\mathrm{Evaluate}<mspace\; width="1em"></mspace>\oint \overrightarrow{\mathrm{B}}.\mathrm{d}\overrightarrow{\mathrm{s}}<mspace\; width="1em"></mspace>\mathrm{over}<mspace\; width="1em"></mspace>\mathrm{the}<mspace\; width="1em"></mspace>\mathrm{surface}<mspace\; width="1em"></mspace>\mathrm{bounded}<mspace\; width="1em"></mspace>\mathrm{loop}<mspace\; width="1em"></mspace>1234.$
$\left(\mathrm{iii}\right)<mspace\; width="1em"></mspace>\mathrm{Use}<mspace\; width="1em"></mspace>\mathrm{equation}<mspace\; width="1em"></mspace>\oint \overrightarrow{\mathrm{E}}.\mathrm{d}\overrightarrow{\mathrm{l}}=-\frac{{\mathrm{d\varphi }}_{\mathrm{B}}}{\mathrm{dt}}<mspace\; width="1em"></mspace>\mathrm{to}<mspace\; width="1em"></mspace>\mathrm{prove}<mspace\; width="1em"></mspace>\frac{{\mathrm{E}}_{\mathrm{o}}}{{\mathrm{B}}_{\mathrm{o}}}=\mathrm{c}.$
$\left(\mathrm{iv}\right)<mspace\; width="1em"></mspace><mspace\; width="1em"></mspace>\mathrm{By}<mspace\; width="1em"></mspace>\mathrm{using}<mspace\; width="1em"></mspace>\oint \overrightarrow{\mathrm{B}}.\mathrm{d}\overrightarrow{\mathrm{s}}<mspace\; width="1em"></mspace>={\mathrm{\mu }}_{\mathrm{o}}\mathrm{I}+{\mathrm{\epsilon }}_{\mathrm{o}}\frac{{\mathrm{d\varphi }}_{\mathrm{E}}}{\mathrm{dt}},<mspace\; width="1em"></mspace>\mathrm{prove}<mspace\; width="1em"></mspace>\mathrm{that}<mspace\; width="1em"></mspace>\mathrm{c}=\frac{1}{\sqrt{{\mathrm{\mu }}_{\mathrm{o}}{\mathrm{\epsilon }}_{\mathrm{o}}}}.$