The energy of electromagnetic wave in vacuum is given by the relation

A plane $EM$ wave travelling in vacuum along $z-$ direction is given by $\vec E = {E_0}\,\,\sin (kz - \omega t)\hat i$ and $\vec B = {B_0}\,\,\sin (kz - \omega t)\hat j$.

$(i)$ Evaluate $\int {\vec E.\overrightarrow {dl} } $ over the rectangular loop $1234$ shown in figure.

$(ii)$ Evaluate $\int {\vec B} .\overrightarrow {ds} $ over the surface bounded by loop $1234$.

$(iii)$ $\int {\vec E.\overrightarrow {dl}  =  - \frac{{d{\phi _E}}}{{dt}}} $ to prove $\frac{{{E_0}}}{{{B_0}}} = c$

$(iv)$ By using similar process and the equation $\int {\vec B} .\overrightarrow {dl}  = {\mu _0}I + { \in _0}\frac{{d{\phi _E}}}{{dt}}$ , prove that  $c = \frac{1}{{\sqrt {{\mu _0}{ \in _0}} }}$ 

The velocity of certain ions that pass undeflected through crossed electric field $E = 7.7\,k\,V /m$ and magnetic field $B = 0.14\,T$ is.....$km/s$

Suppose that the electric field amplitude of an electromagnetic wave is $E_{0}=120\; N / C$ and that its frequency is $v=50.0\; MHz$.

$(a)$ Determine, $B_{0}, \omega, k,$ and $\lambda .$

$(b)$ Find expressions for $E$ and $B$

In the $EM$ wave the amplitude of magnetic field $H_0$ and the amplitude of electric field $E_o$ at any place are related as

The electric field and magnetic field components of an electromagnetic wave going through vacuum is described by

$E _{ x }= E _0 \sin ( kz -\omega t )$

$B _{ y }= B _0 \sin ( kz -\omega t )$

Then the correct relation between $E_0$ and $B_0$ is given by