The electric field associated with an $e.m.$ wave in vacuum is given by $\vec E = \hat i\,40\,\cos \,\left( {kz - 6 \times {{10}^8}\,t} \right)$. where $E$, $z$ and $t$ are in $volt/m$, meter and seconds respectively. The value of wave factor $k$ is ....... $m^{-1}$.

An electromagnetic wave travelling in the $x-$ direction has frequency of $2 \times 10^{14}\,Hz$ and electric field amplitude of $27\,Vm^{-1}$ . From the options given below, which one describes the magnetic field for this wave ?

A linearly polarized electromagnetic wave in vacuum is $E=3.1 \cos \left[(1.8) z-\left(5.4 \times 10^{6}\right) {t}\right] \hat{\text { i }}\, {N} / {C}$ is incident normally on a perfectly reflecting wall at $z=a$. Choose the correct option

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

$(i)$ The average energy density of the wave is given by $U_{av} = \frac{1}{4}{ \in _0}E_0^2 + \frac{1}{4}.\frac{{B_0^2}}{{{\mu _0}}}$

$(ii)$ The time averaged intensity of the wave is given by  $ I_{av}= \frac{1}{2}c{ \in _0}E_0^2$ વડે આપવામાં આવે છે.

There exists a uniform magnetic and electric field of magnitude $1\, T$ and $1\, V/m$ respectively along positive $y-$ axis. A charged particle of mass $1\,kg$ and of charge $1\, C$ is having velocity $1\, m/sec$ along $x-$ axis and is at origin at $t = 0.$ Then the co-ordinates of particle at time $\pi$ seconds will be :-

The magnetic field in a plane electromagnetic wave is given by

${B_y} = \left( {2 \times {{10}^{ - 7}}} \right)\sin \left( {0.5 \times {{10}^3}x + 1.5 \times {{10}^{11}}t} \right)T$

$(a)$ What is the wavelength and frequency of the wave?

$(b)$ Write an expression for the electric field.