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  4. The electric fields of two plane electromagnetic plane waves in vacuum are given by vecE1 = E0 hatj cos(ω t — kx) and vecE2 = E0 hatk cos(ω t - ky) At t = 0, a partide of charge q is at origin with a velocity vecv = 0.8 c hatj (c is the speedof light in vaccum). The instantaneousforce experienced by the partide is :

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Q. The electric fields of two plane electromagnetic plane waves in vacuum are given by
$\vec{E}_{1} = E_{0}\hat{j} \, cos\left(\omega t — kx\right)$ and
$\vec{E_{2}} = E_{0}\hat{k} cos\left(\omega t - ky\right)$
At $t = 0$, a partide of charge q is at origin with a velocity $\vec{v} = 0.8\,c\hat{j}$ (c is the speedof light in vaccum). The instantaneousforce experienced by the partide is :

JEE MainJEE Main 2020Electromagnetic Waves

Solution:

$\vec{E}_{1}=E_{0}\hat{j}\,cos\left(\omega t-kx\right)$
Its corresponding magnetic field will be
$\vec{B_{1}}=\frac{E_{0}}{c}\hat{k}\,cos\left(\omega t-kx\right)$
$\vec{E}_{2}=E_{0}\hat{k}\,cos\left(\omega t-ky\right)$
$\vec{B_{2}}=\frac{E_{0}}{c}\hat{i}\,cos\left(\omega t-ky\right)$
Net force on charge particle
$=q\vec{E_{1}}+q\vec{E_{2}}+q\vec{v}\times B_{1}+q\vec{v}\times\vec{B_{2}}$
$=qE_{0}\,\hat{j}+qE_{0}\,\hat{k}+q\left(0.8c\hat{j}\right)\times\left(\frac{E_{0}}{c}\hat{k}\right)+q\left(0.8c\hat{j}\right)\times\left(\frac{E_{0}}{c}\hat{i}\right)$
$=qE_{0}\,\hat{j}+qE_{0}\,\hat{k}+0.8qE_{0}\,\hat{i}-0.8qE_{0}\,\hat{k}$
$\vec{F}=qE_{0}\left[0.8\,\hat{i}+1\,\hat{j}+0.2\,\hat{k}\right]$