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The electric fields of two plane electromagnetic plane waves in vacuum are given by
$\overrightarrow{\mathrm{E}}_{1}=\mathrm{E}_{0} \hat{\mathrm{j}} \cos (\omega \mathrm{t}-\mathrm{kx})$ and
$\overrightarrow{\mathrm{E}}_{2}=\mathrm{E}_{0} \hat{\mathrm{k}} \cos (\omega \mathrm{t}-\mathrm{ky})$
At $t=0,$ a particle of charge $q$ is at origin with a velocity $\overrightarrow{\mathrm{v}}=0.8 \mathrm{c} \hat{\mathrm{j}}$ ($c$ is the speed of light in vacuum). The instantaneous force experienced by the particle is
$\mathrm{E}_{0} \mathrm{q}(-0.8 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$
$\mathrm{E}_{0} \mathrm{q}(0.8 \hat{\mathrm{i}}-\hat{\mathrm{j}}+0.4 \hat{\mathrm{k}})$
$\mathrm{E}_{0} \mathrm{q}(0.8 \hat{\mathrm{i}}+\hat{\mathrm{j}}+0.2 \hat{\mathrm{k}})$
$\mathrm{E}_{0} \mathrm{q}(0.4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+0.8 \hat{\mathrm{k}})$
Solution
$\overrightarrow{\mathrm{E}}_{1}=\mathrm{E}_{0} \mathrm{j} \cos (\omega \mathrm{t}-\mathrm{kx})$
Its corresponding magnetic field will be
$\overrightarrow{\mathrm{B}}_{1}=\frac{\mathrm{E}_{0}}{\mathrm{c}} \hat{\mathrm{k}} \cos (\omega \mathrm{t}-\mathrm{kx})$
$\overrightarrow{\mathrm{E}}_{2}=\mathrm{E}_{0} \hat{\mathrm{k}} \cos (\omega \mathrm{t}-\mathrm{ky})$
$\overrightarrow{\mathrm{B}}_{2}=\frac{\mathrm{E}_{0}}{\mathrm{c}} \hat{\mathrm{i}} \cos (\omega \mathrm{t}-\mathrm{ky})$
Net force on charge particle
$=\mathrm{q} \overrightarrow{\mathrm{E}}_{1}+\mathrm{q} \overrightarrow{\mathrm{E}}_{2}+\mathrm{q} \overrightarrow{\mathrm{v}} \times \mathrm{B}_{1}+\mathrm{q} \overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{B}}_{2}$
$=\mathrm{q} \mathrm{E}_{0} \hat{\mathrm{j}}+\mathrm{q} \mathrm{E}_{0} \hat{\mathrm{k}}+\mathrm{q}(0.8 \mathrm{c} \hat{\mathrm{j}}) \times\left(\frac{\mathrm{E}_{0}}{\mathrm{c}} \hat{\mathrm{k}}\right)+\mathrm{q}(0.8 \mathrm{c} \hat{\mathrm{j}}) \times\left(\frac{\mathrm{E}_{0}}{\mathrm{c}} \hat{\mathrm{i}}\right)$
$=\mathrm{q} \mathrm{E}_{0} \hat{\mathrm{j}}+\mathrm{q} \mathrm{E}_{0} \hat{\mathrm{k}}+0.8 \mathrm{q} \mathrm{E}_{0} \hat{\mathrm{i}}-0.8 \mathrm{q} \mathrm{E}_{0} \hat{\mathrm{k}}$
$\overrightarrow{\mathrm{F}}=\mathrm{q} \mathrm{E}_{0}[0.8 \hat{\mathrm{i}}+1 \hat{\mathrm{j}}+0.2 \hat{\mathrm{k}}]$
