A particle of charge q and mass m is projected from origin with an initial velocity v =(v_0/(√ 2) x +v_0/(√ 2) y ). There exists a uniform magnetic…

A particle of charge $\displaystyle q$ and mass $\displaystyle m$ is projected from origin with an initial velocity $\displaystyle \vec{v}=\left(\frac{v_0}{\sqrt{2}} \hat{x}+\frac{v_0}{\sqrt{2}} \hat{y}\right)$. There exists a uniform magnetic field $\displaystyle \vec{B}=B_0 \hat{z}$ and a space varying electric field $\displaystyle \vec{E}=E_{\mathrm{o}} \mathrm{e}^{-\lambda x} \hat{x}$ within the region $\displaystyle 0 \leqslant x \leqslant L$. After travelling a distance such that $\displaystyle x$-coordinate has changed from $\displaystyle x=0$ to $\displaystyle x=L$, the change in the kinetic energy is $\displaystyle \_\_\_\_$.