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The region between $y = 0$ and $y = d$ contains a magnetic field $\vec B = B\hat z$ A particle of mass $m$ and charge $q$ enters the region with a velocity $\vec v = v\hat i$. If $d = \frac{{mv}}{{2qB}}$ , the acceleration of the charged particle at the point of its emergence at the other side is
$\frac{{qvB}}{m}\,\left( {\,\frac{{ \hat j + \hat i}}{{\sqrt 2 }}} \right)$
$\frac{{qvB}}{m}{\mkern 1mu} \left( {{\mkern 1mu} \frac{{\sqrt 3 }}{2}{\mkern 1mu} \hat i + \frac{1}{2}\hat j} \right)$
$\frac{{qvB}}{m}\,\left( {\,\frac{{ - \hat j + \hat i}}{{\sqrt 2 }}} \right)$
$\frac{{qvB}}{m}{\mkern 1mu} \left( {\frac{1}{2}\hat j - \frac{{\sqrt 3 }}{2}\hat i} \right)$
Solution
$\overrightarrow{\mathrm{B}}=\mathrm{B} \hat{\mathrm{k}}$
$r=\frac{m v}{q B}$
$r=2 d$
$\sin \theta=\frac{d}{2 d} \Rightarrow \sin \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{6}$
$\vec{a}=\frac{|q| v b}{m}(\cos \theta(-\hat{i})+\sin \theta \hat{j})$
$\vec{a}=\frac{|q| v b}{m}\left(-\frac{\sqrt{3}}{2} i+\frac{1}{2} \hat{j}\right)$
