An electron initially at rest at point O (lying on the cathode) is acted upon by a magnetic field $\vec{B} = B_{0} \hat{k}$ and electric field $\vec{E} = - E_{0} \hat{i}$ This electric field results due to the potential difference V between the cathode and anode separated by a distance d as shown in figure. Dimension of anode & cathode plates are very large compared to ‘d’.

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The electron will just reach the anode in time $t = \frac{mπ}{2 qB}$

The electron will just reach the anode in time $t = \sqrt{\frac{2 md}{qE}}$

The electron will not reach the anode if $B_{0} = \frac{π}{2} \sqrt{\frac{m E_{0}}{q d}}$

The electron will not reach the anode if $B_{0} > \frac{π}{2} \sqrt{\frac{{mE}_{0}}{2 qd}}$

answer is A, C, D.

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Detailed Solution

$ω = \frac{qB}{m}$
The electron will not reach the anode only when it turns by $\frac{π}{2}$ by the time it comes near anode. Let time taken be t.
$\begin{array}{l} θ = ωt \\ \frac{π}{2} = \frac{qB}{m} t \Rightarrow t = \frac{mπ}{2 qB} \end{array}$
Acceleration due to electronic field a $= \frac{qE}{m}$
Horizontal distance curved in time t=d
$\frac{1}{2} {at}^{2} < d \Rightarrow B > \frac{π}{2} \sqrt{\frac{mE}{2 qd}}$