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A rectangular region of dimensions $\mathrm{w} \times l(\mathrm{w}< < l)$ has a constant magnetic field into the plane of the paper as shown. On one side the region is bounded by a screen. On the other side positive ions of mass $\mathrm{m}$ and charge $\mathrm{q}$ are accelerated from rest and towards the screen by a parallel plate capacitor at constant potential difference $\mathrm{V} < 0$, and come out through a small hole in the upper plate. Which one of the following statements is correct regarding the charge on the ions that hit the screen?
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The correct answer is:
Ions with $\mathrm{q} < \frac{2|\mathrm{v}| \mathrm{m}}{\mathrm{B}^{2} \mathrm{w}^{2}}$ will hit the screen.
Ions will hit if $\mathrm{r}>\mathrm{W}$.
$\begin{array}{l}
\mathrm{w}=\mathrm{q} \Delta \mathrm{V} \\
\frac{1}{2} \mathrm{mv}^{2}=\mathrm{q} \mathrm{V} \\
\mathrm{v}=\sqrt{\frac{2 \mathrm{qV}}{\mathrm{m}}} \\
\frac{\mathrm{mv}^{2}}{\mathrm{r}}=\mathrm{qvB}
\end{array}$
$\begin{aligned} \Rightarrow \mathrm{r} &=\frac{\mathrm{mv}}{\mathrm{qB}}=\frac{\mathrm{m}}{\mathrm{qB}} \sqrt{\frac{2 \mathrm{q} \mathrm{V}}{\mathrm{m}}} \\ \mathrm{r}=\frac{1}{\mathrm{~B}} \sqrt{\frac{2 \mathrm{mV}}{\mathrm{q}}} \\ & \frac{1}{\mathrm{~B}} \sqrt{\frac{2 \mathrm{mV}}{\mathrm{q}}}>\mathrm{w} \\ & \frac{2 \mathrm{mV}}{\mathrm{q}}>\mathrm{w}^{2} \mathrm{~B}^{2} \\ \Rightarrow \mathrm{q} & < \frac{2 \mathrm{mV}}{\mathrm{w}^{2} \mathrm{~B}^{2}} \end{aligned}$
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