Question

A charged particle is fired at an angle $\theta$ to a uniform magnetic directed along the $x$-axis. During its motion along a helical path, if the pitch of the helical path is equal to the maximum distance of the particle from the $x$-axis.

• A. $\cos \theta =\frac{1}{\pi }$
• B. $\sin \theta =\frac{1}{\pi }$
• C. $\tan \theta =\frac{1}{\pi }$
• D. $\tan \theta =\pi$

Answer 1

The correct option is D $\tan \theta =\pi$

Pitch,$\left(P\right)=v_{\parallel }\times T=v\cos \theta \times \frac{2\pi m}{qB}$

and radius, $\left(r\right)=\frac{m{v}_{\perp }}{qB}=\frac{mv\sin \theta }{qB}$

Maximum distance of the particle from the $x$-axis $=2r$

$\therefore v\cos \theta \times \frac{2\pi m}{qB}=\frac{2mv\sin \theta }{qB}$

$\Rightarrow \cos \theta \times \pi =\sin \theta$

$\therefore \tan \theta =\pi$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, option (d) is the correct answer.