Question

Two identical charged particles enter a uniform magnetic field with same speed but at angles $30^{\circ }$ and $60^{\circ }$ with field. Let $a,b$ and $c$ be the ratio of their time periods, radii and pitches of the helical paths then-

• A. $abc=\frac{1}{\sqrt{3}}$
• B. $abc>1$
• C. $abc<1$
• D. $a=bc$

Answer 1

Answer: (D)

$T=\frac{2\pi m}{Bq}$
$\therefore a=\frac{T_1}{T_2}=1$
$r=\frac{mv\sin \theta }{Bq}$
$\therefore b=\frac{r_1}{r_2}=\frac{\sin 30^{\circ }}{\sin 60^{\circ }}=\frac{1}{\sqrt{3}}$
$p=(T)(v\cos \theta )$
$\therefore c=\frac{p_1}{p_2}=\frac{\cos 30^{\circ }}{\cos 60^{\circ }}=\sqrt{3}$
Therefore $a=bc$