Question

$\alpha -particles$, deuterons and protons of same energy are moving in a perpendicular magnetic field. The ratio of diameters of their circular paths will be

• A. $1:\sqrt{2}:1$
• B. $\sqrt{2}:1:1$
• C. $2:1:1$
• D. $1:1:2$

Answer 1

The correct option is A $1:\sqrt{2}:1$

In a magnetic field, perpendicular to velocity of particle
$\frac{m{v}^2}{r}=Bqv$
or $r=\frac{mv}{Bq}$
Energy $E=\frac{1}{2}m{v}^2$ or $mv=\sqrt{2mE}$
$\therefore r=\frac{\sqrt{2mE}}{Bq}$
For the same values of $E$ and $B,r\propto \frac{\sqrt{m}}{q}$
For alpha particle $r_{\alpha }=\frac{\sqrt{m_{\alpha }}}{q_{\alpha }}=\frac{\sqrt{4m}}{2e}=\frac{\sqrt{m}}{e}$
For deuterons $r_d=\frac{\sqrt{m_d}}{q_d}=\frac{\sqrt{2m}}{e}=\frac{\sqrt{2m}}{e}$
For proton $r_p=\frac{\sqrt{m_p}}{q_p}=\frac{\sqrt{m}}{e}$
$\therefore r_{\alpha }:r_d:r_p=1:\sqrt{2}:1$
$d_{\alpha }:d_d:d_p=1:\sqrt{2}:1$.