Question

A proton, a deuteron and an $\alpha$- particle are moving with the same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them and the ratio of their speeds are respectively :

• A. $2:1:1$ and $4:2:1$
• B. $1:2:4$ and $2:1:1$
• C. $1:2:4$ and $1:1:2$
• D. $4:2:1$ and $2:1:1$

Answer 1

The correct option is A $2:1:1$ and $4:2:1$

We know that, momentum, $p=mv$

So, $v\propto \frac{1}{m}$ for same momentum.

$\therefore v_1:v_2:v_3=\frac{1}{m_1}:\frac{1}{m_2}:\frac{1}{m_3}$

$\Rightarrow v_1:v_2:v_3=\frac{1}{m}:\frac{1}{2m}:\frac{1}{4m}=4:2:1$

Further, magnetic force, $F=qvB$

So, $F\propto qv$ for same magnetic field.

$\therefore F_1:F_2:F_3=q_1{v}_1:q_2{v}_2:q_3{v}_3$

$\Rightarrow F_1:F_2:F_3=q\times 4:q\times 2:2q\times 1=2:1:1$