Question

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

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

Answer 1

Answer: (B)

$F=q(\vec{v}\times \vec{B})=\frac{q}{m}(\vec{P}\times \vec{B})$
$\Rightarrow F\propto \frac{q}{m}$
thus $F_1:F_2:F_3=\frac{q_1}{m_1}:\frac{q_2}{m_2}:\frac{q_3}{m_3}$
$=\frac{e}{m_p}:\frac{e}{2m_p}:\frac{2e}{4m_p}$
$=\frac{1}{1}:\frac{1}{2}:\frac{2}{4}$
$=2:1:1$
Now for speed calculation
$P=$ constant $\Rightarrow v\propto \frac{1}{m}$
thus $v_1:v_2:v_3=\frac{1}{m_p}:\frac{1}{2m_p}:\frac{1}{4m_p}$
$=\frac{1}{1}:\frac{1}{2}:\frac{1}{4}$
$=4:2:1$