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 is $\_\_\_\_\_\_$ and their speeds are in the ratio$\_\_\_\_\_\_\_.$

• A. $2:1:1and4:2:1$
• B. $1:2:4and2:1:1$
• C. $1:2:4and1:1:2$
• D. $4:2:1and2:1:1$

Answer 1

The correct option is A

$2:1:1and4:2:1$

Step 1. Given Data,

The momentum in a Uniform Magnetic Field is the same for a proton, a deuteron, and a $\alpha$particle.

Let the momentum be $p$.

Let, $F_1$ is Magnetic Force of proton, $F_2$ is Magnetic Force of deuteron, $F_3$ is Magnetic Force of $\alpha$ particle.

Step 2. The ratio of Magnetic Forces acting on a proton, a deuteron, and a $\alpha$particle is,

The force is given by

$F=qvB$ (where, $F$ is Magnetic Force, $q$ is charge, $v$ is velocity, and $B$ is Magnetic Field.)

Since, velocity $=$Momentum$/$mass $\Rightarrow v=\frac{p}{m}$ (where, $m$ is mass, and $p$ is momentum.)

If $M$ is mass of proton, then the mass of a deutron and a $\alpha$particle can be written as $2M$ and $4M$ respectively.

Putting value of Velocity ,$F=\frac{qpB}{m}$

Force of Proton $F_1=\frac{qpB}{M},$

Force of Deuteron $F_2=\frac{qpB}{2M},$

Force of $\alpha$particle $F_3=\frac{qpB}{3M},$

Then the ratio of Magnetic Forces acting on a proton, a deuteron, and a $\alpha$particle is,

$$\begin{gathered} F_1:F_2:F_3=1:\frac{1}{2}:\frac{1}{2} \\ \Rightarrow F_1:F_2:F_3=2:1:1 \end{gathered}$$

Step 3. Ratio of Velocity of a proton, a deuteron, and a $\alpha$particle is,

Velocity of Proton $v_1=\frac{p}{M}$,

Velocity of Deuteron $v_1=\frac{p}{2M}$,

Velocity of $\alpha$particle $v_3=\frac{p}{4M}$,

Then the ratio of Velocity of a proton, a deuteron, and a $\alpha$particle is,

$$\begin{gathered} V_1:V_2:V_3=1:\frac{1}{2}:\frac{1}{4} \\ \Rightarrow V_1:V_2:V_3=4:2:1 \end{gathered}$$

Hence,the ratio of Magnetic Force acting on a proton, a deuteron, and a $\alpha$particle is $2:1:1$ and the ratio of Velocity of a proton, a deuteron, and a $\alpha$particle is $4:2:1$.

Hence, option A is correct.