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

Hint:

Charge and Mass ratios for these three particles are high-yield: Proton ($1, 1$), Deuteron ($1, 2$), Alpha ($2, 4$). The Deuteron and Alpha particle often have the same $q/m$ ratio, leading to identical behavior in certain fields.

Answer 1

The Correct Option is B

Step 1: Force $F = qvB$. Since momentum $p = mv$ is same, $v = p/m$.
Step 2: $F = q(p/m)B = \frac{q}{m}(pB)$. Since $p, B$ are constant, $F \propto q/m$.
Step 3: Ratio of $q/m$: Proton ($e/m$), Deuteron ($e/2m$), $\alpha$ ($2e/4m = e/2m$). Ratio $F$: $1 : 1/2 : 1/2 \Rightarrow 2 : 1 : 1$.
Step 4: Ratio of speeds ($v = p/m \propto 1/m$): Proton ($1/m$), Deuteron ($1/2m$), $\alpha$ ($1/4m$). Ratio $v$: $1 : 1/2 : 1/4 \Rightarrow 4 : 2 : 1$.