Radius Ratio in Magnetic Field - Detailed Solution

Radius Ratio in a Magnetic Field (NEET 2019)

Question: Ionized hydrogen atoms and alpha-particles with the same momentum enter perpendicular to a constant magnetic field, \( B \). What is the ratio of the radii of their paths?

(a) \( 1:4 \)

(b) \( 2:1 \)

(c) \( 1:2 \)

(d) \( 4:1 \)

Detailed Step-by-Step Explanation

Step 1: Recall the Expression for the Radius
A charged particle moving with momentum \( p \) perpendicular to a magnetic field \( B \) follows a circular path with radius given by: \[ r = \frac{p}{qB}, \] where \( q \) is the charge of the particle.

Step 2: Identify the Charges
For an ionized hydrogen atom (a proton), \( q_H = e \). For an alpha particle, \( q_\alpha = 2e \).

Step 3: Express the Radii
Let the momentum \( p \) be the same for both. Then:
For the hydrogen ion (proton): \[ r_H = \frac{p}{eB}, \] and for the alpha particle: \[ r_\alpha = \frac{p}{2eB}. \]

Step 4: Compute the Ratio
The ratio of their radii is: \[ \frac{r_H}{r_\alpha} = \frac{\frac{p}{eB}}{\frac{p}{2eB}} = \frac{2eB}{eB} = 2. \] Thus, the ratio is: \[ r_H : r_\alpha = 2:1. \]

Final Answer: The ratio of the radii of their paths is \( 2:1 \), which corresponds to option (b).