Question

A proton and an alpha particle moving with equal speeds enter normally into a uniform magnetic field. The ratio of times taken by the proton and the alpha particle to make one complete revolution in the magnetic field is

• A. \( 1 : \sqrt{2} \)
• B. \( 1 : 2 \)
• C. \( \sqrt{2} : 1 \)
• D. \( 2 : 1 \)

Hint:

Remember the properties of an alpha particle: Mass is 4 times that of a proton, charge is 2 times that of a proton. \( T \propto \frac{m}{q} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

A charged particle moving perpendicular to a magnetic field executes circular motion. The time period of revolution depends on the charge-to-mass ratio but is independent of speed.
Step 2: Key Formula or Approach:

Time period of revolution: \[ T = \frac{2\pi m}{qB} \] Ratio: \[ \frac{T_p}{T_\alpha} = \frac{m_p / q_p}{m_\alpha / q_\alpha} = \frac{m_p}{m_\alpha} \times \frac{q_\alpha}{q_p} \]
Step 3: Detailed Explanation:

For Proton (\( p \)): Mass \( m_p = m \), Charge \( q_p = e \). For Alpha particle (\( \alpha \)): Mass \( m_\alpha = 4m \) (2 protons + 2 neutrons), Charge \( q_\alpha = 2e \). Substitute into ratio: \[ \frac{T_p}{T_\alpha} = \frac{m}{4m} \times \frac{2e}{e} \] \[ \frac{T_p}{T_\alpha} = \frac{1}{4} \times 2 = \frac{1}{2} \]
Step 4: Final Answer:

The ratio is 1:2.