Question

A proton and an \( \alpha \)-particle enter perpendicularly in a uniform magnetic field with the same velocity. Find out the ratio of their period of revolutions.

Hint:

The period of revolution depends on the mass and charge of the particle. A heavier and more charged particle will have a larger period of revolution.

Answer 1

Step 1: Formula for Period of Revolution.
The period \( T \) of revolution of a charged particle moving in a magnetic field is given by the formula: \[ T = \frac{2 \pi m}{q B} \] where \( m \) is the mass of the particle, \( q \) is the charge of the particle, and \( B \) is the magnetic field strength.
Step 2: Mass and Charge of the Proton and \( \alpha \)-Particle.
- For the proton, the charge \( q_p = +e \) (where \( e \) is the elementary charge) and mass \( m_p \) is the mass of a proton. - For the \( \alpha \)-particle, the charge \( q_{\alpha} = 2e \) (twice the charge of a proton) and mass \( m_{\alpha} = 4m_p \) (four times the mass of a proton).
Step 3: Ratio of Periods.
The ratio of the periods \( T_p \) for the proton and \( T_{\alpha} \) for the \( \alpha \)-particle is given by: \[ \frac{T_{\alpha}}{T_p} = \frac{m_{\alpha} / q_{\alpha}}{m_p / q_p} = \frac{4m_p / 2e}{m_p / e} = 2 \]
Step 4: Conclusion.
The ratio of the period of revolution of the \( \alpha \)-particle to the proton is 2:1.