Question

A proton, a deuteron and an alpha particle of same velocity enter perpendicularly in an uniform magnetic field of \( 10^5 \) tesla. Calculate the ratio of their time periods of revolution.

Hint:

The time period of a charged particle in a magnetic field is proportional to the mass-to-charge ratio \( \frac{m}{q} \).

Answer 1

Step 1: The time period of a charged particle in a magnetic field is given by the formula: \[ T = \frac{2\pi m}{q B} \] Step 2: Given that the velocity and magnetic field are the same for all particles, the ratio of time periods depends on the ratio \( \frac{m}{q} \). Step 3: The mass and charge values of the particles: \[ \text{Proton: } m_p, \quad q_p = e \] \[ \text{Deuteron: } m_d = 2m_p, \quad q_d = e \] \[ \text{Alpha particle: } m_{\alpha} = 4m_p, \quad q_{\alpha} = 2e \] Step 4: Using the formula \( T \propto \frac{m}{q} \): \[ \frac{T_{\text{proton}}}{T_{\text{deuteron}}} = \frac{m_p/e}{2m_p/e} = \frac{1}{2} \] \[ \frac{T_{\text{proton}}}{T_{\alpha}} = \frac{m_p/e}{4m_p/2e} = \frac{1}{4} \] Thus, the ratio of time periods is: \[ 1:2:4 \] \[ \boxed{1:2:4} \]