Question

When an \( \alpha \) particle and a proton are projected into a perpendicular uniform magnetic field, they describe circular paths of the same radius. The ratio of their respective velocities is:

• A. 1:1
• B. 1:4
• C. 2:1
• D. 1:2
• E. 4:1

Hint:

The radius of the path in a magnetic field depends on both the mass and charge of the particle. When considering particles with different charges and masses, the ratio of their velocities can be directly related to these properties if they travel in paths of the same radius.

Answer 1

The Correct Option is D

The radius \( r \) of the circular path in a magnetic field is given by the equation: \[ r = \frac{mv}{qB} \] where: \( m \) = mass of the particle, 
\( v \) = velocity of the particle, 
\( q \) = charge of the particle, 
\( B \) = magnetic field strength. For an \( \alpha \) particle (helium nucleus), which contains two protons and two neutrons: \( m_{\alpha} = 4m_p \) (where \( m_p \) is the mass of a proton), 
\( q_{\alpha} = 2e \) (where \( e \) is the elementary charge). 
For a proton: \( m_p \) and \( q_p = e \). 
Given that the paths have the same radius: \[ \frac{4m_p v_{\alpha}}{2eB} = \frac{m_p v_p}{eB} \]
Simplifying the equation, we find: \[ 2v_{\alpha} = v_p \quad {or} \quad v_{\alpha} = \frac{v_p}{2} \]
Thus, the ratio of their velocities \( v_{\alpha} : v_p \) is: \[ \frac{v_{\alpha}}{v_p} = \frac{1}{2} \] Therefore, the ratio of the velocity of the \( \alpha \) particle to the proton is 1:2.