Question

A proton and an alpha particle moving with energies in the ratio \( 1:4 \) enter a uniform magnetic field of 37 T at right angles to the direction of the field. The ratio of the magnetic forces acting on the proton and the alpha particle is:

• A. \( 1:2 \)
• B. \( 1:4 \)
• C. \( 2:3 \)
• D. \( 1:3 \)

Hint:

For charged particles in a magnetic field, force is given by \( F = q v B \). When energy is proportional to velocity squared, velocity ratios help determine force ratios.

Answer 1

The Correct Option is A

Step 1: Apply Magnetic Force Formula The force on a charged particle in a magnetic field is: \[ F = q v B \] where \( v \) is velocity, \( q \) is charge, and \( B \) is the magnetic field. Step 2: Compute Velocity Ratio From kinetic energy relation, \[ KE = \frac{1}{2} m v^2 \] Solving for \( v \), \[ v \propto \sqrt{KE} \] Since the energy ratio is \( 1:4 \), \[ v_p : v_\alpha = 1:2 \] Step 3: Compute Force Ratio Since \( q_\alpha = 2q_p \) and \( v_\alpha = 2v_p \), \[ F_p : F_\alpha = (q_p v_p B) : (2q_p 2v_p B) = 1:2 \] Thus, the correct answer is \( 1:2 \).