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 \( 1:2 \).

Answer 2

Given:
- Energy ratio of proton to alpha particle, \( E_p : E_\alpha = 1 : 4 \)
- Magnetic field strength, \( B = 37 \, \text{T} \)
- Both particles enter the magnetic field at right angles.

We need to find the ratio of magnetic forces acting on the proton and the alpha particle.

Step 1: Magnetic force on a charged particle moving perpendicular to magnetic field is:
\[ F = q v B \] where \( q \) is charge, \( v \) is velocity, and \( B \) is magnetic field.

Step 2: Find the velocity ratio using the energy relation:
Kinetic energy \( E = \frac{1}{2} m v^2 \), so:
\[ v = \sqrt{\frac{2E}{m}} \]

Given energy ratio:
\[ \frac{E_p}{E_\alpha} = \frac{1}{4} \] Thus:
\[ \frac{v_p}{v_\alpha} = \sqrt{\frac{E_p / m_p}{E_\alpha / m_\alpha}} = \sqrt{ \frac{1 / m_p}{4 / m_\alpha} } = \sqrt{ \frac{m_\alpha}{4 m_p} } \]

Step 3: Mass and charge of particles:
- Proton: charge \( q_p = +e \), mass \( m_p \)
- Alpha particle: charge \( q_\alpha = +2e \), mass \( m_\alpha = 4 m_p \)

Substitute values:
\[ \frac{v_p}{v_\alpha} = \sqrt{ \frac{4 m_p}{4 m_p} } = \sqrt{1} = 1 \]

Step 4: Calculate the ratio of magnetic forces:
\[ \frac{F_p}{F_\alpha} = \frac{q_p v_p B}{q_\alpha v_\alpha B} = \frac{e \times v_p}{2e \times v_\alpha} = \frac{v_p}{2 v_\alpha} = \frac{1}{2} \]

Therefore, the ratio of magnetic forces acting on proton and alpha particle is:
\[ \boxed{1 : 2} \]