Question

A proton and a deuteron (\( q = +e, \, m = 2.0u \)) having the same kinetic energies enter a region of uniform magnetic field \( \vec{B} \), moving perpendicular to \( \vec{B} \). The ratio of the radius \( r_d \) of the deuteron path to the radius \( r_p \) of the proton path is:

• A. 1 : 1
• B. \( 1 : \sqrt{2} \)
• C. \( \sqrt{2} : 1 \)
• D. \( 1 : 2 \)

Answer 1

The Correct Option is C

To determine the ratio of the radius of a deuteron's path (\( r_d \)) to the radius of a proton's path (\( r_p \)) in a uniform magnetic field, we proceed as follows:

1. Recall the formula for the radius of a charged particle's trajectory in a magnetic field:

The radius \( r \) of a charged particle's path, when moving perpendicularly to a magnetic field, is given by \(r = \frac{mv}{qB}\), where:

• \(m\) denotes the particle's mass,
• \(v\) denotes its velocity,
• \(q\) denotes the particle's charge, and
• \(B\) denotes the magnetic field strength.

1. Since the proton and deuteron possess identical kinetic energy, we formulate their respective kinetic energy equations:

The kinetic energy \( KE \) for both particles is expressed as \(KE = \frac{1}{2}mv^2\). Given that their kinetic energies are equal, we equate these expressions and solve for velocity:

\(\frac{1}{2}m_pv_p^2 = \frac{1}{2}m_dv_d^2\)

Here:

• \(m_p\) and \(m_d\) represent the masses of the proton and deuteron, respectively.
• \(v_p\) and \(v_d\) represent the velocities of the proton and deuteron, respectively.

This equality yields the relationship between their velocities:

\(v_p^2 = \frac{m_d}{m_p} v_d^2\)

Therefore, \(v_p = \sqrt{\frac{m_d}{m_p}} v_d\).

1. Apply the radius formula to each particle:

For the proton, \(r_p = \frac{m_pv_p}{q_pB}\).

For the deuteron, \(r_d = \frac{m_dv_d}{q_dB}\).

As the charge \(q = +e\) for both particles, it cancels out during comparison.

1. Calculate the required ratio:

The ratio of the radii is computed as:

\(\frac{r_d}{r_p} = \frac{\left(\frac{m_dv_d}{qB}\right)}{\left(\frac{m_pv_p}{qB}\right)} = \frac{m_d v_d}{m_p v_p}\)

Substituting \(v_p = \sqrt{\frac{m_d}{m_p}} v_d\):

\(\frac{r_d}{r_p} = \frac{m_d v_d}{m_p \left(\sqrt{\frac{m_d}{m_p}} v_d\right)} = \sqrt{\frac{m_d}{m_p}}\)

Given that \(m_d = 2m_p\) (as a deuteron consists of a proton and a neutron), we find:

\(\frac{r_d}{r_p} = \sqrt{2}\)

1. Conclusion:

The ratio of the deuteron's path radius to the proton's path radius is \( \sqrt{2} : 1 \). The final answer is \( \sqrt{2} : 1 \).