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 solve the problem of finding the ratio of the radius of the deuteron path (\( r_d \)) to the radius of the proton path (\( r_p \)) in a uniform magnetic field, we follow a step-by-step approach:

1. Understand the formula for the radius of the path of charged particles in a magnetic field:

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

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

1. Since the proton and deuteron have the same kinetic energy, write their kinetic energy equations:

The kinetic energy \( KE \) for both particles is:

\(KE = \frac{1}{2}mv^2\)

Given that their kinetic energies are equal, set the kinetic energy equations equal and solve for velocity:

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

where:

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

From here, find the relationship between their velocities:

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

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

1. Apply the radius formula for both particles:

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

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

The charge \(q = +e\) for both particles, so it cancels out when comparing

1. Calculate the desired ratio:

The ratio of the radii is given by:

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

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

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

Given that \(m_d = 2m_p\) (for a deuteron which is made up of a proton and a neutron), we have:

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

1. Conclusion:

The ratio of the radii of the paths of the deuteron and proton is \(\sqrt{2} : 1\). Thus, the correct answer is \( \sqrt{2} : 1 \).

Answer 2

The radius of the path of a charged particle moving in a magnetic field is given by:

\( r = \frac{mv}{qB} \),

where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.

Since both particles have the same kinetic energy, \( \frac{1}{2}mv^2 = K \), and the velocity \( v \) can be expressed as:

\( v = \sqrt{\frac{2K}{m}} \).

Thus, the radius of the proton path \( r_p \) is:

\( r_p = \frac{m_p \sqrt{2K/m_p}}{eB} \),

and the radius of the deuteron path \( r_d \) is:

\( r_d = \frac{m_d \sqrt{2K/m_d}}{eB} \).

Since \( m_d = 2m_p \), the ratio of the radii is:

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

Thus, the ratio is \( \sqrt{2} : 1 \), and the correct answer is Option (3).