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:
- 1 : 1
- \( 1 : \sqrt{2} \)
- \( \sqrt{2} : 1 \)
- \( 1 : 2 \)
The Correct Option is C
Approach Solution - 1
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:
- 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.
- 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, \(v_p = \sqrt{\frac{m_d}{m_p}} v_d\)
- 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
- 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:
\(\frac{r_d}{r_p} = \sqrt{2}\)
- Conclusion:
The ratio of the radii of the paths of the deuteron and proton is \(\sqrt{2} : 1\). Thus, the correct answer is \( \sqrt{2} : 1 \).
Approach Solution -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).
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