Question

An α particle and a proton are accelerated from rest through the same potential difference. The ratio of linear momenta acquired by above two particles will be:

• A. √2:1
• B. 2√2:1
• C. 4√2:1
• D. 8:1

Answer 1

The Correct Option is B

To solve this problem, we need to determine the ratio of the linear momenta acquired by an α particle and a proton when both are accelerated from rest through the same potential difference. Let's break down the steps:

Step-by-step Solution: 

1. When a charged particle is accelerated through a potential difference \( V \), the work done on the particle is converted into kinetic energy. The kinetic energy \( K \) gained by the particle is given by: \(K = qV\), where \( q \) is the charge of the particle.
2. The expression for kinetic energy is also given by: \(K = \frac{1}{2}mv^2\), where \( m \) is the mass and \( v \) is the velocity of the particle.
3. From the above two equations, we get: \(qV = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{\frac{2qV}{m}}\)
4. The linear momentum \( p \) of the particle is: \(p = mv = m\sqrt{\frac{2qV}{m}} = \sqrt{2mqV}\)
5. Now, compute the momenta for both the α particle and the proton:
   • For the α particle (He\(^{2+}\)), the charge \( q \) is 2e, and the mass \( m \) is approximately 4 times that of the proton. Therefore, the momentum for the α particle is: \(p_{\alpha} = \sqrt{2 \cdot 4m_p \cdot 2e \cdot V} = \sqrt{16m_peV}\)
   • For the proton, the charge \( q \) is e and the mass \( m \) is \( m_p \). Therefore, the momentum for the proton is: \(p_p = \sqrt{2m_peV}\)
6. Ratio of the linear momenta of α particle to proton: \(\frac{p_{\alpha}}{p_p} = \frac{\sqrt{16m_peV}}{\sqrt{2m_peV}} = \frac{\sqrt{16}}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}\)

Therefore, the ratio of linear momenta acquired by the α particle and the proton is 2√2:1.

Answer 2

The ratio of linear momenta acquired by above two particles,
\(\frac{pα}{pp}=\frac{\sqrt{2(4m)(2eV)}}{{\sqrt{2(m)(eV)}}}\)
\(=\frac{\sqrt{16}}{√2}\)
=\(\frac{4}{√2}\)
\(=\frac{2√2}{1}\)
So, the correct option is (B): \(\frac{2√2}{1}\)