Question

Ratio of de-Broglie wavelengths of a proton and an alpha particle accelerated through the same potential is:

• A. \( 1:2 \)
• B. \( 2\sqrt{2} : 1 \)
• C. \( 2:1 \)
• D. \( \sqrt{8} : 1 \)

Hint:

For particles accelerated through the same potential, \[ \lambda \propto \frac{1}{\sqrt{mq}} \] Heavier mass or higher charge means shorter de-Broglie wavelength.

Answer 1

The Correct Option is D

To find the ratio of the de-Broglie wavelengths of a proton and an alpha particle accelerated through the same potential, we use the de-Broglie wavelength formula: 

\(\lambda = \frac{h}{\sqrt{2mKE}}\),

where:

• \(h\) is Planck's constant.
• \(m\) is the mass of the particle.
• \(KE\) is the kinetic energy of the particle.

For a particle accelerated through a potential \(V\), the kinetic energy is \(KE = qV\), where \(q\) is the charge of the particle.

Thus, the de-Broglie wavelength is given by:

\(\lambda = \frac{h}{\sqrt{2mqV}}\)

Let's denote the de-Broglie wavelengths of the proton and alpha particle as \(\lambda_p\) and \(\lambda_\alpha\) respectively:

\(\lambda_p = \frac{h}{\sqrt{2m_pq_pV}}\)

\(\lambda_\alpha = \frac{h}{\sqrt{2m_\alpha q_\alpha V}}\)

To find the ratio \(\frac{\lambda_p}{\lambda_\alpha}\):

\(\frac{\lambda_p}{\lambda_\alpha} = \frac{\sqrt{2m_\alpha q_\alpha V}}{\sqrt{2m_p q_p V}}\)

The potential \(V\) is the same, so it cancels out:

\(\frac{\lambda_p}{\lambda_\alpha} = \sqrt{\frac{m_\alpha q_\alpha}{m_p q_p}}\)

For a proton:

• Mass \(m_p = 1.67 \times 10^{-27} \, \text{kg}\)
• Charge \(q_p = 1 \times e\)

For an alpha particle:

• Mass \(m_\alpha = 4 \times 1.67 \times 10^{-27} \, \text{kg} = 6.68 \times 10^{-27} \, \text{kg}\)
• Charge \(q_\alpha = 2 \times e\)

Substitute these into the ratio formula:

\(\frac{\lambda_p}{\lambda_\alpha} = \sqrt{\frac{6.68 \times 10^{-27} \times 2}{1.67 \times 10^{-27} \times 1}}\)

\(\frac{\lambda_p}{\lambda_\alpha} = \sqrt{8}\)

Thus, the ratio of the de-Broglie wavelengths of the proton and alpha particle is \(\sqrt{8} : 1\).

Therefore, the correct answer is: \(\sqrt{8} : 1\).