Question

Let \( \lambda_e, \lambda_p, \lambda_d \) be the wavelengths associated with an electron, a proton, and a deuteron, all moving with the same speed. Then the correct relation between them is:

• A. \( \lambda_e>\lambda_p>\lambda_d \)
• B. \( \lambda_p>\lambda_e>\lambda_d \)
• C. \( \lambda_e>\lambda_p>\lambda_d \)
• D. \( \lambda_e = \lambda_p = \lambda_d \)

Hint:

The de Broglie wavelength is inversely proportional to the mass of the particle. Heavier particles have smaller wavelengths.

Answer 1

The Correct Option is A

To determine the relationship between the wavelengths \( \lambda_e, \lambda_p, \lambda_d \) associated with an electron, a proton, and a deuteron all moving at the same speed, we utilize the de Broglie wavelength formula: 

$$ \lambda = \frac{h}{mv} $$

where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity. Given that the speed \( v \) is the same for all three particles, the wavelengths are inversely proportional to their masses:

1. \( \lambda_e = \frac{h}{m_e v} \)

2. \( \lambda_p = \frac{h}{m_p v} \)

3. \( \lambda_d = \frac{h}{m_d v} \)

Where \( m_e, m_p, \) and \( m_d \) are the masses of the electron, proton, and deuteron respectively. Given:

- The mass of an electron \( m_e \) is the smallest.

- The mass of a proton \( m_p \) is larger than the mass of an electron.

- The mass of a deuteron \( m_d \) (approximately twice the mass of a proton) is the largest.

Therefore, since the masses follow \( m_e < m_p < m_d \), the wavelengths satisfy \( \lambda_e > \lambda_p > \lambda_d \). Hence, the correct relation between the wavelengths is:

\( \lambda_e > \lambda_p > \lambda_d \)