Question

Let \( \lambda_e \), \( \lambda_p \), and \( \lambda_d \) 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. \( \lambda_d>\lambda_p>\lambda_e \)
• B. \( \lambda_e>\lambda_p>\lambda_d \)
• C. \( \lambda_p>\lambda_e>\lambda_d \)
• D. \( \lambda_e = \lambda_p = \lambda_d \)

Hint:

The de Broglie wavelength is inversely proportional to the mass of the particle. The more massive the particle, the smaller its wavelength.

Answer 1

The Correct Option is A

To find the correct relation between the wavelengths (\( \lambda \)) of an electron, a proton, and a deuteron moving with the same speed, we use de Broglie's equation: 

\( \lambda = \frac{h}{mv} \)

where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity (speed).

Given that all particles move with the same speed \( v \), the wavelength is inversely proportional to their mass:

\( \lambda \propto \frac{1}{m} \)

Let's compare their masses:

Particle	Mass
Electron (\( e \))	\( 9.11 \times 10^{-31} \) kg
Proton (\( p \))	\( 1.67 \times 10^{-27} \) kg
Deuteron (\( d \))	\( 3.34 \times 10^{-27} \) kg

Comparing the masses, we see:

\( m_e < m_p < m_d \)

Thus, inversely for wavelengths:

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

Therefore, the correct relation is: \( \lambda_d>\lambda_p>\lambda_e \)