Question

A particle having mass 2000 times that of an electron travels with a velocity thrice that of the electron. The ratio of the de Broglie wavelength of the particle to that of the electron is

• A. \( \frac{1}{3000} \)
• B. \( \frac{1}{2000} \)
• C. \( \frac{1}{6000} \)
• D. \( \frac{1}{8000} \)
• E. \( \frac{1}{1500} \)

Hint:

The de Broglie wavelength is inversely proportional to the mass and velocity of the particle. A larger mass or velocity results in a smaller wavelength.

Answer 1

The Correct Option is C

Step 1: The de Broglie wavelength \( \lambda \) of a particle is given by the equation: \[ \lambda = \frac{h}{mv} \] where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.
Step 2: Let the mass of the electron be \( m_e \) and its velocity be \( v_e \), and the mass of the particle be \( 2000m_e \) with velocity \( 3v_e \).
Step 3: The de Broglie wavelength of the electron is: \[ \lambda_e = \frac{h}{m_e v_e} \] The de Broglie wavelength of the particle is: \[ \lambda_p = \frac{h}{(2000m_e)(3v_e)} = \frac{h}{6000m_e v_e} \] Step 4: The ratio of the de Broglie wavelengths is: \[ \frac{\lambda_p}{\lambda_e} = \frac{\frac{h}{6000m_e v_e}}{\frac{h}{m_e v_e}} = \frac{1}{6000} \] Thus, the ratio of the de Broglie wavelength of the particle to that of the electron is \( \frac{1}{6000} \).