Question

What is the ratio of the de Broglie wavelengths of an electron and a proton moving with the same velocity?

• A. \( \dfrac{m_e}{m_p} \)
• B. \( \dfrac{m_p}{m_e} \)
• C. \(1\)
• D. \( \sqrt{\dfrac{m_p}{m_e}} \)

Hint:

From the de Broglie relation \( \lambda = \frac{h}{mv} \), if two particles move with the same velocity, their wavelengths are inversely proportional to their masses.

Answer 1

The Correct Option is B

Concept: According to the de Broglie hypothesis, the wavelength associated with a moving particle is given by \[ \lambda = \frac{h}{mv} \] where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity. Thus, the wavelength is inversely proportional to the mass when velocity is constant.

Step 1: Write the wavelengths of electron and proton. For an electron: \[ \lambda_e = \frac{h}{m_e v} \] For a proton: \[ \lambda_p = \frac{h}{m_p v} \]

Step 2: Find the ratio. \[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{m_e v}}{\frac{h}{m_p v}} \] \[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \]