Question

Calculate the ratio of de Broglie wavelengths of an electron \((\lambda_e)\) and a proton \((\lambda_p)\) moving with the same velocity.

• A. \( \dfrac{m_e}{m_p} \)
• B. \( \dfrac{m_p}{m_e} \)
• C. \(1\)
• D. \( \dfrac{m_e^2}{m_p^2} \)

Hint:

From the de Broglie relation \( \lambda = \frac{h}{mv} \): - If velocity is constant, wavelength is inversely proportional to mass. - A lighter particle always has a larger de Broglie wavelength.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for the relationship between the de Broglie wavelengths of two different particles (electron and proton) when they share the same velocity \( v \).
Step 2: Key Formula or Approach:
According to the de Broglie hypothesis, the wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
Where \( h \) is Planck's constant, \( m \) is mass, and \( v \) is velocity.
Step 3: Detailed Explanation:
For the electron: \[ \lambda_e = \frac{h}{m_e v} \]
For the proton: \[ \lambda_p = \frac{h}{m_p v} \]
To find the ratio \( \frac{\lambda_e}{\lambda_p} \), divide the first expression by the second:
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{m_e v}}{\frac{h}{m_p v}} \]
Since \( h \) and \( v \) are the same for both particles, they cancel out:
\[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \]
Step 4: Final Answer:
The ratio of the wavelengths is \( \frac{m_p}{m_e} \).