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

Step 1: Understanding the Question:
The question asks for the ratio of de Broglie wavelengths for two different particles (electron and proton) given that they share the same velocity \(v\).
Step 2: Key Formula or Approach:
The de Broglie wavelength \(\lambda\) is given by the formula:
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
where \(h\) is Planck's constant, \(m\) is mass, and \(v\) is velocity.
Step 3: Detailed Explanation:
Let \(m_e\) be the mass of the electron and \(m_p\) be the mass of the proton.
Since both move with the same velocity \(v\):
Wavelength of electron: \(\lambda_e = \frac{h}{m_e v}\)
Wavelength of proton: \(\lambda_p = \frac{h}{m_p v}\)
Taking the ratio:
\[ \frac{\lambda_e}{\lambda_p} = \frac{\frac{h}{m_e v}}{\frac{h}{m_p v}} \]
\[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \]
Step 4: Final Answer:
The ratio of the wavelengths is \( \dfrac{m_p}{m_e} \).