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:

Start from the momentum form of the de Broglie relation, wavelength equal to h divided by momentum. Since both particles move at the same speed, compare their momenta through their masses first, then invert that ratio to get the wavelength ratio.

Answer 1

The Correct Option is B

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

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

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

Step 3: Simplify the expression. \[ \frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e} \] \[ \boxed{\frac{\lambda_e}{\lambda_p} = \frac{m_p}{m_e}} \]

Answer 2

Concept:

• The de Broglie relation in its original form is written using momentum, $\lambda = \dfrac{h}{p}$, where $p = mv$ is the momentum of the particle.
• Since wavelength is inversely proportional to momentum, comparing the momenta of the electron and proton directly gives the ratio of their wavelengths, without writing both wavelength expressions separately.

Step 1: Write the momentum of each particle.
Both move with the same velocity $v$:
$p_e = m_e v$, $\quad p_p = m_p v$

Step 2: Find the ratio of momenta.
$\dfrac{p_e}{p_p} = \dfrac{m_e v}{m_p v} = \dfrac{m_e}{m_p}$

Step 3: Use the inverse relation between wavelength and momentum.
Since $\lambda \propto \dfrac{1}{p}$, the wavelength ratio is the reciprocal of the momentum ratio:
$\dfrac{\lambda_e}{\lambda_p} = \dfrac{p_p}{p_e} = \dfrac{m_p}{m_e}$

Final Answer: $\dfrac{\lambda_e}{\lambda_p} = \dfrac{m_p}{m_e}$