Question

Two electrons are moving with different kinetic energies. If the first electron is moving with kinetic energy \(k\) and the second electron with kinetic energy \(4k\), the ratio of their de Broglie wavelengths respectively will be:

• A. \(1:2\)
• B. \(1:4\)
• C. \(2:1\)
• D. \(4:1\)

Hint:

For the same particle, de Broglie wavelength varies inversely as the square root of kinetic energy: \[ \lambda \propto \frac{1}{\sqrt{K}} \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The de Broglie hypothesis proposes that particles like electrons exhibit wave-like properties. The wavelength associated with a particle is called its de Broglie wavelength (\(\lambda\)).
The wavelength is inversely proportional to the momentum (\(p\)) of the particle.
Momentum is related to the kinetic energy (\(K\)) of the particle. For a particle of mass \(m\), \(K = \frac{p^2}{2m}\), which gives \(p = \sqrt{2mK}\).
Step 2: Key Formula or Approach:
The formula for de Broglie wavelength in terms of kinetic energy is:
\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}} \]
Since \(h\) (Planck's constant) and \(m\) (mass of the electron) are constants, we can say:
\[ \lambda \propto \frac{1}{\sqrt{K}} \]
This means the ratio of wavelengths is the inverse of the ratio of the square roots of their kinetic energies.
Step 3: Detailed Explanation:
Given:
Kinetic energy of the first electron, \(K_1 = k\)
Kinetic energy of the second electron, \(K_2 = 4k\)

Let their respective wavelengths be \(\lambda_1\) and \(\lambda_2\).
Using the proportionality:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{K_2}}{\sqrt{K_1}} \]
Substitute the given energy values:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{4k}}{\sqrt{k}} \]
\[ \frac{\lambda_1}{\lambda_2} = \frac{2 \sqrt{k}}{\sqrt{k}} \]
\[ \frac{\lambda_1}{\lambda_2} = \frac{2}{1} \]
The ratio is 2 : 1. This makes sense physically: the faster electron (higher kinetic energy) has a shorter wavelength.
Step 4: Final Answer:
The ratio of their de Broglie wavelengths is 2 : 1.