Question

The de Broglie wavelength of an electron travelling with 20% of velocity of light is

• A. \( 2.4 \times 10^{-11} \, \text{m} \)
• B. \( 1.2 \times 10^{-11} \, \text{m} \)
• C. \( 3.6 \times 10^{-11} \, \text{m} \)
• D. \( 4.8 \times 10^{-11} \, \text{m} \)

Hint:

The de Broglie wavelength can be used to describe the wave-like nature of particles, and this formula is fundamental in quantum mechanics.

Answer 1

The Correct Option is B

The de Broglie wavelength (\( \lambda \)) of a particle is given by the equation: \[ \lambda = \frac{h}{mv} \] Where: - \( h = 6.626 \times 10^{-34} \, \text{J.s} \) (Planck's constant) - \( m = 9.1 \times 10^{-31} \, \text{kg} \) (mass of the electron) - \( v = 0.2c \) (velocity of the electron, 20% of speed of light) Given \( c = 3 \times 10^8 \, \text{m/s} \), we have: \[ v = 0.2 \times 3 \times 10^8 = 6 \times 10^7 \, \text{m/s} \] Substituting the values: \[ \lambda = \frac{6.626 \times 10^{-34}}{(9.1 \times 10^{-31})(6 \times 10^7)} = 1.2 \times 10^{-11} \, \text{m} \] Thus, the de Broglie wavelength is \( 1.2 \times 10^{-11} \, \text{m} \).

Answer 2

Step 1: Understanding the Problem
We need to calculate the de Broglie wavelength of an electron moving at 20% of the speed of light.

Step 2: Given Data
- Velocity of electron, \( v = 0.2c = 0.2 \times 3.0 \times 10^8 \, \text{m/s} = 6.0 \times 10^7 \, \text{m/s} \)
- Mass of electron, \( m = 9.11 \times 10^{-31} \, \text{kg} \)
- Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{Js} \)

Step 3: Formula for de Broglie Wavelength
\[ \lambda = \frac{h}{mv} \]

Step 4: Calculation
\[ \lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 6.0 \times 10^7} = 1.21 \times 10^{-11} \, \text{m} \]

Step 5: Final Answer
The de Broglie wavelength of the electron is approximately \( 1.2 \times 10^{-11} \, \text{m} \).