Question

What is the wavelength associated with an electron moving with a velocity of \(10^6 \, \text{m/s}\)?
(\(h = 6.63 \times 10^{-34} \, \text{Js}\))

• A. 72.7 nm
• B. 7.27 nm
• C. 0.727 nm
• D. 0.0727 nm

Hint:

Electron wavelengths are typically in nanometer range for such speeds.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Moving microscopic particles exhibit wave properties. The wavelength of such a matter wave is given by the de Broglie equation.
: Key Formula or Approach:
De Broglie wavelength: \( \lambda = \frac{h}{mv} \).
Mass of an electron (\( m \)) \( \approx 9.1 \times 10^{-31} \text{ kg} \).
Step 2: Detailed Explanation:
Given:
\( v = 10^{6} \text{ m/s} \).
\( h = 6.63 \times 10^{-34} \text{ Js} \).
\( m = 9.1 \times 10^{-31} \text{ kg} \).
Calculation:
\[ \lambda = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \times 10^{6}} \]
\[ \lambda = \frac{6.63}{9.1} \times 10^{(-34 + 31 - 6)} \text{ m} \]
\[ \lambda \approx 0.728 \times 10^{-9} \text{ m} \]
Converting to nanometers (\( 1 \text{ nm} = 10^{-9} \text{ m} \)):
\[ \lambda \approx 0.727 \text{ nm} \].
Step 3: Final Answer:
The de Broglie wavelength is approximately \( 0.727 \text{ nm} \).