Question

Calculate the de Broglie wavelength of an electron accelerated through a potential difference of \(100\,V\).

• A. \(1.227\,\text{\AA}\)
• B. \(12.27\,\text{\AA}\)
• C. \(0.1227\,\text{\AA}\)
• D. \(122.7\,\text{\AA}\)

Hint:

For electrons accelerated through a potential \(V\), remember the shortcut formula \(\lambda = \frac{12.27}{\sqrt{V}}\,\text{\AA}\). It is widely used in quick numerical problems related to matter waves.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the de Broglie wavelength of an electron when it is given kinetic energy by an accelerating electric potential \( V \).
Step 2: Key Formula or Approach:
The de Broglie wavelength \( \lambda \) for an electron accelerated through a potential difference \( V \) (in Volts) can be calculated using the simplified formula:
\[ \lambda = \frac{12.27}{\sqrt{V}} \, \text{\AA} \]
Alternatively, in nanometers, it is \( \lambda = \frac{1.227}{\sqrt{V}} \, \text{nm} \).
Step 3: Detailed Explanation:
Given the potential difference \( V = 100 \, V \).
Using the formula:
\[ \lambda = \frac{12.27}{\sqrt{100}} \, \text{\AA} \]
\[ \lambda = \frac{12.27}{10} \, \text{\AA} \]
\[ \lambda = 1.227 \, \text{\AA} \]
Step 4: Final Answer:
The de Broglie wavelength is \( 1.227 \, \text{\AA} \).