Question

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

• A. \(0.1227\,\text{\AA}\)
• B. \(1.227\,\text{\AA}\)
• C. \(12.27\,\text{\AA}\)
• D. \(2.27\,\text{\AA}\)

Hint:

For electrons accelerated through a potential \(V\), remember the shortcut formula: \[ \lambda(\text{\AA}) = \frac{12.27}{\sqrt{V}} \] This avoids lengthy substitutions of physical constants.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the de Broglie wavelength (\(\lambda\)) associated with an electron that has been accelerated by a given potential difference (\(V\)).
Step 2: Key Formula or Approach:
The de Broglie wavelength is given by \(\lambda = h/p\), where \(p\) is momentum. For an electron accelerated from rest by a potential \(V\), its kinetic energy is \(eV\). This kinetic energy is also \(p^2/(2m)\). Combining these, we get \(\lambda = h/\sqrt{2meV}\).
For electrons, a very useful shortcut formula exists that combines all the constants (\(h, m_e, e\)):
\[ \lambda (\text{in \AA}) = \frac{12.27}{\sqrt{V (\text{in Volts})}} \] Step 3: Detailed Explanation:
(i) Identify the given potential difference:
\[ V = 100 \, \text{V} \] (ii) Substitute the value into the shortcut formula:
\[ \lambda = \frac{12.27}{\sqrt{100}} \, \text{\AA} \] (iii) Calculate the wavelength:
\[ \lambda = \frac{12.27}{10} \, \text{\AA} \] \[ \lambda = 1.227 \, \text{\AA} \] Step 4: Final Answer:
The de Broglie wavelength of the electron is \(1.227\,\text{\AA}\).