Question

The minimum wavelength of X-rays produced by 20 keV electrons is nearly

• A. 0.62 \AA
• B. 1.8 \AA
• C. 3.2 \AA
• D. 6.5 \AA

Hint:

The minimum wavelength of X-rays decreases as the energy of the incident electrons increases, following the inverse relationship $\lambda_{\text{min}} \propto \frac{1}{E}$.

Answer 1

The Correct Option is A

Let’s break this down step by step to calculate the minimum wavelength of X-rays produced by 20 keV electrons and determine why option (1) is the correct answer.
Step 1: Understand the concept of minimum wavelength of X-rays
The minimum wavelength ($\lambda_{\text{min}}$) of X-rays produced by electrons occurs when all the kinetic energy of the electron is converted into the energy of the X-ray photon. This is given by:
\[ E = \frac{hc}{\lambda_{\text{min}}} \]
A practical formula in electron volts and angstroms is:
\[ \lambda_{\text{min}} (\text{in \AA}) = \frac{12398}{E (\text{in eV})} \]
where $hc \approx 12398 \, \text{eV \AA}$.
Step 2: Identify the given values and calculate the wavelength

• Energy of electrons, $E = 20 \, \text{keV} = 20 \times 10^3 \, \text{eV} = 20000 \, \text{eV}$

\[ \lambda_{\text{min}} = \frac{12398}{20000} \approx 0.6199 \, \text{\AA} \]
This is approximately 0.62 \AA.
Step 3: Confirm the correct answer
The calculated minimum wavelength is 0.62 \AA, which matches option (1). The term “nearly” in the question accounts for slight rounding.
Thus, the correct answer is (1) 0.62 \AA.