An electron beam is acceleration by a potential difference V to hit a metallic target to produce X-rays. It produces continuous as well as characteristic X-rays. If `lambda_(min)` is the smallest possible wavelength of X-rays in the spectrum, the variation of `log lambda_(min)` with `log v` is correctly represented in

Options

A. Option1 Modulation index = 0.8, side frequency bands are at 1180 kHz and 1220 kHz
B. Option2 Modulation index = 0.8, side frequency bands are at 1180 kHz and 1220 kHz
C. Option3 Modulation index = 0.2, side frequency bands are at 1200 kHz and 1180 kHz
D. Option4 Modulation index = 5, side frequency bands are at 21/2 kHz and 18.8 kHz

Answer

To solve the problem, we need to analyze the relationship between the minimum wavelength of X-rays produced by an electron beam and the potential difference through which the electrons are accelerated. Let's break down the solution step by step.

### Step-by-Step Solution

1. **Understanding the Energy of the Electron Beam**:
The energy \( E \) of the electrons accelerated through a potential difference \( V \) is given by:
\[ E = eV \]
where \( e \) is the charge of the electron.

2. **Relating Energy to Wavelength**:
The energy of a photon can also be expressed in terms of its wavelength \( \lambda \) using the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant and \( c \) is the speed of light.

3. **Finding the Minimum Wavelength**:
The minimum wavelength \( \lambda_{\text{min}} \) corresponds to the maximum energy of the photon produced when the entire kinetic energy of the electron is converted into a photon. Thus, we can equate the two expressions for energy:
\[ eV = \frac{hc}{\lambda_{\text{min}}} \]
Rearranging this gives:
\[ \lambda_{\text{min}} = \frac{hc}{eV} \]

4. **Taking Logarithms**:
To analyze the relationship between \( \lambda_{\text{min}} \) and \( V \), we take the logarithm of both sides:
\[ \log \lambda_{\text{min}} = \log \left( \frac{hc}{eV} \right) \]
This can be expanded using logarithmic properties:
\[ \log \lambda_{\text{min}} = \log(hc) - \log(e) - \log(V) \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \log V = \log(hc) - \log(e) - \log \lambda_{\text{min}} \]
This can be rewritten as:
\[ \log V = -\log \lambda_{\text{min}} + \text{constant} \]
where the constant includes \( \log(hc) - \log(e) \).

6. **Identifying the Relationship**:
This equation shows that \( \log V \) is linearly related to \( -\log \lambda_{\text{min}} \). It indicates that as \( \lambda_{\text{min}} \) increases, \( V \) decreases, which means the slope is negative.

7. **Graphical Representation**:
When plotting \( \log V \) on the y-axis against \( \log \lambda_{\text{min}} \) on the x-axis, the graph will have a negative slope, indicating an inverse relationship.

### Conclusion
The variation of \( \log \lambda_{\text{min}} \) with \( \log V \) is represented by a straight line with a negative slope.