If `lambda_(min)` is minimum wavelength produced in X-ray tube and `lamda_(kalpha)` is the wavelength of `K_(alpha)` line. As the operating tube voltage is increased.

Options

A. `(lambda_(k)-lambda_(min))` increases
B. `(lambda_(k)-lambda_(min))` decreases
C. `lambda_(kalpha)` increases
D. `lambda_(kalpha)` decreases

Answer

To solve the problem, we need to analyze how the minimum wavelength (\( \lambda_{min} \)) and the wavelength of the K-alpha line (\( \lambda_{K\alpha} \)) change as the operating voltage of an X-ray tube is increased.

### Step-by-Step Solution:

1. **Understanding Minimum Wavelength (\( \lambda_{min} \))**:
- The minimum wavelength produced in an X-ray tube is given by the equation:
\[ \lambda_{min} = \frac{hc}{eV} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, \( e \) is the charge of an electron, and \( V \) is the operating voltage of the tube.
- As the voltage \( V \) increases, \( \lambda_{min} \) decreases because it is inversely proportional to \( V \).

2. **Understanding K-alpha Wavelength (\( \lambda_{K\alpha} \))**:
- The wavelength of the K-alpha line is characteristic of the material of the target in the X-ray tube and does not change with the operating voltage. Therefore, \( \lambda_{K\alpha} \) remains constant as \( V \) increases.

3. **Effect of Increasing Voltage**:
- As the operating voltage \( V \) increases, \( \lambda_{min} \) decreases while \( \lambda_{K\alpha} \) remains constant.
- This means that the difference between \( \lambda_{K\alpha} \) and \( \lambda_{min} \) increases.

4. **Graphical Representation**:
- In the graph of intensity (\( I \)) versus wavelength (\( \lambda \)), the curve will shift as \( \lambda_{min} \) decreases. The peak intensity will occur at a wavelength longer than \( \lambda_{min} \) but shorter than \( \lambda_{K\alpha} \).
- The graph will show that as \( V \) increases, the minimum wavelength shifts to the left (decreases), while the K-alpha line remains at the same position.

5. **Conclusion**:
- As the operating tube voltage is increased, \( \lambda_{min} \) decreases, and \( \lambda_{K\alpha} \) remains constant, leading to an increase in the difference between \( \lambda_{K\alpha} \) and \( \lambda_{min} \).

### Final Answer:
The difference between \( \lambda_{K\alpha} \) and \( \lambda_{min} \) increases as the operating tube voltage is increased.

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