In coolidge tube, if f and `lambda` represent the frequency and wavelength of `K_(alpha)` -line for a metal of atomic number Z, then identify the statement which represents a straight line

Options

A. `sqrtf versus Z`
B. `(1)/(sqrtlambda) versus Z`
C. `f versus Z`
D. `lambda versus Z`

Explanation

To solve the problem regarding the Coolidge tube and the relationship between frequency (f), wavelength (λ), and atomic number (Z) for the Kα line, we need to analyze the given options step by step.

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- The frequency (f) and wavelength (λ) of the Kα line are related by the equation:
\[ f = \frac{c}{\lambda} \]
where \( c \) is the speed of light.

2. **Analyzing the Options**:
- We have four options to consider:
1. \( \sqrt{f} \) versus \( Z \)
2. \( \frac{1}{\sqrt{\lambda}} \) versus \( Z \)
3. \( f \) versus \( Z \)
4. \( \lambda \) versus \( Z \)

3. **Finding Relationships**:
- From the known relationship \( f = aZ - b \) (where \( a \) and \( b \) are constants), we can derive:
\[ \sqrt{f} = \sqrt{aZ - b} \]
This indicates that \( \sqrt{f} \) versus \( Z \) can be represented as a straight line if we express it in the form \( y = mx + c \).

4. **Using the Wavelength Relation**:
- We can also express \( f \) in terms of \( \lambda \):
\[ f = \frac{c}{\lambda} \implies \sqrt{f} = \sqrt{\frac{c}{\lambda}} = \frac{\sqrt{c}}{\sqrt{\lambda}} \]
- Rearranging gives:
\[ \sqrt{f} = \frac{\sqrt{c}}{\sqrt{\lambda}} \implies \frac{1}{\sqrt{\lambda}} = \frac{1}{\sqrt{c}} \sqrt{f} \]
- This shows that \( \frac{1}{\sqrt{\lambda}} \) versus \( Z \) can also be represented as a straight line.

5. **Conclusion**:
- From the analysis, we find that both \( \sqrt{f} \) versus \( Z \) and \( \frac{1}{\sqrt{\lambda}} \) versus \( Z \) represent straight lines. Thus, the correct options are:
- Option A: \( \sqrt{f} \) versus \( Z \)
- Option B: \( \frac{1}{\sqrt{\lambda}} \) versus \( Z \)

### Final Answer:
The statements that represent a straight line are:
- \( \sqrt{f} \) versus \( Z \) (Option A)
- \( \frac{1}{\sqrt{\lambda}} \) versus \( Z \) (Option B)