Radius of central zone of circular zone plate is 3.2 mm. Wavelength of incident light is . 5893 Å Source is at a distance of 6 m. Then the distance of first image will be

Options

A. a. 18 m
B. b. 12 m
C. c. 24 m
D. d. 36 m

Explanation

To solve the problem, we need to find the distance of the first image formed by a circular zone plate given the radius of the central zone and the wavelength of the incident light.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of the central zone, \( r = 3.2 \, \text{mm} = 3.2 \times 10^{-3} \, \text{m} \)
- Wavelength of the incident light, \( \lambda = 5893 \, \text{Å} = 5893 \times 10^{-10} \, \text{m} \)
- Distance of the source, \( d = 6 \, \text{m} \)

2. **Formula for the Focal Length of the Zone Plate**:
The focal length \( f \) of the first zone in a circular zone plate is given by the formula:
\[ f = \frac{r^2}{\lambda} \]

3. **Substituting the Values**:
Substitute the values of \( r \) and \( \lambda \) into the formula:
\[ f = \frac{(3.2 \times 10^{-3})^2}{5893 \times 10^{-10}} \]

4. **Calculating the Numerator**:
Calculate \( (3.2 \times 10^{-3})^2 \):
\[ (3.2 \times 10^{-3})^2 = 10.24 \times 10^{-6} = 1.024 \times 10^{-5} \, \text{m}^2 \]

5. **Calculating the Focal Length**:
Now substitute this value into the equation for \( f \):
\[ f = \frac{1.024 \times 10^{-5}}{5893 \times 10^{-10}} = \frac{1.024 \times 10^{-5}}{5.893 \times 10^{-7}} \approx 17.38 \, \text{m} \]

6. **Rounding the Result**:
The calculated focal length \( f \approx 17.38 \, \text{m} \) can be rounded to approximately \( 18 \, \text{m} \).

7. **Conclusion**:
The distance of the first image formed by the zone plate is approximately \( 18 \, \text{m} \).

### Final Answer:
The distance of the first image is \( 18 \, \text{m} \).