Estimate how large can be the aperture opening to work with laws of ray optics using a monochromatic light of wavelength 450 nm, to a distance of around 20 m.

Options

A. 6 mm
B. 3 mm
C. 2 mm
D. 8 mm

Explanation

To estimate how large the aperture opening can be to work with the laws of ray optics using monochromatic light of wavelength 450 nm at a distance of around 20 m, we can follow these steps:

### Step 1: Understand the Given Values
We are given:
- Wavelength of light, \( \lambda = 450 \, \text{nm} = 450 \times 10^{-9} \, \text{m} \)
- Distance, \( Z_f = 20 \, \text{m} \)

### Step 2: Use the Formula for Aperture Size
The relationship between the aperture size \( a \), the distance \( Z_f \), and the wavelength \( \lambda \) is given by the formula:
\[ Z_f = \frac{a^2}{\lambda} \]
We need to rearrange this formula to solve for \( a \):
\[ a^2 = Z_f \cdot \lambda \]
\[ a = \sqrt{Z_f \cdot \lambda} \]

### Step 3: Substitute the Values
Now, substitute the known values into the equation:
\[ a = \sqrt{20 \, \text{m} \cdot (450 \times 10^{-9} \, \text{m})} \]

### Step 4: Calculate the Value Inside the Square Root
First, calculate the product:
\[ 20 \cdot 450 \times 10^{-9} = 9000 \times 10^{-9} \, \text{m}^2 \]

### Step 5: Take the Square Root
Now, take the square root:
\[ a = \sqrt{9000 \times 10^{-9}} = \sqrt{9 \times 10^{-6}} = 3 \times 10^{-3} \, \text{m} \]

### Step 6: Convert to Millimeters
Convert the result from meters to millimeters:
\[ 3 \times 10^{-3} \, \text{m} = 3 \, \text{mm} \]

### Conclusion
The maximum aperture size \( a \) that can be used while still adhering to the laws of ray optics is:
\[ \boxed{3 \, \text{mm}} \]

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