What is the Fresnel distance for an aperture of 3 mm and wavelength 400 nm?

Options

A. 8.5 m
B. 22.5 m
C. 10 m
D. 15 m

Answer

To find the Fresnel distance for an aperture of 3 mm and a wavelength of 400 nm, we can follow these steps:

### Step 1: Understand the formula for Fresnel distance
The Fresnel distance (ZF) is given by the formula:
\[ Z_F = \frac{b^2}{\lambda} \]
where:
- \( b \) is the width of the aperture,
- \( \lambda \) is the wavelength of the light.

### Step 2: Convert the given values to appropriate units
- The width of the aperture \( b = 3 \, \text{mm} = 3 \times 10^{-3} \, \text{m} \)
- The wavelength \( \lambda = 400 \, \text{nm} = 400 \times 10^{-9} \, \text{m} \)

### Step 3: Substitute the values into the formula
Now, substituting the values into the formula:
\[ Z_F = \frac{(3 \times 10^{-3})^2}{400 \times 10^{-9}} \]

### Step 4: Calculate \( b^2 \)
Calculating \( b^2 \):
\[ (3 \times 10^{-3})^2 = 9 \times 10^{-6} \]

### Step 5: Substitute \( b^2 \) into the formula
Now substituting \( b^2 \) into the formula:
\[ Z_F = \frac{9 \times 10^{-6}}{400 \times 10^{-9}} \]

### Step 6: Simplify the expression
Now simplifying the expression:
\[ Z_F = \frac{9 \times 10^{-6}}{400 \times 10^{-9}} = \frac{9}{400} \times 10^{3} \]

### Step 7: Calculate the numerical value
Calculating the numerical value:
\[ Z_F = \frac{9 \times 10^{3}}{400} = 22.5 \, \text{m} \]

### Final Answer
Thus, the Fresnel distance \( Z_F \) is:
\[ Z_F = 22.5 \, \text{m} \]

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