Light of wavelength `5,500 Å` is illuminated on a slit. Calculate the slit width if Fresnel distance for diffraction set-up is 4 m.

Answer

To solve the problem of calculating the slit width given the Fresnel distance and the wavelength of light, we can follow these steps:

### Step 1: Understand the Given Values
- Wavelength of light, \( \lambda = 5500 \, \text{Å} = 5500 \times 10^{-10} \, \text{m} \)
- Fresnel distance, \( z_f = 4 \, \text{m} \)

### Step 2: Use the Formula for Fresnel Distance
The relationship between the Fresnel distance \( z_f \), the slit width \( D \), and the wavelength \( \lambda \) is given by the formula:
\[ z_f = \frac{D^2}{\lambda} \]

### Step 3: Rearrange the Formula to Solve for Slit Width \( D \)
To find the slit width \( D \), we can rearrange the formula as follows:
\[ D^2 = z_f \cdot \lambda \]
\[ D = \sqrt{z_f \cdot \lambda} \]

### Step 4: Substitute the Given Values into the Equation
Now, substituting the values of \( z_f \) and \( \lambda \):
\[ D = \sqrt{4 \, \text{m} \cdot (5500 \times 10^{-10} \, \text{m})} \]

### Step 5: Calculate the Value Inside the Square Root
Calculating the product:
\[ D = \sqrt{4 \cdot 5500 \times 10^{-10}} = \sqrt{22000 \times 10^{-10}} = \sqrt{2.2 \times 10^{-6}} \, \text{m} \]

### Step 6: Calculate the Square Root
Now, calculating the square root:
\[ D \approx 1.48 \times 10^{-3} \, \text{m} \]

### Step 7: Convert to Millimeters
To express the width in millimeters:
\[ D \approx 1.48 \, \text{mm} \]

### Final Answer
The slit width \( D \) is approximately \( 1.48 \, \text{mm} \).

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