In a single slit diffraction set-up, the screen is placed at a distance of 80 cm from a slit. Light of `5,000 Å` is illuminated on the slit, the diffraction pattern is oberved on the screen. Determine the aperture of the slit if distance between first and second minima is 2.1 mm.

Answer

To determine the aperture (width) of the slit in a single slit diffraction setup, we can follow these steps:

### Step 1: Understand the Problem
We are given:
- Distance from the slit to the screen (D) = 80 cm = 0.8 m
- Wavelength of light (λ) = 5000 Å = 5000 × 10^(-10) m
- Distance between the first and second minima (Δy) = 2.1 mm = 2.1 × 10^(-3) m

### Step 2: Use the Formula for Minima in Single Slit Diffraction
The position of the minima in a single slit diffraction pattern is given by:
\[ y_n = \frac{n \lambda D}{b} \]
where:
- \( y_n \) is the position of the nth minima,
- \( n \) is the order of the minima (1 for the first minima, 2 for the second minima),
- \( λ \) is the wavelength of light,
- \( D \) is the distance from the slit to the screen,
- \( b \) is the width of the slit.

### Step 3: Write the Positions of the First and Second Minima
For the first minima (n=1):
\[ y_1 = \frac{1 \cdot \lambda D}{b} \]

For the second minima (n=2):
\[ y_2 = \frac{2 \cdot \lambda D}{b} \]

### Step 4: Calculate the Difference Between the Two Minima
The distance between the first and second minima is given by:
\[ \Delta y = y_2 - y_1 \]
Substituting the expressions for \( y_1 \) and \( y_2 \):
\[ \Delta y = \frac{2 \lambda D}{b} - \frac{1 \lambda D}{b} \]
\[ \Delta y = \frac{\lambda D}{b} \]

### Step 5: Rearrange the Equation to Solve for b
From the equation above, we can rearrange it to find the width of the slit (b):
\[ b = \frac{\lambda D}{\Delta y} \]

### Step 6: Substitute the Known Values
Now, substituting the known values:
- \( λ = 5000 \times 10^{-10} \) m
- \( D = 0.8 \) m
- \( \Delta y = 2.1 \times 10^{-3} \) m

\[ b = \frac{(5000 \times 10^{-10}) \times (0.8)}{2.1 \times 10^{-3}} \]

### Step 7: Calculate b
Calculating the above expression:
\[ b = \frac{(5000 \times 0.8) \times 10^{-10}}{2.1 \times 10^{-3}} \]
\[ b = \frac{4000 \times 10^{-10}}{2.1 \times 10^{-3}} \]
\[ b = \frac{4000}{2.1} \times 10^{-7} \]
\[ b \approx 1904.76 \times 10^{-7} \]
\[ b \approx 1.905 \times 10^{-4} \text{ m} \]

### Final Answer
Thus, the width of the slit is approximately:
\[ b \approx 1.9 \times 10^{-4} \text{ m} \]

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