In a single slit diffraction pattern, a light of wavelength `6000 dot A` is used. The distance between the first and third minima in the diffraction pattern is found to be 3 mm when the screen in placed 50cm away from slits. The width of the slit is _________ `times 10^(-4)`m.

Answer

To find the width of the slit in a single slit diffraction pattern, we can follow these steps:

### Step 1: Understand the Problem
We are given:
- Wavelength of light, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \)
- Distance between the first and third minima, \( y_3 - y_1 = 3 \, \text{mm} = 3 \times 10^{-3} \, \text{m} \)
- Distance from the slit to the screen, \( D = 50 \, \text{cm} = 0.5 \, \text{m} \)

We need to find the width of the slit, denoted as \( d \).

### 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 the formula:

\[ y_n = \frac{n \lambda D}{d} \]

where \( n \) is the order of the minima.

### Step 3: Calculate the Positions of the First and Third Minima
For the first minimum (\( n = 1 \)):
\[ y_1 = \frac{1 \cdot \lambda D}{d} \]

For the third minimum (\( n = 3 \)):
\[ y_3 = \frac{3 \cdot \lambda D}{d} \]

### Step 4: Find the Difference Between the Third and First Minima
The difference between the positions of the third and first minima is given as:
\[ y_3 - y_1 = \frac{3 \lambda D}{d} - \frac{1 \lambda D}{d} = \frac{(3 - 1) \lambda D}{d} = \frac{2 \lambda D}{d} \]

Setting this equal to the given distance:
\[ \frac{2 \lambda D}{d} = 3 \times 10^{-3} \]

### Step 5: Rearranging to Solve for \( d \)
Rearranging the equation to solve for \( d \):
\[ d = \frac{2 \lambda D}{3 \times 10^{-3}} \]

### Step 6: Substitute the Known Values
Substituting the known values into the equation:
\[ d = \frac{2 \cdot (6000 \times 10^{-10}) \cdot (0.5)}{3 \times 10^{-3}} \]

### Step 7: Calculate the Value
Calculating the numerator:
\[ 2 \cdot 6000 \cdot 0.5 = 6000 \]
Thus,
\[ d = \frac{6000 \times 10^{-10}}{3 \times 10^{-3}} = \frac{6000 \times 10^{-10}}{3 \times 10^{-3}} = \frac{6000}{3} \times 10^{-7} = 2000 \times 10^{-7} = 2 \times 10^{-4} \, \text{m} \]

### Final Answer
The width of the slit is \( 2 \times 10^{-4} \, \text{m} \).