In youngs double slit experiment the fringes are formed at a distance of 1m from double slits of sepration 0.12mm .calculate the distance of 4th dark band from the centre of screen ?(wavelength is 12000A).

Answer

To solve the problem of finding the distance of the 4th dark band from the center of the screen in Young's double slit experiment, we will follow these steps:

### Step 1: Understand the Formula for Dark Fringes
The distance of the m-th dark fringe from the center of the screen is given by the formula:
\[ x_m = \frac{(2m - 1) \lambda D}{2d} \]
where:
- \( x_m \) is the distance of the m-th dark fringe,
- \( m \) is the order of the dark fringe (for the 4th dark band, \( m = 4 \)),
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the separation between the slits.

### Step 2: Convert Given Values to Consistent Units
1. **Wavelength (\( \lambda \))**: Given as 12000 Å (angstroms).
- Convert to centimeters:
\[ \lambda = 12000 \, \text{Å} = 12000 \times 10^{-10} \, \text{m} = 12 \times 10^{-5} \, \text{cm} \]

2. **Distance to Screen (\( D \))**: Given as 1 m.
- Convert to centimeters:
\[ D = 1 \, \text{m} = 100 \, \text{cm} \]

3. **Separation between Slits (\( d \))**: Given as 0.12 mm.
- Convert to centimeters:
\[ d = 0.12 \, \text{mm} = 0.012 \, \text{cm} \]

### Step 3: Substitute Values into the Formula
Now, substitute the values into the formula for the 4th dark fringe (\( m = 4 \)):
\[ x_4 = \frac{(2 \times 4 - 1) \lambda D}{2d} \]
Substituting the known values:
\[ x_4 = \frac{(8 - 1) \times (12 \times 10^{-5}) \times 100}{2 \times 0.012} \]
\[ x_4 = \frac{7 \times (12 \times 10^{-5}) \times 100}{2 \times 0.012} \]

### Step 4: Calculate the Distance
Now, calculate the numerator and denominator:
- **Numerator**:
\[ 7 \times (12 \times 10^{-5}) \times 100 = 7 \times 12 \times 10^{-3} = 84 \times 10^{-3} = 0.084 \, \text{cm} \]

- **Denominator**:
\[ 2 \times 0.012 = 0.024 \, \text{cm} \]

Now, divide the numerator by the denominator:
\[ x_4 = \frac{0.084}{0.024} = 3.5 \, \text{cm} \]

### Final Answer
The distance of the 4th dark band from the center of the screen is:
\[ \boxed{3.5 \, \text{cm}} \]