The distance of nth bright fringe to the nth dark fringe in Young's experiment is equal to

Options

A. `(3lambdaD)/(2d)`
B. `(2lambdaD)/d`
C. `(lambdaD)/(2d)`
D. `(lambdaD/d)`

Explanation

To find the distance between the nth bright fringe and the nth dark fringe in Young's double slit experiment, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for the position of the nth bright fringe:**
The position of the nth bright fringe (y_b) is given by the formula:
\[ y_b = \frac{n \lambda D}{d} \]
where:
- \( n \) is the order of the bright fringe,
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the two slits.

2. **Identify the formula for the position of the nth dark fringe:**
The position of the nth dark fringe (y_d) is given by the formula:
\[ y_d = \frac{(2n - 1) \lambda D}{2d} \]
This can also be expressed as:
\[ y_d = \left(n - \frac{1}{2}\right) \frac{\lambda D}{d} \]

3. **Calculate the distance between the nth bright fringe and the nth dark fringe:**
To find the distance (Δy) between the nth bright fringe and the nth dark fringe, we subtract the position of the dark fringe from the position of the bright fringe:
\[ \Delta y = y_b - y_d \]

4. **Substituting the formulas:**
\[ \Delta y = \frac{n \lambda D}{d} - \left(n - \frac{1}{2}\right) \frac{\lambda D}{d} \]

5. **Simplifying the expression:**
\[ \Delta y = \frac{n \lambda D}{d} - \left(\frac{n \lambda D}{d} - \frac{1}{2} \frac{\lambda D}{d}\right) \]
\[ \Delta y = \frac{n \lambda D}{d} - \frac{n \lambda D}{d} + \frac{1}{2} \frac{\lambda D}{d} \]
\[ \Delta y = \frac{1}{2} \frac{\lambda D}{d} \]

6. **Final Result:**
Thus, the distance between the nth bright fringe and the nth dark fringe is:
\[ \Delta y = \frac{1}{2} \frac{\lambda D}{d} \]