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 1: Understand the positions of bright and dark fringes
In Young's experiment, the positions of the bright and dark fringes can be calculated using the formulas:
- The position of the nth bright fringe (B_n) from the center is given by:
\[ B_n = \frac{n \lambda D}{d} \]
where \( \lambda \) is the wavelength of light, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.

- The position of the nth dark fringe (D_n) from the center is given by:
\[ D_n = \frac{(2n - 1) \lambda D}{2d} \]

### Step 2: Calculate the distance between the nth bright fringe and nth dark fringe
To find the distance \( \Delta B \) 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:
\[ \Delta B = B_n - D_n \]

### Step 3: Substitute the formulas for \( B_n \) and \( D_n \)
Substituting the expressions we derived:
\[ \Delta B = \frac{n \lambda D}{d} - \frac{(2n - 1) \lambda D}{2d} \]

### Step 4: Simplify the expression
To simplify, we can find a common denominator:
\[ \Delta B = \frac{2n \lambda D}{2d} - \frac{(2n - 1) \lambda D}{2d} \]
\[ \Delta B = \frac{(2n - (2n - 1)) \lambda D}{2d} \]
\[ \Delta B = \frac{\lambda D}{2d} \]

### Final Result
Thus, the distance between the nth bright fringe and the nth dark fringe is:
\[ \Delta B = \frac{\lambda D}{2d} \]

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