nth bright fringe if red light `(lambda_(1)=7500" Å")` coincides with (n +1)th bright fringe of green light `(lambda_(2)=6000" Å")` The value of n, is

Options

A. 4
B. 5
C. 3
D. 2

Explanation

To solve the problem of finding the value of \( n \) where the \( n \)-th bright fringe of red light coincides with the \( (n + 1) \)-th bright fringe of green light, we can follow these steps:

### Step 1: Understand the fringe condition
The condition for bright fringes in interference is given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
where \( y_n \) is the position of the \( n \)-th bright fringe, \( \lambda \) is the wavelength of the light, \( D \) is the distance to the screen, and \( d \) is the separation between the slits.

### Step 2: Set up the equation for the red and green light
For red light with wavelength \( \lambda_1 = 7500 \, \text{Å} \) (or \( 7500 \times 10^{-10} \, \text{m} \)), the position of the \( n \)-th bright fringe is:
\[ y_n = \frac{n \lambda_1 D}{d} \]

For green light with wavelength \( \lambda_2 = 6000 \, \text{Å} \) (or \( 6000 \times 10^{-10} \, \text{m} \)), the position of the \( (n + 1) \)-th bright fringe is:
\[ y_{n+1} = \frac{(n + 1) \lambda_2 D}{d} \]

### Step 3: Set the two positions equal
Since the \( n \)-th bright fringe of red light coincides with the \( (n + 1) \)-th bright fringe of green light, we can set the two equations equal:
\[ \frac{n \lambda_1 D}{d} = \frac{(n + 1) \lambda_2 D}{d} \]

### Step 4: Simplify the equation
We can cancel \( D \) and \( d \) from both sides:
\[ n \lambda_1 = (n + 1) \lambda_2 \]

### Step 5: Substitute the values of the wavelengths
Substituting \( \lambda_1 = 7500 \, \text{Å} \) and \( \lambda_2 = 6000 \, \text{Å} \):
\[ n \cdot 7500 = (n + 1) \cdot 6000 \]

### Step 6: Expand and rearrange the equation
Expanding the right side:
\[ 7500n = 6000n + 6000 \]
Now, rearranging the equation gives:
\[ 7500n - 6000n = 6000 \]
\[ 1500n = 6000 \]

### Step 7: Solve for \( n \)
Dividing both sides by 1500:
\[ n = \frac{6000}{1500} = 4 \]

### Conclusion
The value of \( n \) is \( 4 \).

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