In Young's double slit experiment, we get `60` fringes in the field of view of monochromatic light of wavelength `4000Å`. If we use monochromatic light of wavelength `6000Å`, then the number of fringes obtained in the same field of view is

Options

A. 60
B. 90
C. 40
D. 1.5

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the number of fringes and fringe width
In Young's double slit experiment, the number of fringes (n) in a given field of view is given by the formula:
\[ n = \frac{L}{\beta} \]
where \( L \) is the length of the region and \( \beta \) is the fringe width.

### Step 2: Express the fringe width in terms of wavelength
The fringe width \( \beta \) is given by:
\[ \beta = \frac{\lambda D}{d} \]
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. Since \( D \) and \( d \) remain constant in both cases, we can express the relationship between the two scenarios.

### Step 3: Set up the equation for both cases
Let:
- \( n_1 \) = number of fringes with wavelength \( \lambda_1 = 4000 \, \text{Å} \) (given as 60)
- \( n_2 \) = number of fringes with wavelength \( \lambda_2 = 6000 \, \text{Å} \)

From the relationship established earlier:
\[ n_1 \beta_1 = n_2 \beta_2 \]

Substituting the expression for fringe width:
\[ n_1 \left( \frac{\lambda_1 D}{d} \right) = n_2 \left( \frac{\lambda_2 D}{d} \right) \]

### Step 4: Simplify the equation
Since \( D \) and \( d \) are constant, they cancel out:
\[ n_1 \lambda_1 = n_2 \lambda_2 \]

### Step 5: Substitute known values
Now we can substitute the known values into the equation:
\[ 60 \times 4000 \, \text{Å} = n_2 \times 6000 \, \text{Å} \]

### Step 6: Solve for \( n_2 \)
Rearranging the equation gives:
\[ n_2 = \frac{60 \times 4000}{6000} \]

Calculating this:
\[ n_2 = \frac{240000}{6000} = 40 \]

### Conclusion
The number of fringes obtained with a wavelength of \( 6000 \, \text{Å} \) in the same field of view is \( 40 \).