In young's double slit experiment the separation `d` between the slits is `2mm`, the wavelength `lambda` of the light used is `5896 Å` and distance `D` between the screen and slits is `100 cm`. It is found that the angular width of the fringes is `0.20^(@)`. To increases the fringe angular width to `0.21^(@)`(with same `lambda` and `D`) the separtion between the slits needs to be changed to

Options

A. `2.1 mm`
B. `1.7 mm`
C. `1.9 mm`
D. `1.8 mm`

Explanation

To solve the problem, we will use the formula for the angular width of the fringes in Young's double slit experiment. The angular width (θ) of the fringes is given by the formula:

\[ \theta = \frac{\lambda D}{d} \]

Where:
- \( \theta \) is the angular width of the fringe,
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the separation between the slits.

### Step 1: Convert the given values to consistent units
- Wavelength \( \lambda = 5896 \, \text{Å} = 5896 \times 10^{-10} \, \text{m} = 5.896 \times 10^{-7} \, \text{m} \)
- Separation between the slits \( d = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- Distance from the slits to the screen \( D = 100 \, \text{cm} = 1 \, \text{m} \)

### Step 2: Calculate the initial angular width \( \theta_1 \)
Using the formula:

\[ \theta_1 = \frac{\lambda D}{d} \]

Substituting the values:

\[ \theta_1 = \frac{(5.896 \times 10^{-7} \, \text{m})(1 \, \text{m})}{2 \times 10^{-3} \, \text{m}} = \frac{5.896 \times 10^{-7}}{2 \times 10^{-3}} = 2.948 \times 10^{-4} \, \text{radians} \]

To convert radians to degrees:

\[ \theta_1 \approx 0.20^\circ \]

### Step 3: Set up the equation for the new angular width \( \theta_2 = 0.21^\circ \)
We want to find the new separation \( d' \) that gives us \( \theta_2 = 0.21^\circ \).

Using the same formula:

\[ \theta_2 = \frac{\lambda D}{d'} \]

### Step 4: Set up the ratio of the two angular widths
From the two equations for angular widths, we can set up the ratio:

\[ \frac{\theta_1}{\theta_2} = \frac{d'}{d} \]

Substituting the known values:

\[ \frac{0.20}{0.21} = \frac{d'}{2 \times 10^{-3}} \]

### Step 5: Solve for \( d' \)
Cross-multiplying gives:

\[ d' = 2 \times 10^{-3} \times \frac{0.20}{0.21} \]

Calculating \( d' \):

\[ d' = 2 \times 10^{-3} \times \frac{20}{21} = 2 \times 10^{-3} \times 0.95238 \approx 1.90476 \times 10^{-3} \, \text{m} \approx 1.90 \, \text{mm} \]

### Final Answer
The new separation between the slits \( d' \) needs to be approximately \( 1.90 \, \text{mm} \).

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