In Young.s double slit experiment the separation d between the slits is 2mm, the wavelength `lambda` of the light used is `5896 A^(0)` and distance D between the screen and slits is 100 cm. It is found that the angular width of the fringes is `0.20^(0)`. To increase the fringe angular width to `0.21^(0)` (with same `lambda` and D) the separation between the slits needs to be changed to

Options

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

Answer

To solve the problem step by step, we will use the relationship between the angular width of the fringes, the wavelength of light, the distance between the slits, and the distance to the screen.

### Step 1: Understand the relationship
In Young's double slit experiment, the angular width (AW) of the fringes is given by the formula:
\[ \text{AW} = \frac{\lambda}{d} \]
where:
- \(\lambda\) = wavelength of light
- \(d\) = separation between the slits

### Step 2: Write down the known values
From the problem, we have:
- Initial separation between the slits, \(d_1 = 2 \text{ mm} = 2 \times 10^{-3} \text{ m}\)
- Wavelength, \(\lambda = 5896 \text{ Å} = 5896 \times 10^{-10} \text{ m}\)
- Initial angular width, \(\text{AW}_1 = 0.20^\circ\)
- New angular width, \(\text{AW}_2 = 0.21^\circ\)

### Step 3: Convert angular widths to radians
To use the formula, we need to convert the angular widths from degrees to radians:
\[ \text{AW}_1 = 0.20^\circ \times \frac{\pi}{180} = \frac{0.20 \pi}{180} \text{ radians} \]
\[ \text{AW}_2 = 0.21^\circ \times \frac{\pi}{180} = \frac{0.21 \pi}{180} \text{ radians} \]

### Step 4: Set up the equation
Using the relationship for angular width, we can set up the equation for the two cases:
\[ \text{AW}_1 \cdot d_1 = \text{AW}_2 \cdot d_2 \]
Substituting the known values:
\[ \left(\frac{0.20 \pi}{180}\right) \cdot (2 \times 10^{-3}) = \left(\frac{0.21 \pi}{180}\right) \cdot d_2 \]

### Step 5: Solve for \(d_2\)
Now, we can solve for \(d_2\):
\[ d_2 = \frac{\left(\frac{0.20 \pi}{180}\right) \cdot (2 \times 10^{-3})}{\left(\frac{0.21 \pi}{180}\right)} \]
The \(\pi\) and \(\frac{1}{180}\) cancel out:
\[ d_2 = \frac{0.20 \cdot (2 \times 10^{-3})}{0.21} \]
Calculating this gives:
\[ d_2 = \frac{0.40 \times 10^{-3}}{0.21} \approx 1.90476 \times 10^{-3} \text{ m} \approx 1.90 \text{ mm} \]

### Step 6: Conclusion
The new separation between the slits \(d_2\) required to increase the angular width to \(0.21^\circ\) is approximately \(1.90 \text{ mm}\).