In double slit experiment, the angular width of the fringes is `0.20^@` for the sodium light `(lambda=5890Å)`. In order to increase the angular width of the fringes by `10%`, the necessary change in the wavelength is

Options

A. (a) Increase of `589Å`
B. (b) Decrease of `589Å`
C. (c) Increase of `6479Å`
D. (d) Zero

Explanation

To solve the problem, we need to find the necessary change in the wavelength to increase the angular width of the fringes by 10%. Let's break down the steps:

### Step 1: Understand the relationship between wavelength and angular width
In a double slit experiment, the angular width (θ) of the fringe is given by the formula:
\[ \theta = \frac{\lambda}{d} \]
where:
- \( \theta \) is the angular width of the fringe,
- \( \lambda \) is the wavelength of the light,
- \( d \) is the distance between the slits.

### Step 2: Calculate the initial angular width
Given that the initial angular width \( \theta_1 = 0.20^\circ \), we need to convert this angle into radians for calculations:
\[ \theta_1 = 0.20^\circ \times \frac{\pi}{180} \approx 0.00349 \text{ radians} \]

### Step 3: Determine the new angular width
To increase the angular width by 10%, we calculate the new angular width \( \theta_2 \):
\[ \theta_2 = \theta_1 + 0.10 \times \theta_1 = 1.10 \times \theta_1 \]
\[ \theta_2 = 1.10 \times 0.00349 \approx 0.003839 \text{ radians} \]

### Step 4: Set up the equation for the new wavelength
Using the relationship \( \theta = \frac{\lambda}{d} \), we can express the new wavelength \( \lambda_2 \) in terms of the new angular width:
\[ \theta_2 = \frac{\lambda_2}{d} \]
From the original equation, we have:
\[ \theta_1 = \frac{\lambda_1}{d} \]
Thus, we can express \( d \) in terms of \( \lambda_1 \) and \( \theta_1 \):
\[ d = \frac{\lambda_1}{\theta_1} \]
Substituting this into the equation for \( \theta_2 \):
\[ \theta_2 = \frac{\lambda_2}{\frac{\lambda_1}{\theta_1}} \implies \lambda_2 = \theta_2 \cdot \frac{\lambda_1}{\theta_1} \]

### Step 5: Calculate the new wavelength
Substituting the values:
\[ \lambda_2 = \theta_2 \cdot \frac{\lambda_1}{\theta_1} = 0.003839 \cdot \frac{5890 \times 10^{-10}}{0.00349} \]
Calculating this gives:
\[ \lambda_2 \approx 0.003839 \cdot 1.684 \times 10^{-6} \approx 6.46 \times 10^{-7} \text{ m} \approx 6460 \text{ Å} \]

### Step 6: Calculate the change in wavelength
The change in wavelength \( \Delta \lambda \) is:
\[ \Delta \lambda = \lambda_2 - \lambda_1 = 6460 \text{ Å} - 5890 \text{ Å} = 570 \text{ Å} \]

### Final Answer
The necessary change in the wavelength to increase the angular width of the fringes by 10% is \( 570 \text{ Å} \).

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