In Young's double-slit experiment the angular width of a fringe formed on a distant screen is `1^(@)`. The wavelength of light used is `6000 Å`. What is the spacing between the slits?

Options

A. 344 mm
B. 0.1344mm
C. 0.0344 mm
D. 0.034 mm

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between angular width, fringe width, and slit separation
In Young's double-slit experiment, the angular width of a fringe (θ) is related to the fringe width (β) and the distance between the slits (d) by the formula:
\[ \theta = \frac{\beta}{D} \]
where \(D\) is the distance from the slits to the screen.

### Step 2: Express fringe width in terms of wavelength and slit separation
The fringe width (β) can be expressed as:
\[ \beta = \frac{\lambda D}{d} \]
where \(λ\) is the wavelength of light used.

### Step 3: Substitute fringe width into the angular width formula
Substituting the expression for fringe width into the angular width formula gives:
\[ \theta = \frac{\lambda D}{dD} = \frac{\lambda}{d} \]
Thus, we can rearrange this to find the slit separation:
\[ d = \frac{\lambda}{\theta} \]

### Step 4: Convert the angular width from degrees to radians
The angular width given is \(1^\circ\). We need to convert this to radians:
\[ \theta = 1^\circ = \frac{\pi}{180} \text{ radians} \]

### Step 5: Substitute the known values into the equation
Now we can substitute the values of \(λ\) and \(θ\) into the equation for \(d\):
- Wavelength \(λ = 6000 \text{ Å} = 6000 \times 10^{-10} \text{ m} = 6 \times 10^{-7} \text{ m}\)
- Angular width \(θ = \frac{\pi}{180} \text{ radians}\)

Substituting these values into the equation for \(d\):
\[ d = \frac{6 \times 10^{-7}}{\frac{\pi}{180}} \]

### Step 6: Calculate the value of d
Now, calculate \(d\):
\[ d = 6 \times 10^{-7} \times \frac{180}{\pi} \]
Using \(\pi \approx 3.14\):
\[ d \approx 6 \times 10^{-7} \times \frac{180}{3.14} \approx 6 \times 10^{-7} \times 57.3 \approx 3.44 \times 10^{-5} \text{ m} \]

### Step 7: Convert the result to millimeters
To convert meters to millimeters:
\[ d = 3.44 \times 10^{-5} \text{ m} = 0.0344 \text{ mm} \]

### Final Answer
The spacing between the slits is approximately \(0.0344 \text{ mm}\).

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