In a Young's double slit esperiment, the angular width of a fringe formed on a distant screen is `1^(@)` . The slit separation is 0.01 mm. The wavelength of the light is

Options

A. 0.174 nm
B. 0.174 Ã…
C. 0.174 `mu m`
D. `0.174 xx 10^(-4)m`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between angular width, wavelength, and slit separation.
In a Young's double slit experiment, the angular width of the fringe (θ) is related to the wavelength (λ) and the slit separation (d) by the formula:
\[ \theta = \frac{\lambda}{d} \]

### Step 2: Rearrange the formula to find the wavelength.
To find the wavelength (λ), we can rearrange the formula:
\[ \lambda = d \cdot \theta \]

### Step 3: Substitute the given values.
We are given:
- Slit separation, \( d = 0.01 \, \text{mm} = 0.01 \times 10^{-3} \, \text{m} = 1 \times 10^{-5} \, \text{m} \)
- Angular width, \( \theta = 1^\circ \)

### Step 4: Convert the angular width from degrees to radians.
To use the formula, we need to convert degrees to radians:
\[ \theta \text{ (in radians)} = \frac{\pi}{180} \text{ radians} \]
Thus,
\[ \theta = \frac{1 \times \pi}{180} \approx 0.01745 \, \text{radians} \]

### Step 5: Substitute the values into the wavelength formula.
Now we can substitute the values of \( d \) and \( \theta \) into the formula for wavelength:
\[ \lambda = (1 \times 10^{-5} \, \text{m}) \cdot \left(\frac{\pi}{180}\right) \]

### Step 6: Calculate the wavelength.
Now, we calculate:
\[ \lambda = 1 \times 10^{-5} \cdot \frac{\pi}{180} \approx 1 \times 10^{-5} \cdot 0.01745 \approx 1.745 \times 10^{-7} \, \text{m} \]

### Step 7: Convert the wavelength to micrometers.
To express the wavelength in micrometers:
\[ \lambda \approx 0.1745 \, \mu m \]

### Final Answer:
The wavelength of the light is approximately \( 0.1745 \, \mu m \).

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