What will be the angle of diffracting for the first minimum due to Fraunhofer diffraction with sources of light of wavelength `550nm` and slit of width `0.55mm`?

Options

A. (a)0.001 rad
B. (b)0.01 rad
C. (c)1 rad
D. (d)0.1 rad

Answer

To find the angle of diffraction for the first minimum due to Fraunhofer diffraction, we can use the formula related to single-slit diffraction. The condition for the first minimum is given by:

\[ d \sin \theta = n \lambda \]

where:
- \( d \) is the width of the slit,
- \( \theta \) is the angle of diffraction,
- \( n \) is the order of the minimum (for the first minimum, \( n = 1 \)),
- \( \lambda \) is the wavelength of the light.

### Step 1: Identify the given values
- Wavelength \( \lambda = 550 \, \text{nm} = 550 \times 10^{-9} \, \text{m} \)
- Slit width \( d = 0.55 \, \text{mm} = 0.55 \times 10^{-3} \, \text{m} \)

### Step 2: Substitute the values into the formula
Using the formula for the first minimum:

\[ d \sin \theta = n \lambda \]

Substituting \( n = 1 \):

\[ 0.55 \times 10^{-3} \sin \theta = 1 \times 550 \times 10^{-9} \]

### Step 3: Solve for \( \sin \theta \)
Rearranging the equation to solve for \( \sin \theta \):

\[ \sin \theta = \frac{550 \times 10^{-9}}{0.55 \times 10^{-3}} \]

### Step 4: Calculate \( \sin \theta \)
Calculating the right-hand side:

\[ \sin \theta = \frac{550 \times 10^{-9}}{0.55 \times 10^{-3}} = \frac{550}{0.55} \times 10^{-6} = 1000 \times 10^{-6} = 0.001 \]

### Step 5: Calculate \( \theta \)
Now, we can find \( \theta \) using the inverse sine function:

\[ \theta = \sin^{-1}(0.001) \]

Since \( \sin \theta \) is small, we can approximate:

\[ \theta \approx 0.001 \, \text{radians} \]

### Final Answer
The angle of diffraction for the first minimum is approximately:

\[ \theta \approx 0.001 \, \text{radians} \]