In a Fraunhofer diffraction at single slit of width d with incident light of wavelength 5500 Ã…, the first minimum is observed, at angle `30^(@)`. The first secondary maximum is observed at an angle `theta=`

Options

A. `"sin"^(-1)((1)/(sqrt(2)))`
B. `"sin"^(-1)((1)/(4))`
C. `"sin"^(-1)((3)/(4))`
D. `"sin"^(-1)((sqrt(3))/(2))`

Explanation

To solve the problem of finding the angle for the first secondary maximum in a Fraunhofer diffraction pattern at a single slit, we will follow these steps:

### Step 1: Understand the given information
We are given:
- Wavelength of light, \( \lambda = 5500 \, \text{Å} = 5500 \times 10^{-10} \, \text{m} \)
- Angle for the first minimum, \( \theta_1 = 30^\circ \)

### Step 2: Use the formula for the first minimum
In a single slit diffraction pattern, the position of the first minimum is given by the formula:
\[ \sin \theta_1 = \frac{\lambda}{d} \]
Where \( d \) is the width of the slit.

### Step 3: Calculate the slit width \( d \)
Substituting the known values into the formula:
\[ \sin(30^\circ) = \frac{1}{2} \]
Thus,
\[ \frac{1}{2} = \frac{5500 \times 10^{-10}}{d} \]
Rearranging to find \( d \):
\[ d = \frac{5500 \times 10^{-10}}{\frac{1}{2}} = 11000 \times 10^{-10} \, \text{m} = 1.1 \times 10^{-6} \, \text{m} \]

### Step 4: Use the formula for the first secondary maximum
The position of the first secondary maximum occurs at:
\[ \sin \theta = \frac{3}{2} \cdot \frac{\lambda}{d} \]

### Step 5: Substitute the values to find \( \theta \)
Substituting the values of \( \lambda \) and \( d \):
\[ \sin \theta = \frac{3}{2} \cdot \frac{5500 \times 10^{-10}}{1.1 \times 10^{-6}} \]
Calculating the right-hand side:
\[ \sin \theta = \frac{3}{2} \cdot \frac{5500 \times 10^{-10}}{1.1 \times 10^{-6}} = \frac{3 \cdot 5500}{2 \cdot 11000} = \frac{16500}{22000} = \frac{3}{4} \]

### Step 6: Find the angle \( \theta \)
Now, we can find \( \theta \):
\[ \theta = \sin^{-1} \left( \frac{3}{4} \right) \]

### Final Answer
Thus, the angle for the first secondary maximum is:
\[ \theta = \sin^{-1} \left( \frac{3}{4} \right) \]