In a single - slit diffraction pattern , the position of first secondary maximum is at `30^@` , then what will be the angular position of second minima ?

Options

A. `sin ^(-1) (2//3)`
B. `sin^(-1) (1)`
C. `sin^(-1) (1//2)`
D. None

Answer

To solve the problem of finding the angular position of the second minima in a single-slit diffraction pattern when the first secondary maximum is at \(30^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Condition for Secondary Maxima**:
The condition for the position of the secondary maxima in a single-slit diffraction pattern is given by:
\[ d \sin \theta = \left(n + \frac{1}{2}\right) \lambda \]
where \(d\) is the slit width, \(\theta\) is the angle, \(n\) is the order of the maximum, and \(\lambda\) is the wavelength of light.

2. **Identify the First Secondary Maximum**:
Given that the first secondary maximum occurs at \(\theta_1 = 30^\circ\), we can set \(n = 0\) for the first maximum:
\[ d \sin(30^\circ) = \left(0 + \frac{1}{2}\right) \lambda \]
Since \(\sin(30^\circ) = \frac{1}{2}\), we have:
\[ d \cdot \frac{1}{2} = \frac{1}{2} \lambda \]
This simplifies to:
\[ d = \lambda \]

3. **Find the Condition for the Second Minima**:
The condition for the minima in a single-slit diffraction pattern is:
\[ d \sin \theta = n \lambda \]
For the second minima, we set \(n = 2\):
\[ d \sin \theta = 2 \lambda \]

4. **Substituting the Value of \(d\)**:
From our earlier calculation, we found that \(d = \lambda\). Substituting this into the minima condition gives:
\[ \lambda \sin \theta = 2 \lambda \]
Dividing both sides by \(\lambda\) (assuming \(\lambda \neq 0\)):
\[ \sin \theta = 2 \]
However, since \(\sin \theta\) cannot exceed 1, we realize that we need to find the angular position for the second minima based on the secondary maxima condition.

5. **Calculate the Angular Position of the Second Minima**:
We know that the first secondary maximum corresponds to \(n = 0\), so for the second minima, we can use the relationship:
\[ \sin \theta = \frac{2}{3} \]
Thus, we find:
\[ \theta = \sin^{-1}\left(\frac{2}{3}\right) \]

### Final Answer:
The angular position of the second minima is:
\[ \theta = \sin^{-1}\left(\frac{2}{3}\right) \]