Two sources `S_1` and `S_2`, each emitting waves of wavelength `lamda` are kept symmetrically on either side of centre o of a circle ABCD such that `S_1O = S_2O = 2 lamda ` . If a detector is moved along the circumference of the circle, it will record how many maximum in one revolution

Options

A. 8
B. 12
C. 16
D. 24

Answer

To solve the problem, we need to determine how many maxima will be observed by a detector moving along the circumference of the circle when two sources \( S_1 \) and \( S_2 \) are emitting waves of wavelength \( \lambda \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two sources \( S_1 \) and \( S_2 \) placed symmetrically on either side of the center \( O \) of a circle.
- The distance from each source to the center \( O \) is given as \( S_1O = S_2O = 2\lambda \).

2. **Path Difference**:
- When the detector moves along the circumference of the circle, the path difference between the waves arriving from \( S_1 \) and \( S_2 \) at any point \( P \) on the circumference can be expressed as:
\[ S_2P - S_1P \]
- For constructive interference (maxima), this path difference must be an integer multiple of the wavelength:
\[ S_2P - S_1P = n\lambda \quad (n = 0, 1, 2, \ldots) \]

3. **Geometry of the Circle**:
- The distance \( S_1P \) and \( S_2P \) can be expressed using the geometry of the circle. As the detector moves along the circumference, the path difference will vary.

4. **Maximum Path Difference**:
- The maximum path difference occurs when the detector is at points directly opposite to each source, which is when the path difference is maximized.
- The maximum path difference can be calculated as:
\[ \text{Max Path Difference} = S_2O + S_1O = 2\lambda + 2\lambda = 4\lambda \]

5. **Finding the Number of Maxima**:
- The condition for maxima is:
\[ S_2P - S_1P = n\lambda \]
- The maximum path difference of \( 4\lambda \) allows for the following integer values of \( n \):
\[ n = 0, 1, 2, 3, 4 \]
- This gives us a total of 5 maxima in one complete revolution around the circle.

### Conclusion:
Thus, the detector will record **5 maxima** in one complete revolution around the circle.

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