Two beams of light having intensities 9I and 4I interfere to produce fringe pattern on a screen P, Q and R are three points on the screen at which the phase differences between the interfering beams are `30^(@), 45^(@)" and "60^(@)` and the intensities are `I_(P), I_(Q)" and "I_(R )` respectivley. Arrange the diffrence between the intensities in ascending order

Options

A. `(I_(P)-I_(Q)), (I_(P)-I_(R )), (I_(Q)-I_(R ))`
B. `(I_(P)-I_(Q)), (I_(Q)-I_(R )), (I_(P)-I_(R ))`
C. `(I_(P)-I_(R )), (I_(Q)-I_(R )), (I_(P)-I_(Q ))`
D. `(I_(Q)-I_(R )), (I_(P)-I_(Q )), (I_(P)-I_(R ))`

Answer

To solve the problem, we will calculate the intensities at points P, Q, and R using the formula for the intensity of two interfering beams. Then we will find the differences between the intensities and arrange them in ascending order.

### Step-by-Step Solution:

1. **Identify the Intensities of the Two Beams:**
- Let the intensities of the two beams be \( I_1 = 9I \) and \( I_2 = 4I \).

2. **Use the Intensity Formula:**
- The formula for the intensity at a point due to two interfering beams is:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
where \( \phi \) is the phase difference.

3. **Calculate Intensity at Point P (Phase Difference = 30°):**
- For point P:
\[ I_P = 9I + 4I + 2 \sqrt{9I \cdot 4I} \cos(30° \]
- Calculate \( \cos(30°) = \frac{\sqrt{3}}{2} \):
\[ I_P = 13I + 2 \cdot 6I \cdot \frac{\sqrt{3}}{2} \]
- Simplifying:
\[ I_P = 13I + 6\sqrt{3}I \]

4. **Calculate Intensity at Point Q (Phase Difference = 45°):**
- For point Q:
\[ I_Q = 9I + 4I + 2 \sqrt{9I \cdot 4I} \cos(45°) \]
- Calculate \( \cos(45°) = \frac{1}{\sqrt{2}} \):
\[ I_Q = 13I + 2 \cdot 6I \cdot \frac{1}{\sqrt{2}} \]
- Simplifying:
\[ I_Q = 13I + \frac{12I}{\sqrt{2}} = 13I + 6\sqrt{2}I \]

5. **Calculate Intensity at Point R (Phase Difference = 60°):**
- For point R:
\[ I_R = 9I + 4I + 2 \sqrt{9I \cdot 4I} \cos(60°) \]
- Calculate \( \cos(60°) = \frac{1}{2} \):
\[ I_R = 13I + 2 \cdot 6I \cdot \frac{1}{2} \]
- Simplifying:
\[ I_R = 13I + 6I = 19I \]

6. **Calculate Differences Between Intensities:**
- \( I_P - I_Q \):
\[ I_P - I_Q = (13I + 6\sqrt{3}I) - (13I + 6\sqrt{2}I) = (6\sqrt{3} - 6\sqrt{2})I \]
- \( I_P - I_R \):
\[ I_P - I_R = (13I + 6\sqrt{3}I) - 19I = (6\sqrt{3} - 6I) \]
- \( I_Q - I_R \):
\[ I_Q - I_R = (13I + 6\sqrt{2}I) - 19I = (6\sqrt{2} - 6I) \]

7. **Arrange the Differences in Ascending Order:**
- We need to compare \( 6(\sqrt{3} - \sqrt{2})I \), \( 6(\sqrt{2} - 1)I \), and \( 6(\sqrt{3} - 1)I \).
- Since \( \sqrt{3} \approx 1.732 \) and \( \sqrt{2} \approx 1.414 \):
- \( \sqrt{3} - \sqrt{2} \) is positive.
- \( \sqrt{2} - 1 \) is also positive.
- \( \sqrt{3} - 1 \) is positive and larger than the others.
- Thus, the order is:
- \( I_P - I_Q < I_Q - I_R < I_P - I_R \).

### Final Answer:
The differences in intensities in ascending order are:
\[ I_P - I_Q < I_Q - I_R < I_P - I_R \]