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Step-by-Step Solution
Step 1: Write the Formula for Resultant Intensity
The resultant intensity when two light beams of intensities $I_1$ and $I_2$ interfere with a phase difference $\phi$ is given by:
$ I_{\text{resultant}} = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi. $
Step 2: Identify the Given Values
From the question:
$I_1 = I$
$I_2 = 9I$
At point $P$, $\phi = \frac{\pi}{2}$
At point $Q$, $\phi = \pi$
Step 3: Calculate the Resultant Intensity at Point P
Substitute $I_1 = I$, $I_2 = 9I$, and $\phi = \frac{\pi}{2}$ into the formula:
$ I_P = I + 9I + 2\sqrt{I \cdot 9I}\cos\left(\frac{\pi}{2}\right). $
Since $\cos\left(\frac{\pi}{2}\right) = 0$, the third term is zero. Also, $\sqrt{I \cdot 9I} = 3I.$ Hence:
$ I_P = 10I + 2 \times 3I \times 0 = 10I. $
Step 4: Calculate the Resultant Intensity at Point Q
Substitute $I_1 = I$, $I_2 = 9I$, and $\phi = \pi$ into the formula:
$ I_Q = I + 9I + 2\sqrt{I \cdot 9I}\cos(\pi). $
Here, $\cos(\pi) = -1$, so:
$ I_Q = 10I + 2 \times 3I \times (-1) = 10I - 6I = 4I. $
Step 5: Find the Difference in Intensities
The difference between the resultant intensities at $P$ and $Q$ is:
$ I_P - I_Q = 10I - 4I = 6I. $
This matches the correct answer, which is $6I$.
