Slit 1 of Young's double-slit experiment is wider then slit 2, so that the light from slits are given as `A_(1) = A_(0) sin omega t and A_(2)= 3 A_(0)sin (ω+π / 3), `, The resultant amplitude and intensity, at a point where the path difference between them is zero, are A and I, respectively. Then

Options

A. `A = sqrt (13) A_(0)`
B. `A = 4 A_(0)`
C. `I prop 16 A_(0)^(2)`
D. `I prop 13 A_(0)^(2)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the Amplitudes
We are given the amplitudes from the two slits:
- \( A_1 = A_0 \sin(\omega t) \)
- \( A_2 = 3 A_0 \sin(\left(\omega + \frac{\pi}{3}\right)) \)

### Step 2: Use the Formula for Resultant Amplitude
The resultant amplitude \( A \) at a point where the path difference is zero can be calculated using the formula:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]
where \( \phi \) is the phase difference. Since the path difference is zero, we can use the phase difference given in \( A_2 \), which is \( \frac{\pi}{3} \).

### Step 3: Substitute the Values
Substituting the values of \( A_1 \) and \( A_2 \):
- \( A_1 = A_0 \)
- \( A_2 = 3 A_0 \)

Now, substituting into the formula:
\[ A = \sqrt{(A_0)^2 + (3 A_0)^2 + 2 (A_0)(3 A_0) \cos\left(\frac{\pi}{3}\right)} \]

### Step 4: Calculate Each Term
Calculating each term:
1. \( A_1^2 = (A_0)^2 = A_0^2 \)
2. \( A_2^2 = (3 A_0)^2 = 9 A_0^2 \)
3. \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)
4. \( 2 A_1 A_2 \cos\left(\frac{\pi}{3}\right) = 2 (A_0)(3 A_0) \cdot \frac{1}{2} = 3 A_0^2 \)

### Step 5: Combine the Terms
Now, substituting these values back into the amplitude formula:
\[ A = \sqrt{A_0^2 + 9 A_0^2 + 3 A_0^2} = \sqrt{13 A_0^2} \]
Thus, we have:
\[ A = \sqrt{13} A_0 \]

### Step 6: Calculate the Intensity
The intensity \( I \) is proportional to the square of the amplitude:
\[ I \propto A^2 \]
Substituting \( A^2 \):
\[ I \propto (A)^2 = 13 A_0^2 \]

### Conclusion
From the calculations, we find:
- The resultant amplitude \( A = \sqrt{13} A_0 \)
- The intensity \( I \propto 13 A_0^2 \)

### Final Answers
Thus, the correct options are:
- \( A = \sqrt{13} A_0 \) (Option A)
- \( I \propto 13 A_0^2 \) (Option D)