Assertion:- When a width of one of the slits of Young's double slit experiment is double that of the other the brighter frings are nine times brighter then the dark fringes.
Reason:- The amplitudes of the wave is proportional to the width of the slit.

Options

A. If both Assertion `&` Reason are True `&` the Reason is a correct explanation of the Assertion.
B. If both Assertion `&` Reason are True but Reason is not a correct explanation of the Assertion.
C. If Assertion is True but the Reason is False.
D. If both Assertion `&` Reason are False.

Answer

To solve the question, we need to analyze both the assertion and the reason provided in the context of Young's double slit experiment.

### Step 1: Understand the Assertion
The assertion states that when the width of one of the slits is double that of the other, the brighter fringes are nine times brighter than the dark fringes.

1. **Brightness and Intensity**: Brightness is related to intensity, which is defined as power per unit area. In the context of Young's double slit experiment, the intensity of the light at a point on the screen is determined by the amplitudes of the waves coming from the two slits.

### Step 2: Analyze the Reason
The reason states that the amplitudes of the wave are proportional to the width of the slit.

2. **Amplitude and Slit Width**: In wave optics, the amplitude of the wave emerging from a slit is indeed proportional to the width of that slit. If one slit has a width 'd' and the other has a width '2d', the amplitude from the wider slit will be greater.

### Step 3: Calculate the Intensity Ratio
3. **Intensity Calculation**:
- Let the amplitude from the first slit (width d) be A.
- The amplitude from the second slit (width 2d) will be 2A.
- The intensity is proportional to the square of the amplitude:
- Intensity from the first slit, \( I_1 \propto A^2 \)
- Intensity from the second slit, \( I_2 \propto (2A)^2 = 4A^2 \)

4. **Resulting Intensity at Interference**:
- The resultant intensity at points of constructive interference will be \( I = I_1 + I_2 + 2\sqrt{I_1 I_2} \).
- The ratio of the intensities can be calculated, but for the assertion to be true, we need to check if the brighter fringes are indeed nine times brighter than the dark fringes.

### Step 4: Conclusion on Assertion and Reason
5. **Final Evaluation**:
- The assertion claims that the brighter fringes are nine times brighter than the dark fringes, which is not supported by the calculations. The intensity ratio does not yield a result of nine times.
- The reason correctly states that amplitude is proportional to the width of the slit, but since the assertion is false, the overall conclusion is that both the assertion and reason are incorrect.

### Final Answer
Both the assertion and the reason are false. Therefore, the correct option is D.

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