Ratio of amplitudes of the waves coming from two slits having widths in the ratio `4 : 1 `will be

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Answer

To find the ratio of amplitudes of the waves coming from two slits with widths in the ratio of 4:1, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Widths of the Slits**:
Let the width of the first slit (S1) be \( w_1 = 4x \) and the width of the second slit (S2) be \( w_2 = x \), where \( x \) is a common factor.

2. **Calculate the Areas of the Slits**:
The area of a slit is proportional to its width. Therefore, the areas of the slits can be expressed as:
- Area of S1, \( A_1 \propto w_1 = 4x \)
- Area of S2, \( A_2 \propto w_2 = x \)

3. **Determine the Ratio of Areas**:
The ratio of the areas of the two slits is:
\[ \frac{A_1}{A_2} = \frac{4x}{x} = 4 \]

4. **Relate Area to Intensity**:
The intensity of light (I) is proportional to the area of the slit. Thus, we have:
\[ \frac{I_1}{I_2} = \frac{A_1}{A_2} = 4 \]

5. **Use the Relationship Between Intensity and Amplitude**:
Intensity is directly proportional to the square of the amplitude (A):
\[ I \propto A^2 \]
Therefore, we can write:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]

6. **Substituting the Intensities**:
From the intensity ratio, we have:
\[ \frac{A_1^2}{A_2^2} = 4 \]

7. **Taking the Square Root**:
To find the ratio of amplitudes, we take the square root of both sides:
\[ \frac{A_1}{A_2} = \sqrt{4} = 2 \]

8. **Final Ratio of Amplitudes**:
Thus, the ratio of the amplitudes of the waves coming from the two slits is:
\[ \frac{A_1}{A_2} = 2:1 \]

### Conclusion:
The ratio of amplitudes of the waves coming from the two slits having widths in the ratio 4:1 is \( 2:1 \).