A long horizontal slit is placed 1 mm above a horizontal plane mirror. The interference between the light coming directly from the slit and the after reflection is seen on a screen 1 m away from the slit. If the mirror reflects only 64% of the light falling on it, the ratio of the maximum to the minimum intensity in the interference pattern observed on the screen is

Options

A. `8:1`
B. `3:1`
C. `81:1`
D. `9:1`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the intensities
Let the intensity of the light from the slit be \( I_1 \). The intensity of the light after reflection from the mirror, which reflects 64% of the light, will be:
\[ I_2 = 0.64 I_1 \]

### Step 2: Relate intensities to amplitudes
Since intensity is proportional to the square of the amplitude, we can express the relationship between the amplitudes \( A_1 \) and \( A_2 \) as follows:
\[ \frac{I_2}{I_1} = \frac{A_2^2}{A_1^2} \]
Substituting for \( I_2 \):
\[ \frac{0.64 I_1}{I_1} = \frac{A_2^2}{A_1^2} \]
This simplifies to:
\[ \frac{A_2^2}{A_1^2} = 0.64 \]
Taking the square root gives:
\[ \frac{A_2}{A_1} = \sqrt{0.64} = \frac{4}{5} \]

### Step 3: Calculate maximum and minimum intensities
The maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) in the interference pattern can be calculated using the amplitudes:
\[ I_{\text{max}} = (A_1 + A_2)^2 \]
\[ I_{\text{min}} = (A_1 - A_2)^2 \]

Substituting \( A_2 = \frac{4}{5} A_1 \):
\[ I_{\text{max}} = \left(A_1 + \frac{4}{5} A_1\right)^2 = \left(\frac{9}{5} A_1\right)^2 = \frac{81}{25} A_1^2 \]
\[ I_{\text{min}} = \left(A_1 - \frac{4}{5} A_1\right)^2 = \left(\frac{1}{5} A_1\right)^2 = \frac{1}{25} A_1^2 \]

### Step 4: Find the ratio of maximum to minimum intensity
Now, we can find the ratio of maximum to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{\frac{81}{25} A_1^2}{\frac{1}{25} A_1^2} = \frac{81}{1} = 81 \]

### Conclusion
Thus, the ratio of the maximum to the minimum intensity in the interference pattern observed on the screen is:
\[ \boxed{81} \]