Consider the situation of the previous problem, if the mirror reflects only 64% of the light energy falling on it, what will be the ratio of the maximum to the minimum intensity in the interference pattern observed on the screen ?

Explanation

To solve the problem, we need to determine the ratio of the maximum intensity (I_max) to the minimum intensity (I_min) in the interference pattern when a mirror reflects only 64% of the light energy falling on it.

### Step-by-Step Solution:

1. **Understanding the Reflection Percentage**:
- The mirror reflects 64% of the light energy. This means that the intensity of the reflected light (I1) is 64% of the incident intensity (I0).
- Therefore, we can express this as:
\[ I_1 = 0.64 I_0 \]
- The intensity of the light that is not reflected (I2) is then:
\[ I_2 = I_0 - I_1 = I_0 - 0.64 I_0 = 0.36 I_0 \]

2. **Finding the Ratio of Intensities**:
- The ratio of the intensities I1 to I2 can be written as:
\[ \frac{I_1}{I_2} = \frac{0.64 I_0}{0.36 I_0} = \frac{0.64}{0.36} \]
- Simplifying this gives:
\[ \frac{I_1}{I_2} = \frac{64}{36} = \frac{16}{9} \]

3. **Calculating the Amplitude Ratio**:
- Since intensity is proportional to the square of the amplitude (R), we have:
\[ \frac{R_1^2}{R_2^2} = \frac{16}{9} \]
- Taking the square root gives:
\[ \frac{R_1}{R_2} = \sqrt{\frac{16}{9}} = \frac{4}{3} \]

4. **Finding Maximum and Minimum Intensities**:
- The maximum intensity (I_max) in the interference pattern is given by:
\[ I_{max} = (R_1 + R_2)^2 \]
- The minimum intensity (I_min) is given by:
\[ I_{min} = (R_1 - R_2)^2 \]

5. **Calculating I_max and I_min**:
- Let \( R_1 = 4k \) and \( R_2 = 3k \) for some constant \( k \).
- Then:
\[ I_{max} = (4k + 3k)^2 = (7k)^2 = 49k^2 \]
\[ I_{min} = (4k - 3k)^2 = (k)^2 = k^2 \]

6. **Finding the Ratio of Maximum to Minimum Intensity**:
- The ratio of maximum to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = \frac{49k^2}{k^2} = 49 \]

### Final Answer:
The ratio of the maximum to the minimum intensity in the interference pattern is \( 49:1 \).