If `I_1/I_2` =4 then find the value of `(I_max - I_min)/ (I_max+I_min)`

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Write down the given data
We are given the ratio of intensities \( \frac{I_1}{I_2} = 4 \). This implies:
- \( I_1 = 4k \)
- \( I_2 = k \)
for some constant \( k \).

### Step 2: Calculate \( I_{max} \)
The formula for maximum intensity \( I_{max} \) in terms of \( I_1 \) and \( I_2 \) is:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{max} = (\sqrt{4k} + \sqrt{k})^2 = (2\sqrt{k} + \sqrt{k})^2 = (3\sqrt{k})^2 = 9k \]

### Step 3: Calculate \( I_{min} \)
The formula for minimum intensity \( I_{min} \) is:
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting the values:
\[ I_{min} = (\sqrt{4k} - \sqrt{k})^2 = (2\sqrt{k} - \sqrt{k})^2 = (\sqrt{k})^2 = k \]

### Step 4: Calculate \( I_{max} - I_{min} \) and \( I_{max} + I_{min} \)
Now we can find \( I_{max} - I_{min} \) and \( I_{max} + I_{min} \):
\[ I_{max} - I_{min} = 9k - k = 8k \]
\[ I_{max} + I_{min} = 9k + k = 10k \]

### Step 5: Find the value of \( \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \)
Now we can substitute these values into the expression:
\[ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{8k}{10k} = \frac{8}{10} = \frac{4}{5} \]

### Final Answer
Thus, the value of \( \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \) is \( \frac{4}{5} \).

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