If `I_1/I_2` =9 then find the value of `(I_max - I_min)/ (I_max+I_min)` `A ̅`

Answer

To solve the problem where we need to find the value of \((I_{\text{max}} - I_{\text{min}}) / (I_{\text{max}} + I_{\text{min}})\) given that \(\frac{I_1}{I_2} = 9\), we can follow these steps:

### Step 1: Understand the given ratio
We are given that \(\frac{I_1}{I_2} = 9\). This implies that:
\[ I_1 = 9k \quad \text{and} \quad I_2 = k \]
for some constant \(k\).

### Step 2: Calculate \(I_{\text{max}}\) and \(I_{\text{min}}\)
Using the formulas for maximum and minimum intensity in interference patterns:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

### Step 3: Substitute the values of \(I_1\) and \(I_2\)
Substituting \(I_1 = 9k\) and \(I_2 = k\):
\[ I_{\text{max}} = (\sqrt{9k} + \sqrt{k})^2 = (3\sqrt{k} + \sqrt{k})^2 = (4\sqrt{k})^2 = 16k \]
\[ I_{\text{min}} = (\sqrt{9k} - \sqrt{k})^2 = (3\sqrt{k} - \sqrt{k})^2 = (2\sqrt{k})^2 = 4k \]

### Step 4: Calculate \(I_{\text{max}} - I_{\text{min}}\) and \(I_{\text{max}} + I_{\text{min}}\)
Now we can find:
\[ I_{\text{max}} - I_{\text{min}} = 16k - 4k = 12k \]
\[ I_{\text{max}} + I_{\text{min}} = 16k + 4k = 20k \]

### Step 5: Find the required ratio
Now we can find the ratio:
\[ \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} = \frac{12k}{20k} = \frac{12}{20} = \frac{3}{5} \]

### Final Answer
Thus, the value of \(\frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}}\) is \(\frac{3}{5}\).

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