Two coherent light beams of intensity ` I` and `4I` are superposed. If the maximum and minimum possible intensities in the resulting beam are `I_("max") and I_("min")`, then `I_("max")-I_("min")=nI` Find the value of nI. Find the value of n.

Answer

To solve the problem, we need to find the maximum and minimum intensities when two coherent light beams of intensities \( I_1 = I \) and \( I_2 = 4I \) are superposed.

### Step-by-Step Solution:

1. **Understanding the Superposition of Intensities**:
When two coherent light beams are superimposed, the resultant intensity can be calculated using the formula:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \theta \]
where \( \theta \) is the phase difference between the two beams.

2. **Calculating Maximum Intensity \( I_{\text{max}} \)**:
The maximum intensity occurs when \( \cos \theta = 1 \):
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
Substituting \( I_1 = I \) and \( I_2 = 4I \):
\[ I_{\text{max}} = I + 4I + 2\sqrt{I \cdot 4I} \]
\[ I_{\text{max}} = 5I + 2\sqrt{4I^2} = 5I + 4I = 9I \]

3. **Calculating Minimum Intensity \( I_{\text{min}} \)**:
The minimum intensity occurs when \( \cos \theta = -1 \):
\[ I_{\text{min}} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]
Substituting \( I_1 = I \) and \( I_2 = 4I \):
\[ I_{\text{min}} = I + 4I - 2\sqrt{I \cdot 4I} \]
\[ I_{\text{min}} = 5I - 4I = I \]

4. **Finding the Difference \( I_{\text{max}} - I_{\text{min}} \)**:
Now we can find the difference:
\[ I_{\text{max}} - I_{\text{min}} = 9I - I = 8I \]

5. **Relating to the Given Equation**:
We are given that:
\[ I_{\text{max}} - I_{\text{min}} = nI \]
From our calculation, we have:
\[ 8I = nI \]
Dividing both sides by \( I \) (assuming \( I \neq 0 \)):
\[ n = 8 \]

### Final Answer:
The value of \( n \) is \( 8 \).