Two sources with intensity `4I_0` , and `9I_0` , interfere at a point in medium. The minimum intensity would be

Options

A. `5 I_0`
B. `4 I_0`
C. `2 I_0`
D. `I_0`

Answer

To find the minimum intensity when two sources with intensities \(4I_0\) and \(9I_0\) interfere, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Intensities**:
We have two sources with intensities:
- \(I_1 = 4I_0\)
- \(I_2 = 9I_0\)

2. **Use the Formula for Minimum Intensity**:
The formula for the minimum intensity \(I_{\text{min}}\) when two waves interfere is given by:
\[ I_{\text{min}} = \left(\sqrt{I_1} - \sqrt{I_2}\right)^2 \]

3. **Calculate the Square Roots**:
- Calculate \(\sqrt{I_1}\):
\[ \sqrt{I_1} = \sqrt{4I_0} = 2\sqrt{I_0} \]
- Calculate \(\sqrt{I_2}\):
\[ \sqrt{I_2} = \sqrt{9I_0} = 3\sqrt{I_0} \]

4. **Substitute into the Minimum Intensity Formula**:
Now substitute the square roots back into the formula:
\[ I_{\text{min}} = \left(2\sqrt{I_0} - 3\sqrt{I_0}\right)^2 \]

5. **Simplify the Expression**:
- Calculate the difference:
\[ 2\sqrt{I_0} - 3\sqrt{I_0} = -1\sqrt{I_0} \]
- Now square the result:
\[ I_{\text{min}} = (-\sqrt{I_0})^2 = I_0 \]

6. **Final Result**:
Therefore, the minimum intensity at the point of interference is:
\[ I_{\text{min}} = I_0 \]

### Conclusion:
The minimum intensity when two sources with intensities \(4I_0\) and \(9I_0\) interfere is \(I_0\).

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