When two waves of intensities `l_1 and l_2` coming from coherent sources interfere at a point P, where phase difference is `phi`, then resultant intensity `(l_(res))` at point P would be

Options

A. `l_(res)=l_1-l_2-2sqrt(l_1l_2)`
B. `l_(res)=l_1+l_2+2sqrt(l_1l_2)cosphi`
C. `l_(res)=l_1+l_2-2sqrt(l_1l_2)cos^2phi`
D. `l_(res)=(l_1)^2+(l_2)^2-2(l_1l_2)cos^2phi`

Answer

To find the resultant intensity \( I_{\text{res}} \) at point P when two waves of intensities \( I_1 \) and \( I_2 \) interfere with a phase difference \( \phi \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship Between Intensity and Amplitude**:
- The intensity \( I \) of a wave is related to its amplitude \( A \) by the formula:
\[ I \propto A^2 \]
- This means that if we know the intensity, we can find the amplitude as:
\[ A = \sqrt{k \cdot I} \]
- Here, \( k \) is a constant that depends on the medium.

2. **Expressing Amplitudes in Terms of Intensities**:
- For the two waves, we can express their amplitudes \( A_1 \) and \( A_2 \) in terms of their respective intensities:
\[ A_1 = \sqrt{k \cdot I_1} \]
\[ A_2 = \sqrt{k \cdot I_2} \]

3. **Resultant Amplitude Calculation**:
- When two waves interfere, the resultant amplitude \( A_{\text{res}} \) can be calculated using the formula for the resultant of two vectors:
\[ A_{\text{res}} = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos \phi} \]
- Substituting the expressions for \( A_1 \) and \( A_2 \):
\[ A_{\text{res}} = \sqrt{(\sqrt{k \cdot I_1})^2 + (\sqrt{k \cdot I_2})^2 + 2 (\sqrt{k \cdot I_1})(\sqrt{k \cdot I_2}) \cos \phi} \]
- Simplifying this gives:
\[ A_{\text{res}} = \sqrt{k \cdot I_1 + k \cdot I_2 + 2k \sqrt{I_1 I_2} \cos \phi} \]
- Factoring out \( k \):
\[ A_{\text{res}} = \sqrt{k \left( I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos \phi \right)} \]

4. **Calculating Resultant Intensity**:
- Now, to find the resultant intensity \( I_{\text{res}} \), we use the relationship between intensity and amplitude:
\[ I_{\text{res}} \propto A_{\text{res}}^2 \]
- Thus,
\[ I_{\text{res}} = k A_{\text{res}}^2 \]
- Substituting for \( A_{\text{res}}^2 \):
\[ I_{\text{res}} = k \left( k \left( I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos \phi \right) \right) \]
- Since \( k \) cancels out, we get:
\[ I_{\text{res}} = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos \phi \]

### Final Result:
The resultant intensity \( I_{\text{res}} \) at point P is given by:
\[ I_{\text{res}} = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos \phi \]