Waves of same amplitude and same frequency from two coherent source overlap at a point. The ratio of the resultant intensity when they arrive in phase to that when they arrive with `90^(@)` phase difference is

Options

A. `1:1`
B. `sqrt(2) : 1`
C. `2 : 1`
D. `4 : 1`

Answer

To solve the problem, we need to calculate the resultant intensity of two coherent waves arriving at a point under two different conditions: when they are in phase and when they have a phase difference of 90 degrees.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have two coherent sources emitting waves of the same amplitude (A) and frequency. We need to find the ratio of resultant intensities when the waves arrive in phase (0 degrees) and when they arrive with a phase difference of 90 degrees.

2. **Resultant Amplitude When In Phase:**
When the two waves are in phase (phase difference θ = 0 degrees), the resultant amplitude (R1) can be calculated using the formula:
\[ R_1 = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(0)} \]
Since \(A_1 = A_2 = A\), we have:
\[ R_1 = \sqrt{A^2 + A^2 + 2A \cdot A \cdot 1} = \sqrt{2A^2 + 2A^2} = \sqrt{4A^2} = 2A \]

3. **Resultant Amplitude When Out of Phase by 90 Degrees:**
When the two waves have a phase difference of 90 degrees (θ = 90 degrees), the resultant amplitude (R2) is given by:
\[ R_2 = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(90)} \]
Again, since \(A_1 = A_2 = A\), we have:
\[ R_2 = \sqrt{A^2 + A^2 + 2A \cdot A \cdot 0} = \sqrt{2A^2} = \sqrt{2}A \]

4. **Calculating Intensities:**
The intensity (I) of a wave is proportional to the square of its amplitude:
\[ I \propto R^2 \]
Therefore, the intensity when the waves are in phase (I1) is:
\[ I_1 \propto (R_1)^2 = (2A)^2 = 4A^2 \]
And the intensity when the waves are at 90 degrees (I2) is:
\[ I_2 \propto (R_2)^2 = (\sqrt{2}A)^2 = 2A^2 \]

5. **Finding the Ratio of Intensities:**
Now, we can find the ratio of the resultant intensities:
\[ \text{Ratio} = \frac{I_1}{I_2} = \frac{4A^2}{2A^2} = 2 \]

6. **Final Result:**
The ratio of the resultant intensity when the waves arrive in phase to that when they arrive with a 90-degree phase difference is:
\[ \text{Ratio} = 2:1 \]