Question

Find the resultant intensity of two waves having intensities $9I$ and $4I$ when waves are (i) coherent and (ii) non-coherent.

Hint:

Always convert intensities to amplitudes first when dealing with interference problems.

Answer 1

Step 1: Relation between intensity and amplitude.
\[ I \propto A^2 \quad \Rightarrow \quad A = \sqrt{I} \] So, for intensities $9I$ and $4I$: \[ A_1 = \sqrt{9I} = 3\sqrt{I}, \quad A_2 = \sqrt{4I} = 2\sqrt{I} \]
Step 2: Case (i) Coherent waves.
For constructive interference: \[ A = A_1 + A_2 = 3\sqrt{I} + 2\sqrt{I} = 5\sqrt{I} \] \[ I_{max} = A^2 = (5\sqrt{I})^2 = 25I \] For destructive interference: \[ A = A_1 - A_2 = (3\sqrt{I} - 2\sqrt{I}) = \sqrt{I} \] \[ I_{min} = A^2 = (\sqrt{I})^2 = I \]
Step 3: Case (ii) Non-coherent waves.
Here, resultant intensity is simply the sum: \[ I = 9I + 4I = 13I \]
Step 4: Conclusion.
- Coherent case: $I_{max} = 25I$, $I_{min} = I$.
- Non-coherent case: $I = 13I$.