First slide

Coherent sources and their importance in interference

Question

Three harmonic waves having equal frequency $v$ and same intensity $I_{0}$ , have phase angles $0, \frac{π}{4}$ and $- \frac{π}{4}$ respectively. When they are superimposed the intensity of the resultant wave is close to:

Moderate

A

$0.2 I_{0}$
B

$3 I_{0}$
C

$5.8 I_{0}$
D

$I_{0}$

Solution

$\begin{array}{l} {\vec{A}}_{1} = A \cos (\frac{π}{4}) \hat{i} + A \sin (\frac{π}{4}) \hat{j} \\ {\vec{A}}_{2} = A \hat{i} \\ {\vec{A}}_{3} = A \cos (\frac{π}{4}) \hat{i} - A \sin (\frac{π}{4}) \hat{j} \\ ∴ {\vec{A}}_{net} = (1 + \sqrt{2}) A \hat{i} \\ A_{net} = (1 + \sqrt{2}) A \end{array}$
$\begin{array}{l} We know that, Intensity \propto (Amplitude)^{2} \\ \Rightarrow I_{net} = K A_{net}^{2} = (1 + \sqrt{2})^{2} {A_{0}}^{2} K \\ But, I_{0} = {KA}_{0}^{2} \\ ∴ I_{net} = (1 + \sqrt{2})^{2} I_{0} = 5.8 I_{0} \end{array}$

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