Question

Two radio stations broadcast their programmes at the same amplitude $A$ and at slightly different frequencies ${\omega }_1$ and ${\omega }_2$ respectively where ${\omega }_1–{\omega }_2=1\text{kHz}$. A detector receives the signals from the two stations simultaneously. It can only detect signals of Intensity $>2A^2$. Find the interval between successive maxima of the intensity of the signal received by the detector.

• A. $2\times {10}^{-3}\text{s}$
• B. $4\times {10}^{-3}\text{s}$
• C. $1.5\times {10}^{-3}\text{s}$
• D. ${10}^{-3}\text{s}$

Answer 1

The correct option is D ${10}^{-3}\text{s}$

Let $y_1=A\sin \left(2\pi {\omega }_{1}t\right)$ and $y_2=A\sin \left(2\pi {\omega }_{2}t\right)$
$y=y_1+y_2$
$=A\sin \left(2\pi {\omega }_{1}t\right)+A\sin \left(2\pi {\omega }_{2}t\right)$
$=2A\sin 2\pi \frac{({\omega }_2+{\omega }_1)}{2}t\cos 2\pi \frac{({\omega }_2-{\omega }_1)}{2}t$
$A=2A\cos 2\pi \frac{({\omega }_2-{\omega }_1)}{2}t$
$=2A\cos \pi ({\omega }_2-{\omega }_1)t$
$I\propto A^2=4A^2{\cos }^2\pi ({\omega }_2-{\omega }_1)t$
For $I$ to be maximum, $\cos \pi ({\omega }_2-{\omega }_1)t=\pm 1$
$\pi ({\omega }_2-{\omega }_1)t=0,\pi ,2\pi ,...$
$T=t_2-t_1=\frac{1}{{\omega }_2-{\omega }_1}={10}^{-3}\text{s}$