Question

Two radio stations broadcast their programmes at the same amplitude $A$ and at slightly different frequencies ${\omega }_1$ and ${\omega }_2$ respectively where ${\omega }_1-{\omega }_2=1kHz$. 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. $2\times {10}^{-3}s$
• B. $4\times {10}^{-3}s$
• C. $1.5\times {10}^{-3}s$
• D. ${10}^{-3}s$

Answer 1

Answer: (D)

${10}^{-3}s$

$y_1=Asin\left(2\pi {\omega }_{1}t\right)and{y}_2=Asin\left(2\pi {\omega }_{2}t\right)$
$y=y_1+y_2$

$=Asin\left(2\pi {\omega }_{1}t\right)+Asin\left(2\pi {\omega }_{2}t\right)$
$=2Asin2\pi \frac{({\omega }_2+{\omega }_1)}{2}tcos2\pi \frac{({\omega }_2+{\omega }_1)}{2}t$
$A{}^{\prime }=2Acos2\pi \frac{({\omega }_2+{\omega }_1)}{2}t$
$=2Acos\pi ({\omega }_2-{\omega }_1)t$

$I\propto A{{}^{\prime }}^2=4A^{2}co{s}^2\pi ({\omega }_2-{\omega }_1)t$

For $I$ to be maximum $cos\pi ({\omega }_2-{\omega }_1)t=\pm 1$
$\pi ({\omega }_2-{\omega }_1)t=0,\pi ,2\pi ,....$
$T=t_2-t_1=\frac{1}{{\omega }_2-{\omega }_1}$ $={10}^{-3}s$