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  • (a) What do you understand by coherent sources? (b) Derive the expression for the intensity distribution in the interference pattern on the screen when two harmonic waves having same amplitude (a) and phase difference (<img alt="\Phi" src="ht

(a) What do you understand by coherent sources?

(b) Derive the expression for the intensity distribution in the interference pattern on the screen when two harmonic waves having same amplitude (a) and phase difference (\Phi) are superposed on each other. Hence obtain the conditions for constructive and destructive interference.

 

 

 
 
 
 
 

Answers (1)

(a) Coherent sources are the sources that emit continuous waves of the same frequency and with zero phase difference or constant phase difference.

(b) Consider 2 coherent sources S_{1} and S_{2}. Consider a point P ,

displacement produced by source S_{1} at P is 

    y_1=acos\omega t

Then by S_{2} at P is 

    y_2=acos(\omega t+\phi)

Total displacement, y=y_{1}+y_{2}

 =A cos\omega t+Acos(\omega t+\phi)

=2A cos \frac{\phi}{2}cos(\omega t+\frac{\phi}{2})

The resultant amplitude is 

=2A cos \frac{\phi}{2}

Resultant intensity = Square of amplitude 

 =4A cos^2 \frac{\phi}{2}

=4I_0 cos^2 \frac{\phi}{2}         

For constructive interference 

\\I=4I_0 \\\Rightarrow cos^2 \frac{\phi}{2}=1\\\\\Rightarrow\phi=0,\pm 2\pi, \ 4\pi,..............

For destructive interference

\\I=0 \\\Rightarrow cos^2 \frac{\phi}{2}=0\\\\\Rightarrow\phi=\,pi\pm 3\pi, \ 5\pi,..............

Posted by

Safeer PP

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