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(a) Two monochromatic waves emanating from two coherent sources have the displacements represented by y_(1)=acosomegat and y_(2)=acos(omegat+phi), where phi is the phase difference between the two displacements . Show that the resultant intensity at a point due to their superposition is given by I=4I_(0)cos^(2)phi//2, where I_(0)=a^(2). (b) Hence obtain the conditions for constructive and destructive interference. |
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Answer» Solution :(a) The resultant displacement is GIVEN by : `y=y_(1)+y_(2)` `=acosomegat+acos(omegat+phi)` `=acosomegat(1+cosphi)-asinomegatsinphi` Put `Rcostheta=a(1+cosphi)` `Rsintheta=asinphi` `:.R^(2)=a^(2)(1+cos^(2)phi+2cosphi)+a^(2)SIN^(2)phi` `=2a^(2)(1+cosphi)=4a^(2)cos^(2).(phi)/(2)` `:.I=R^(2)4a^(2)cos^(2).(phi)/(2)=4I_(0)cos^(2).(phi)/(2)` For CONSTRUCTIVE interference, `cos.(phi)/(2)=+-1` or `(phi)/(2)=npi` or `phi=2npi` For DESTRUCTIVE interference `cos.(phi)/(2)=0` or `(phi)/(2)=(2n+1).(pi)/(2)` or `phi=(2n+1)pi` |
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