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Question

In young's double slit experiment β is the fringe width and Io is the intensity of the central bright fringe. What will be the intensity at a distance x from the central bright fringe?

A
Io cos (xβ)
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B
Io cos2(xβ)
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C
Io cos2(πxβ)
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D
(Io4) cos2πxβ
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Solution

The correct option is A Io cos2(πxβ)
If δ is the phase difference between two waves with intensities I1 and I2, then the total intensity is given by
I=I1+I2+2I1I2cosδ
At the central fringe, the phase difference, δ=0. Also given Io as the intensity of central bright fringe.
Hence we have,
I=I1+I2+2I1I2
Also for the points between maxima and minima, I1=I2=I1
Therefore Io=I1+I1+2I1=4I1

I1=Io4

Now at distance x from the center, the path difference = πxD
Therefore phase difference, δ=2πλπxD

Hence intensity at that point is given by

I=I1+I1+2I1cos2πxdλD

i.e, I=Io4+Io4+2Io4cos2πxdλD

I=Io2[1+cos2πxdλD]

I=Iocos2πxdλD [2cos2x=1+cos2x]

Also we have, fringe width β=λDd

Hence we have, I=Iocos2πxβ

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