Optics Question 848
Question: In an interference arrangement similar to Young’s double-slit experiment, the slits $ S _{1} $ and $ S _{2} $ are illuminated with coherent microwave sources, each of frequency 106 Hz. The sources are synchronized to have zero phase difference. The slits are separated by a distance d = 150.0 m. The intensity I $ ( \theta ) $ is measured as a function of $ \theta $ , where $ \theta $ is defined as shown. If $ I _{0} $ is the maximum intensity, then I $ ( \theta ) $ for $ $ is given by
Options:
A) $ I(\theta )=I _{0}/2 $ for $ \theta =30^{o} $
B) $ I(\theta )=I _{0}/4 $ for $ \theta =90^{o} $
C) $ I(\theta )=I _{0} $ for $ \theta =0^{o} $
D) $ I=\theta $ is constant for all values of $ \theta $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] We know that $ I(\theta )=I _{0}{{\cos }^{2}}\frac{\delta }{2} $
where $ \delta =\frac{2\pi d\tan \theta }{\lambda } $
$ I(\theta )=I _{0}{{\cos }^{2}}( \frac{\pi d\tan \theta }{\pi } )=I _{0}{{\cos }^{2}}( \frac{\pi \times 150\times \tan \theta }{3\times 10^{8}/10^{6}} ) $
$ =I _{0}{{\cos }^{2}}( \frac{\pi }{2}\tan \theta ) $
For $ \theta =30^{o} $ ; $ I(\theta )=I _{o}{{\cos }^{2}}( \frac{\pi }{2\sqrt{3}} ) $ For $ \theta =90^{o} $ ; $ I(\theta )=I _{o}{{\cos }^{2}}(\infty ) $ For $ \theta =0^{o} $
$ I(\theta )=I _{0} $
$ I(\theta ) $ is not constant.
Alternatively, when $ \theta $ is zero the path difference between wave originating from $ S _{1} $ and that from $ S _{2} $ will be zero.
This corresponds to a maxima.
