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Question
Figure (16-E7) shows two coherent sources S1 and S2 which emit sound of wavelength X in phase. The separation between the sources is 3λ. A circular wire °f large radius is placed in such a way that S1,S2 lies in its plane and the middle point of S1,S2 is at the centre of the wire. Find the angular positions θ on the wire for which constructive interference takes place.
Figure (16-E7) shows two coherent sources S1 and S2 which emit sound of wavelength X in phase. The separation between the sources is 3λ. A circular wire °f large radius is placed in such a way that S1,S2 lies in its plane and the middle point of S1,S2 is at the centre of the wire. Find the angular positions θ on the wire for which constructive interference takes place.
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Solution
As shown in the figure,
(S1P)2=(S1x)2+(Px)2......(1)
(S2P)2=(S2M)2+(PM)2......(2)
From (1)and (2),
(S1P)2−(S2P)2=(Px)2−(PM)2
=(1.5λ+R cos θ)2−(R cosθ−1.5λ)2
=6λ R cos θ
⇒(S1P−S2P)2=6λ R cos θ2R
For constructive interference,
(S1P−S2P)2=x=3λ cos θ=nλ
⇒cos θ=n3
⇒θ=cos−1(n3)
where, n =0,1,2,....
⇒θ=0∘,48∘,2∘,7∘,5∘,90∘ and similar points in other quadrants.
As shown in the figure,
(S1P)2=(S1x)2+(Px)2......(1)
(S2P)2=(S2M)2+(PM)2......(2)
From (1)and (2),
(S1P)2−(S2P)2=(Px)2−(PM)2
=(1.5λ+R cos θ)2−(R cosθ−1.5λ)2
=6λ R cos θ
⇒(S1P−S2P)2=6λ R cos θ2R
For constructive interference,
(S1P−S2P)2=x=3λ cos θ=nλ
⇒cos θ=n3
⇒θ=cos−1(n3)
where, n =0,1,2,....
⇒θ=0∘,48∘,2∘,7∘,5∘,90∘ and similar points in other quadrants.
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