(a) State the importance of coherent sources in the phenomenon of interference.✌️
(b) In Young’s double slit experiment to produce interference pattern, obtain the
conditions for constructive and destructive interference. Hence, deduce the expression✌️
for the fringe width.
(c) How does the fringe width get affected, if the entire experimental apparatus of Young ✌️
is immersed in water?
Answers
Answer:
a. If coherent sources are not taken, the phase difference between two interfering waves, will change continuously and a sustained interference pattern will not be obtained. ... Thus, coherent sources provide sustained interference pattern.
b.Young's double slit experiment produces the interference pattern. The typical arrangement of double slit experiment is shown in figure.
Here S= point source, S
1
,S
2
are the two slits, d is the distance between slits, PM is the screen where pattern will be observed and D is the distance between slits and screen.
c.If the Young's apparatus is immersed in water, the effect on fringe width will be narrower. The wavelength of light is less in water than in air. Hence, the fringe width will decrease.
Answer:
If coherent sources are not taken. the phase difference between two interfering waves, meeting at any point will change continuously and a sustained interference pattern will not be obtained. Thus. coherent sources provide sustained interference pattern.If coherent sources are not taken, the phase difference between two interfering waves, will change continuously and a sustained interference pattern will not be obtained. Thus, coherent sources provide sustained interference pattern. Suppose two coherent waves travel in the same direction along a straight line, the frequency of each wave is ω/2π and amplitudes of electric field are a1 and a2 respectively. If at any time t, the electric fields of waves at a point are y1 and y2 respectively and phase difference is ϕ, then equation of waves may be expressed as According to Young’s principle of superposition, the resultant displacement at that point will be y = y1 + y2 ..(iii) Substituting values of y y 1 2 and from (i) and (ii) in (iii), we get where A and q are new constants. Then equation (iv) gives y = Acosθsinωt + Asinθcosωt = A sin(ωt + θ) ...(vii) This is the equation of the resultant disturbance. Clearly the amplitude of resultant disturbance is A and phase difference from first wave is θ. The values of A and θ are determined by (v) and (vi). Squaring (v) and (vi) and then adding, we get As the intensity of a wave is proportional to its amplitude i.e. I ∝ A2 or I = KA2 watt/m2 / where K is a constant which depends on properties of medium and the frequency of wave. in interference the frequenciess of two waves are same and medium is same, therefore for convenience, we may take, K =1, then the units of in tensity I will not be watt/m2 but arbitrary. ∴ Intensity of resultant wave Clearly the intensity of resultant wave at any point depends on the amplitudes of individual waves and the phase difference between the waves at the point. Constructive Interference: For maximum intensity at any point cosϕ = +1 Clearly the maximum intensity is obtained in the region of superposition at those points where waves meet in the same phase or the phase difference between the waves is even multiple of π or path difference between them is the integral multiple of λ and maximum intensity is (a1 + a2)2 which is greater than the sum intensities of individual waves by an amount 2a1a2 . Destructive Interference: For minimum in tensity at any point cosϕ = -1 Clearly, the minimum intensity is obtained in the region of superposition at those points where waves meet in opposite phase or the phase difference between the waves is odd multiple of p or path difference between the waves is odd multiple of λ/2 and minimum intensity =(a1 - a2 )2 which is less than the sum of intensities of the individual waves by an amount 2a1a2. From equations (xii) and (xvi) it is clear that the intensity 2a1a2 is transferred from positions of minima to maxima. This implies that the interference is based on conservation of energy. Variation of Intensity of light with position x is shown in fig. Read more on Sarthaks.com - https://www.sarthaks.com/57742/state-the-importance-of-coherent-sources-in-the-phenomenon-of-interference