ex explain the theory of interference band and find the equation of bandwidth
Answers
Answer:
ANSWER
Expression for bandwidth of interference fringes in youngs double experience : Let d be the distance between two coherent sources A and B of wavelength λ. A screen XY is placed parallel to AB at a distance D from the coherent sources. C is the mid point of AB. O is a point on the screen equidistant from A and B.P is a point at a distance x from O, as shown in figure. Waves from A and B meet at P in phase or out of phase depending upon the path difference between two waves.
Interference bandwidth
Draw AM perpendicular to BP.
The path difference S=BP−AP
AP=MP
∴δ=BP−AP=BP−MP=BM
In right angled △ABM,BM=dsinθ
If Q is small, sinθ=θ
∴ the path difference δ=θd
In right angled triangle COP,
tanθ=
CP
OP
=
D
x
For small values of θ,tanθ=θ
∴ the path difference δ=
D
xd
Bright fringes : By the principle of interference, condition for constructive interference is the path difference =nλ
∴μ
D
xd
=λ
where n=0,1,2,..... indicates the order of the bright fringes.
∴x=
d
D
λ
This equation gives the distance of the n
th
bright fringe from the point O.
Dark fringes : By the principle of interference, condition for destructive interference is the path difference =(2n−1)
2
λ
.
where n=1,2,3,.... indicate the order of the dark fringes.
∴x=
d
D
(2n−1)
2
λ
This equation gives the distance of the nth dark fringe from the point O. Thus, on the screen alternate dark and bright bands are seen on either side of the central bright band.
Bandwidth (β) : The distance between any two consecutive bright or dark bands is called bandwidth. The distance between (n+1)
th
and n
th
order consecutive bright fringes from O is given by
x
(n+1)
−x
n
=
d
D
(n+1)λ−
d
D
nλ=
d
D
λ
Bandwidth, β=
d
D
λ.
solution
Explanation:
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