3,8.
Diffraction at a Straight Edge
M
Consider a straight edge AD and an illuminated
narrow slit S parallel to each other and perpendicular to
the plane of the paper. P is apoint on the screen MN,
placed perpendicular to the plane of the paper, Below X
the point P is the geometrical shadow and above P is the B
illuminated portion. XY is the incident wavefront at the
straight edge. S P
b
Let P' be a point distant x above P on the screen.
JoinP'S cutting the wavefront at B, then B is the pole of
the wavefront with reference to the point P'. The D
intensity at P' will depend mainly on the number of half
period strips enclosed between the points A and B. The
point P' will be of maximum intensity, if the number of
half period strips enclosed between B and A is odd and Fig. 7
the intensity at P' willbe minimum if the number of half
period strips enclosed between Band A is even.
Positions of maximum and minimum intensity
Let the distance between the slit and the straight edge = a
The distance between the straight edge and the screen =b
The distance PP'=x
The path differenceð = AP' - BP'
= (b' +x)2 -(SP'-SB)
1
- (8' +x)2 -(a+b)² +2+a
2 2 +a
= bl1+:
262 2(a+b)
x +a=
2b 2(a+b) 2b atb)
Or
2 b(a+b)
The pointP'will be of maximumintensity ifð= (2n+l)
2
2b(a + b) =(2n+1)2
Or 2 (2n+1)(a+b)bl
or |(2n+1)\a+b)bl
where x, is the distance of the nth bright band from P. The point P' will be of minimum intensiy i
-n
2b(a+b)
Or 2n(a+6) b
or |2n (a+b) b2
where x, is the distance of the nth dark band from P.
Thus. diffraction bands of varying intensity are observed above the
geometrical shadowie. above P.
Intensity at a point inside the geometrical shadow
IfP' isa point below P and B is the
M
new pole of the wavefront with reference
to the point P', then the half period strips
below B are cut off by the obstacle and
only the uncovered half_period strips
above B will be effective in producing S
b
the illumination at P'. As P' moves P 4
farther from P, more number of half B
period strips above B are also cut off and D
the intensity gradually falls. Thus, within p
the geometrical shadow, the intensity
gradually falls off depending on the
position of P' with re_pect to P. Fig. &
Cio 9shows the intensitydistribution on the
PM, alternate bright and dark bands of gradually screen due tointensity
diminishing a straight will
edge.be Inobserved
the illuminated portion
and the intensity
fallssoff graduallyin the region of the geometrical shadow.
M M
Illuminated
Portion
Screen
S P
A
Geometrical
Shadow
Intensity
N N
Fig. 9
Expression for fringe width
The fringe width B is the distance between any two consecutive bright or dark fringes. Since for a dark
dband or fringe
|2n(a+b) ba
a
Fringe width ß = Xn+l n
2(a+b) bi
a
|2(a+ b) b
-Nn+I-yak where k=
0.41k
wian of Istdark fringe = X, -x,-(/2 -/1) k=
0.31 k
"un or 2nddark fringe = X; -X, (3-2 ) k=
Wiath of 3ddark fringe = X, -X, = (W4-3 ) k=0.26 ketc.
bands produced are not equally
iS Shows that the fringe width goes on decreasing and hence the
Spaced.
