742
15 Optics
Therefore, the condition for maximum intensity at P is
xd
L
= mλ
(bright fringes)
(3)
and the condition for zero intensity at P is
xd
L
=
m +
1
2
λ
(dark fringes)
(4)
m is called the order of fringe system. On the axis, at y = 0, m = 0 and
we have the intensity maximum. The central bright band on the screen is
flanked on either side by a series of b bright and dark bands corresponding to
m = 1, 2, 3, . . ., the mth bright fringe being at a distance y m from the axis:
y m =
mλL
d
(5)
and the (m + 1)th bright fringe being at a distance
y (m+1) =
(m + 1) λL
d
(6)
The separation of the fringes β called the bandwidth is given by
y = y (m+1) − y m = β =
λL
d
(7)
β =
λL
d
=
612 × 10 −9 × 4
2 × 10 −3
= 1.124 × 10
−3 m = 1.124 mm
15.22 Let a transparent plate of thickness t and refractive index μ be introduced
in the path of one of the two interfering beams of monochromatic light,
Fig. 15.30. A ray travelling from S 1 to O covers a distance t in the plate
while the rest of the distance (S 1 O − t) is covered in air. The effective optical
path length would be
μt + (S 1 O − t) or S 1 O + (μ − 1)t
The optical path length for the ray emanating from S 2 would be S 2 O. Clearly
S 1 O + (μ − 1)t > S 2 O
Consequently, the central fringe corresponding to zero path difference is not
formed at O, the normal position of the central fringe in the absence of the
plate. The new position of the central fringe would be at O such that
S 1 O
+ (μ − 1)t = S 2 O
15 Optics
Therefore, the condition for maximum intensity at P is
xd
L
= mλ
(bright fringes)
(3)
and the condition for zero intensity at P is
xd
L
=
m +
1
2
λ
(dark fringes)
(4)
m is called the order of fringe system. On the axis, at y = 0, m = 0 and
we have the intensity maximum. The central bright band on the screen is
flanked on either side by a series of b bright and dark bands corresponding to
m = 1, 2, 3, . . ., the mth bright fringe being at a distance y m from the axis:
y m =
mλL
d
(5)
and the (m + 1)th bright fringe being at a distance
y (m+1) =
(m + 1) λL
d
(6)
The separation of the fringes β called the bandwidth is given by
y = y (m+1) − y m = β =
λL
d
(7)
β =
λL
d
=
612 × 10 −9 × 4
2 × 10 −3
= 1.124 × 10
−3 m = 1.124 mm
15.22 Let a transparent plate of thickness t and refractive index μ be introduced
in the path of one of the two interfering beams of monochromatic light,
Fig. 15.30. A ray travelling from S 1 to O covers a distance t in the plate
while the rest of the distance (S 1 O − t) is covered in air. The effective optical
path length would be
μt + (S 1 O − t) or S 1 O + (μ − 1)t
The optical path length for the ray emanating from S 2 would be S 2 O. Clearly
S 1 O + (μ − 1)t > S 2 O
Consequently, the central fringe corresponding to zero path difference is not
formed at O, the normal position of the central fringe in the absence of the
plate. The new position of the central fringe would be at O such that
S 1 O
+ (μ − 1)t = S 2 O