CHAPTER
Ray Optics and Optical
Instruments
9
Reflection
Spherical Mirrors
ra
cid
ted
en
i r
r
re
ay
tr
fle
c
n
in
e
M
In vector form r= e − 2(e.n)n
Object
Real: Point from which rays actually diverge.
Virtual: Point towards which rays appear to converge.
Image
Image is decided by reflected or refracted rays only. The point of
image for a mirror is that point towards which the rays reflected
from the mirror actually converge (real image).
OR
From which the reflected rays appear to diverge (virtual image).
Characteristics of Reflection by a Plane Mirror
The size of the image is the same as that of the object.
For a real object the image is virtual and for a virtual object
the image is real.
For a fixed incident light ray, if the mirror is rotated through an
angle q the reflected ray turns through an angle 2q in the same
sense.
360
Number of images (n) in inclined mirror. Find
=m
θ
If m is even, then n = m – 1, for all positions of object.
If m is odd, then n = m, if object is not on bisector and
n = m – 1, if object is bisector
If m is fraction then n = nearest even number
20
M
M'
principal
axis
M'
C
θ
M
C F
spherical
surface
spherical
mirror
Mirror Formula:
1 1 1
=
+ .
f v u
y
Laws of Reflection
The incident ray the reflected ray and normal to the surface of
reflection at the point of incidence lie in the same plane, This
plane is called the plane of incidence (also plane of reflection).
The angle of incidence and the angle of reflection are equal
∠i = ∠r
normal
P
C F
M
concave
mirror
P
M
convex
mirror
f = focal length u = object distance
v = image distance
Note: Valid only for paraxial rays.
Transverse Magnification: mt =
h2
v
= −
h1
u
h2 = height of image h1 = height of object
(both perpendicular to the principal axis of mirror)
Longitudinal magnification: ml =
Length of image
Length of object
For small object ml = –m2t
Velocity of image of Moving Object
(Spherical Mirror)
Velocity component along axis (Longitudinal velocity)
M
O
When an object is coming from infinity towards the focus of
concave mirror
JEE (XII) Module-3 PW
A
1 1 1
1 dv 1 du
v2
+ =
∴− 2
− 2
= 0 ⇒ v IM = − 2 v OM = −m 2 v OM
v u f
v dt u dt
u
VIM
=
i
dv
= velocity of image with respect to mirror
dt
N
AIR
B
GLASS
r
t
du
VOM
= = velocity of object with respect to mirror.
dt
N'
Optical power of a mirror (in Diopters) = −
1
f
where f = focal length (in meters) with sign.
Laws of Refraction
(i) Incident ray, refracted ray and normal always lie in the same
plane.
(ii) The product of refractive index and sine of angle of incidence
at a point in a medium is constant. m1 sin i = m2 sin r (Snell’s
law)
i
x
i
D
Lateral shift x =
t sin(i − r )
; t = thickness of slab
cos r
Notes: Emergent ray will not be parallel to the incident ray if the
medium on both the sides are different.
Apparent Depth of Submerged Object: (h’ < h)
r
2
1
n
e
90°
c
Optical Power
1 > 2
1
i
hʹ
h
r
2
Oʹ
r
i
r
O
In vector form (eˆ × nˆ ).rˆ =
0
For near normal incidence h′ =
Snell’s Law
v
µ
λ
sin i 1
=µ2 = 2 = 1 = 1 In vector form µ1 | eˆ × nˆ |=µ 2 | rˆ × nˆ |
µ1 v2 λ 2
sin r
Notes: Frequency of light does not change during refraction.
Deviation of a Ray Due to Refraction
µ2
h
µ1
1
∆x = Apparent shift
= t 1 −
µ
=1
O
always in direction of incident ray.
=1
Oʹ
∆x
t
Notes: h and h′ are always measured from surface.
Critical Angle & Total Internal Reflection (TIR)
i
N
N
N
r
r
Rarer
1
d
C
i
1ʹ
i >C
Denser
angle of deviation, δ = i − r (clockwise)
Refraction Through a Parallel Slab
Emergent ray is parallel to the incident ray, if medium is same on
both sides.
P
W Ray Optics and Optical Instruments
S
Nʹ
Nʹ
Nʹ
Conditions of TIR
Ray is going from denser to rarer medium.
Angle of incidence should be greater than the critical angle (i > C).
21
λ
−1 µ R
−1 vD
=
C sin
=
sin
=
sin −1 D
Critical angle
vR
µD
λR
θ
δ3
δ1δ2
R
θ
Refraction Through Prism
mean ray
P
V
A
δ
i
r
iʹ
For small angled prism (A ≤ 10º);
rʹ
=
ω
Q
R
d = (i + i¢) – (r + r¢)
Refraction at Spherical Surface
r + r¢ = A
d = i + i¢ – A
µ 2 µ1 µ 2 − µ1
− =
v
u
R
v, u and R are to be kept with sign as
v = PI
u = –PO
R = PC
(a)
Variation of d versus i
δmax
(b) m =
δmin
i
i2
angle of
incidence
There is one and only one angle of incidence for which the
angle of deviation is minimum. When d = dmin then i = i′ and
r = r′, the ray passes symmetrically about the prism, and then
A + δmin
sin
2
µ=
, where m = absolute R.I. of glass.
A
sin
2
Notes: When the prism is dipped in a medium then µ = R.I. of
glass w.r.t. medium.
For a thin prism (A ≤ 10º) ; δ = (µ – 1)A
Dispersion of Light
The angular splitting of a ray of white light into a number of
components when it is refracted in a medium other than air is
called Dispersion of Light.
O
P
C
I
+ve
1
1
(a)
1
1 1 1
1
1
− = , (b) = (µ − 1) −
+ve v u
f
f
R1 R2
(c) Magnification m =
v
u
Power of Lenses
Reciprocal of focal length in meter is known as power of lens.
SI unit: Dioptre (D)
Power of lens:
=
P
1
=
f ( m)
100
dioptre
f (cm)
Combination of Lenses
Two thin lens are placed in contact to each other
f1
f2
Angle between the rays of the extreme colours in the refracted
(dispersed) light is called Angle of Dispersion. θ = δv – δr
Dispersive power (w) of the medium of the material of prism.
angular dispersion
ω=
devitation of mean ray (yellow)
22
µ1v
µ2u
2
1
Lens Formula
i1
mV, mR and my are R.I. of material for violet, red and yellow
colours respectively.
angle of
deviation
δV − δ R µV − µ R
µV + µ R
; µy
=
=
δ
µ y −1
2
Power of combination. P = P1 + P2 ⇒
1 1 1
= +
F f1 f 2
JEE (XII) Module-3 PW
Use sign convention while solving numericals.
Newton‘s Formula
For Compound Microscope
u
x1
O
v
f
F1
f
f = x1 x2
x2
F2
Silvering of Lens
Silvering of one surface of lens (use Peq = 2Pl + Pm)
When plane surface is silvered O
f =
Tube length L = v0 + | ue |
v D
− 0×
When final image is formed at infinity M =
and L
u0 f e
= v0 + fe
Astronomical Telescope
Magnifying power when final image is formed at D, M
v
D
=
− 0 1 +
u0
fe
I
x1 = distance of object from first focus; x2 = distance of image from
second focus.
When image is formed at infinity M = D/f
R
2(µ − 1)
Magnifying power when final image is formed at D:
f
f
f
M =
− 0 =0 1 + e
ue
fe
D
Tube length: L = f0 + | ue |
When final image is formed at infinity: M =
length L = f0 + fe
When convex surface is slivered O
f =
R
2µ
f0
and tube
fe
Limit of resolution for microscope:
1.22λ
1
=
2µ sin θ Resolving power
Limit of resolution for telescope:
Optical Instruments
For Simple Microscope
1.22λ
1
=
a
Resolving power
Magnifying power when image is formed at D: M = 1 + D/f
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