9.32
2. The distances measured in the direction of
incident
light
are
a
positive.
3. The distances measured in the opposite direction
of incident light are negative.
Assumptions used in the study of refraction at a
and
sphericalsurtace:
tan a
NM
NM
:
OP
OM
PHYSICS-X1
Pis close
to MM
M
NM NM
p
tan
y
tan
MI P
NM NM
YMC
PC
From Snell's law of refraction,
1. The object taken is a point object placed on the
principal avis.
F
2. The aperture of the spherical refracting surface
sin
i=H2 sin r
As i and r are small, so
is small.
sini=i and
3. The incident and refracted rays make small
angles with the principal axis so that the sines or
sinr=r
tangents of these angles may be taken equal to
the angles themselves.
P la +71=H2lv-B]
or
Retraction at a conver spherical surtace:
)The object lies in rarer medium and the image
or
fornned is real. In Fig. 9.55, APB is a convex refracting
surface which separates a rarer medium of refractive or
index u, from
a
denser medium of refractive index ,
Let P be the pole, C be the centre of curvature and
R= PC be the radius of curvature of this surface.
Suppose a point object O is placed on the principal
axis in the rarer medium. Starting from the point object
0,a ray ON is incident at an angle i. After refraction, it
bends towards the normal CN at an angle of refraction
surface and passes undeviated. The two refracted rays
meet at point I. So lis the real image of point object O.
O
Denser F2
OP
Using new Cartesian sign convention, we find
r. Another ray OP is incident normally on the convex
Rarer 1
L#2-H2H1
PI PC
or
Object distance,
Image distance,
OP=-u
Radius of curvature,
PC = + R
Pl=+
Hi,2-z
or
R
NOTE If first medium is air,
then
P M
=
land 2 =4,
we have
A-R
Fig. 9.55 Refraction from rarer to denser medium,
(i) The object lies in
image fornmed is virtual.
the
rarer
medium and the
When the
object
when the image is real
Draw NM perpendicular to the principal axis. Let
a,Band y be the angles, as shown in Fig. 9.55.
Rarer
Denser
In ANOC, i is an exterior angle, therefore,
M
Similarly, from
A
NIC,
we
have
-R
Y=r+ß
or
Suppose all the rays are pararial. Then he angles Fig 9 56 Refraction from rarer to denser medium,
,r, a, B and y will be small.
when the image is virtual.
O in the
ANI)
CPIMCAI
INSA
NI
Pul the mvex
lom
d4mint
h r rtnm
Iayhapyarlo
arla, thaluw
the ulneial
MP
lyet
i-Y
the rays are pondval.
Suppise all
y wll suall
lhn
mglew
liy.
Relha tlon lum
9.
NM
NM
nedum
denser lo 1aret
wrhieyn lhe Image s peal,
be
,,a,land
lan
TIhen
are purarinl.
Shuppume all the rayn
h,u, and y
NAI NM
M I
Thenn ue
nyis
wll he nanal.
NM N
OM (
NM NM
NM
NM
MI
P
NM
NM
lan
Fm Snell's lw ot tetie lhn,
1 7 (M C
f»r ulran lim frnn
Ium 'ull'% law »l elrau litm,
meulun, we hae
demr u
nAr
An andr
ae nnall aIgle,
t»
NM, NM
NAI, NA
or'l"
NM NM
or
Carlesian
Using new
sign
convention, we
aC Or |""|MCP
find that
Objct distance,
IP=-
Image distanc,
1PC=
Radius of curvature,
4
PI
R
Usinyg
R
-
R
(ui) The object lies in the
image formed is real. Fig.
refracting
surface which is
denser
a
proint object
From ANOC, y = i + a
From ANIC, r=ß+Y
O
sign
Object distance,
Image distance,
OP
Radius of curvature,
CP
converntion,
we
have
- u
Pl
- R
R
rarer
medium.
point object Olies in the denwr medium
The two refracted rays neet at point I. So l is the real
image of the
Cartesian
convex
lowards the
The
new
CP
medium and the
9.57 shows
convex
the
OP
Du
R
(iv) The object lies in the denser medium and the
or
i=y-
image formed is virtual. If the point object Oplaced on
the principal axis lies close to the pole of the refracting
.31
surface, hen the
vo refracted rays appear
as shown in Fig. 9,58. So
Trom the e
point 1,
lo
come
Iis the
9.23
CONCAVE
REFRACION AT A
SPHERICAL SURFACE
Virual
image of the point object
From ANOC, i1y=a or isuFrom ANIC, r+y=ß or r=-Y
Denser 2
Rarer P
.-a
MP
conventions
37.
By stating lhe sigubetween
and
ass.
sump
0uject distanC
a concave shl
lakes place (ið from c a
the refruction
and (i) from o
denser medium
derive tlhe relation
distance amd radius of curvature
used,
surface wlhen
rarer lo oplically
denser lo oplically rarer
medium.
the ass
For new Cartcsian sign convention
tion used, refer to the answer of the previous questi
Relral1l
and
n a v e splrical surlte,
stion.
() The object lies in the rarer mediun. In Fig. 9sa
refracting surface separating twNO
APB is a concave
media of refractive indices j, and
Fig. 9.58
A
Refraction from denser to rarer medium
H2
W
Rarer P
when the image is virtual.
Denser
Suppose all the rays are parnxial. Then the angles
,r,a,p and y will be small.
a
tan
a=NM_
aOM OP
BtanB=
Y tan y
M
C
NM
I: Mis elose to P]
NM_ NM
IM
B
IP
NM NM
fig. 9.59
From Snel's law of refraction, for refraction from
denser to rarer medium, we have
sin
P2
Refraction at a concave surface when
the object lies in the rarer medium.
CP
CM
i=H
sin
r
As i and r are small angles, so
sini=i and
sinr=r
Let
P= Pole of the concave surface APB
C
Centre of curvature of the concave
surface
O=Point object placed on the principal axis
I= Virtualimageof point object O
In ANOC,y is an exterior angle, therefore
Or
or
2 (a-7)=H, (B-7)
NM NM
P2OP C
|7PCP
Y=a+i
or
i=y-a
Similarly, from A NIC, we have
=B+r
or
r=y-B
Suppose all the rays are paraxial. hen the angles i,
or
or
r,a, ß and y will be small.
.a
IP OP
CP
Using the new Cartesian sign convention, we have
Object distance,
NM
tan a =-
OM
IP-v
Radius of curvature,
CP=-R
OP
NM1
tan = NM_
M
OP=-
Image distance,
NM
Ytan y
P
NM NM
CM
CP
From Snell's law of refraction,
PSin i=H, sin r
-
-
14
-
R
As i and i are small angles, so
sini
or
R
i and sinr=r
:
Mis close to P
OPTICS
RAY
INSTRUMENTS
AND OPTICAL
9.35
NM-NM NM
M
2la +1=#,1B+y]
Or
,lY-a]=p,Y-B
NM, NM-u.NM, NM
PC
P2OP
Or
PCP OP
or
OP
Using new Cartesian sign
convention,
Object distance,
OP-
Image distance,
IP=-v
we
PC
IP
PCP
Using new Cartesian sign convention,
find
Radius of curvature, CP=- R
Object distance,
Image distance,
OP=-
Radius of curvature,
PC=+ R
or
IP=-v
or
R
R
(i) The object lies in the denser medium. As shown
For Your Knowledge
in Fig. 9.60, when the point object O is placed in the
denser medium, the refracted rays appear to
diverge
from a point I in the denser medium. So l is the virtual
image of the point object O.
For both convex and concave spherical surfaces, the
refraction formulae are same, only proper signs of u, v
and R are to be used.
refraction from
formula is
For
i=a+
r=ß+7
rarer lo
denser mediun, the refraction
...1)
R
Denser Hh
find
-14
-R
From A NOC,
From ANIC
we
Rarer-H1
PM
For refraction from denser to rarer medium, we inter
Change
and u, and obtain the refraction formula,
I f the rarer medium is air (4, = 1) and the denser
medium has refractive index
(i.e., 4, = p), then for
refraction from rarer to denser medium, from (1) we
get the relation:
Fig.9.60 Refraction at a concave surface when the
object lies in the denser medium.
Suppose al the rays are paraxial. Then the angles i,
r,a, andy will be small.
a
tan a =iV
OP
tan p= NM
IM
NM
I: Mis cdose to P]
NM
NM
MC
PC
H, sin
sinr= r
positive for a
surface, v will be negative
R/(4- 1). In that case, the
convex
if the value of uis less than
image will be formed in air and will be virtual.
r
As i and r are small angles, so
sinii and
object placed in air, the refraction formula (3) is
applicable, ie. _=
an
As Ris
denser to rarer medium, we have
Sini
put
4)
R
For
we
R
IP
From Snell's law of refraction, for refraction from
2
For refraction from denser to rarer medium,
4 H 1 / p in (2) and get the relation
1/1 (1/g)-1
NM
OM
Ytan y
(3)
R
As R is negative for a concave
will also be negative for all
surface,
the value of v
values
of u. Thus
negative
image will always be formed in air and will be virtual.
The factor
is
called
power factor
of the
spherical refracting surface. It gives a measure of the
degree to which the refracting surface can converge or
diverge the rays of light passing through it.
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