Important Formulas: Ray Optics and Optical Instruments | Physics Class 12 - NEET PDF Download

 Page 2


           
             
 
       
Differentiating w.r.t time , we  get
v
(im)x
 = -v
(om)x 
; v
(im)y
 = v
(om)y 
; v
(im)z
 = v
(om)z ,
3. Spherical  Mirror
1
v
+
1
u
=
2
R
=
1
f
..... Mirror formula
x co–ordinate of centre of  Curvature and  focus of Concave
mirror are negative and those for Convex mirror are positive.
In case of mirrors since light rays reflect back in - X direction,
therefore -ve sign of v indicates real image and +ve
sign of v indicates virtual image
(b) Lateral magnification (or transverse magnification)
m= 
h
h
2
1
m = ?
v
u
.
(d) On differentiating (a) we get
dv
du
=
?
v
u
2
2
 .
(e) On differentiating (a) with respect to time we get
dv
dt
v
u
du
dt
? ?
2
2
,where 
dv
dt
 is the velocity of image along Principal
axis and
du
dt
 is the velocity of object along Principal axis. Negative
sign implies that the image , in case of mirror, always moves
in the direction opposite to that of object.This discussion is
for velocity with respect to mirror and along the x axis.
(f) Newton's  Formula: XY = f 
2
X  and Y are the distances ( along the principal axis ) of the object
and image respectively from the principal focus. This formula can
be used when the distances are mentioned or asked from the
focus.
(g) Optical power of a mirror (in Diopters) = 
f
1
f = focal length with sign and in meters.
(h) If object lying along the principal axis is not of very small size, the
longitudinal magnification =
1 2
1 2
u u
v v
?
?
     (it will always be inverted)
Page 3


           
             
 
       
Differentiating w.r.t time , we  get
v
(im)x
 = -v
(om)x 
; v
(im)y
 = v
(om)y 
; v
(im)z
 = v
(om)z ,
3. Spherical  Mirror
1
v
+
1
u
=
2
R
=
1
f
..... Mirror formula
x co–ordinate of centre of  Curvature and  focus of Concave
mirror are negative and those for Convex mirror are positive.
In case of mirrors since light rays reflect back in - X direction,
therefore -ve sign of v indicates real image and +ve
sign of v indicates virtual image
(b) Lateral magnification (or transverse magnification)
m= 
h
h
2
1
m = ?
v
u
.
(d) On differentiating (a) we get
dv
du
=
?
v
u
2
2
 .
(e) On differentiating (a) with respect to time we get
dv
dt
v
u
du
dt
? ?
2
2
,where 
dv
dt
 is the velocity of image along Principal
axis and
du
dt
 is the velocity of object along Principal axis. Negative
sign implies that the image , in case of mirror, always moves
in the direction opposite to that of object.This discussion is
for velocity with respect to mirror and along the x axis.
(f) Newton's  Formula: XY = f 
2
X  and Y are the distances ( along the principal axis ) of the object
and image respectively from the principal focus. This formula can
be used when the distances are mentioned or asked from the
focus.
(g) Optical power of a mirror (in Diopters) = 
f
1
f = focal length with sign and in meters.
(h) If object lying along the principal axis is not of very small size, the
longitudinal magnification =
1 2
1 2
u u
v v
?
?
     (it will always be inverted)
4. Refraction  of  Light
 vacuum. ? ? ?
speed of light in vacuum
speed of light in medium
c
v
.
4.1 Laws  of  Refraction (at  any  Refracting  Surface)
(b)
r Sin
i Sin
 = Constant  for any pair of media and for light of a given
wave length. This is known as Snell's Law. More precisely,
Sin i
Sin r
  = 
n
n
2
1
 = 
v
v
1
2
 = 
?
?
1
2
4.2 Deviation  of  a  Ray  Due  to  Refraction
Deviation ( ?) of ray incident at ? ?i and refracted at ? ?r is given by ? = |i
 
?
 
r|.
5. Principle of  Reversibility  of  Light  Rays
A ray travelling along the path of the reflected ray is reflected along the
path of the incident ray. A refracted  ray  reversed  to travel back along  its
path will get  refracted along the path of the incident  ray. Thus the incident
and refracted rays are mutually reversible.
7. Apparent  Depth and shift of  Submerged  Object
At near normal incidence (small angle of incidence i) apparent depth (d ?)
is given by:
d ?=
relative
n
d
 ? n
relative
 = 
) refraction of medium of . I . R ( n
) incidence of medium of . I . R ( n
r
i
Apparent  shift =  d 
?
?
?
?
?
?
?
?
?
rel
n
1
1
Refraction   through   a Composite  Slab  (or  Refraction through  a
number  of  parallel  media, as  seen  from  a  medium  of  R. I.  n
0
)
Apparent  depth (distance of final image from final surface)
=  
t
n
rel
1
1
 + 
t
n
rel
2
2
 + 
t
n
rel
3
3
 
+......... + 
rel n
n
n
t
      
Page 4


           
             
 
       
Differentiating w.r.t time , we  get
v
(im)x
 = -v
(om)x 
; v
(im)y
 = v
(om)y 
; v
(im)z
 = v
(om)z ,
3. Spherical  Mirror
1
v
+
1
u
=
2
R
=
1
f
..... Mirror formula
x co–ordinate of centre of  Curvature and  focus of Concave
mirror are negative and those for Convex mirror are positive.
In case of mirrors since light rays reflect back in - X direction,
therefore -ve sign of v indicates real image and +ve
sign of v indicates virtual image
(b) Lateral magnification (or transverse magnification)
m= 
h
h
2
1
m = ?
v
u
.
(d) On differentiating (a) we get
dv
du
=
?
v
u
2
2
 .
(e) On differentiating (a) with respect to time we get
dv
dt
v
u
du
dt
? ?
2
2
,where 
dv
dt
 is the velocity of image along Principal
axis and
du
dt
 is the velocity of object along Principal axis. Negative
sign implies that the image , in case of mirror, always moves
in the direction opposite to that of object.This discussion is
for velocity with respect to mirror and along the x axis.
(f) Newton's  Formula: XY = f 
2
X  and Y are the distances ( along the principal axis ) of the object
and image respectively from the principal focus. This formula can
be used when the distances are mentioned or asked from the
focus.
(g) Optical power of a mirror (in Diopters) = 
f
1
f = focal length with sign and in meters.
(h) If object lying along the principal axis is not of very small size, the
longitudinal magnification =
1 2
1 2
u u
v v
?
?
     (it will always be inverted)
4. Refraction  of  Light
 vacuum. ? ? ?
speed of light in vacuum
speed of light in medium
c
v
.
4.1 Laws  of  Refraction (at  any  Refracting  Surface)
(b)
r Sin
i Sin
 = Constant  for any pair of media and for light of a given
wave length. This is known as Snell's Law. More precisely,
Sin i
Sin r
  = 
n
n
2
1
 = 
v
v
1
2
 = 
?
?
1
2
4.2 Deviation  of  a  Ray  Due  to  Refraction
Deviation ( ?) of ray incident at ? ?i and refracted at ? ?r is given by ? = |i
 
?
 
r|.
5. Principle of  Reversibility  of  Light  Rays
A ray travelling along the path of the reflected ray is reflected along the
path of the incident ray. A refracted  ray  reversed  to travel back along  its
path will get  refracted along the path of the incident  ray. Thus the incident
and refracted rays are mutually reversible.
7. Apparent  Depth and shift of  Submerged  Object
At near normal incidence (small angle of incidence i) apparent depth (d ?)
is given by:
d ?=
relative
n
d
 ? n
relative
 = 
) refraction of medium of . I . R ( n
) incidence of medium of . I . R ( n
r
i
Apparent  shift =  d 
?
?
?
?
?
?
?
?
?
rel
n
1
1
Refraction   through   a Composite  Slab  (or  Refraction through  a
number  of  parallel  media, as  seen  from  a  medium  of  R. I.  n
0
)
Apparent  depth (distance of final image from final surface)
=  
t
n
rel
1
1
 + 
t
n
rel
2
2
 + 
t
n
rel
3
3
 
+......... + 
rel n
n
n
t
      
Apparent  shift =  t
1
 
?
?
?
?
?
?
?
?
?
rel 1
n
1
1 + 
 
t
2 
?
?
?
?
?
?
?
?
?
rel 2
n
1
1 +........+
?
?
?
?
?
?
?
?
?
rel n
n
n
1
8. Critical  Angle  and  Total  Internal  Reflection ( T. I. R.)
?C = sin
 ?1 
n
n
r
d
(i) Conditions  of   T. I. R.
(a) light is incident on the interface from denser medium.
(b) Angle of incidence should be greater than the critical
angle (i > c).
9. Refraction  Through  Prism
9.1 Characteristics of a prism
? = (i + e) ? (r
1
 + r
2
)  and  r
1
 + r
2
 = A
? ? ? ? ? ? = i + e ? A.
9.2 Variation of ? versus i
Page 5


           
             
 
       
Differentiating w.r.t time , we  get
v
(im)x
 = -v
(om)x 
; v
(im)y
 = v
(om)y 
; v
(im)z
 = v
(om)z ,
3. Spherical  Mirror
1
v
+
1
u
=
2
R
=
1
f
..... Mirror formula
x co–ordinate of centre of  Curvature and  focus of Concave
mirror are negative and those for Convex mirror are positive.
In case of mirrors since light rays reflect back in - X direction,
therefore -ve sign of v indicates real image and +ve
sign of v indicates virtual image
(b) Lateral magnification (or transverse magnification)
m= 
h
h
2
1
m = ?
v
u
.
(d) On differentiating (a) we get
dv
du
=
?
v
u
2
2
 .
(e) On differentiating (a) with respect to time we get
dv
dt
v
u
du
dt
? ?
2
2
,where 
dv
dt
 is the velocity of image along Principal
axis and
du
dt
 is the velocity of object along Principal axis. Negative
sign implies that the image , in case of mirror, always moves
in the direction opposite to that of object.This discussion is
for velocity with respect to mirror and along the x axis.
(f) Newton's  Formula: XY = f 
2
X  and Y are the distances ( along the principal axis ) of the object
and image respectively from the principal focus. This formula can
be used when the distances are mentioned or asked from the
focus.
(g) Optical power of a mirror (in Diopters) = 
f
1
f = focal length with sign and in meters.
(h) If object lying along the principal axis is not of very small size, the
longitudinal magnification =
1 2
1 2
u u
v v
?
?
     (it will always be inverted)
4. Refraction  of  Light
 vacuum. ? ? ?
speed of light in vacuum
speed of light in medium
c
v
.
4.1 Laws  of  Refraction (at  any  Refracting  Surface)
(b)
r Sin
i Sin
 = Constant  for any pair of media and for light of a given
wave length. This is known as Snell's Law. More precisely,
Sin i
Sin r
  = 
n
n
2
1
 = 
v
v
1
2
 = 
?
?
1
2
4.2 Deviation  of  a  Ray  Due  to  Refraction
Deviation ( ?) of ray incident at ? ?i and refracted at ? ?r is given by ? = |i
 
?
 
r|.
5. Principle of  Reversibility  of  Light  Rays
A ray travelling along the path of the reflected ray is reflected along the
path of the incident ray. A refracted  ray  reversed  to travel back along  its
path will get  refracted along the path of the incident  ray. Thus the incident
and refracted rays are mutually reversible.
7. Apparent  Depth and shift of  Submerged  Object
At near normal incidence (small angle of incidence i) apparent depth (d ?)
is given by:
d ?=
relative
n
d
 ? n
relative
 = 
) refraction of medium of . I . R ( n
) incidence of medium of . I . R ( n
r
i
Apparent  shift =  d 
?
?
?
?
?
?
?
?
?
rel
n
1
1
Refraction   through   a Composite  Slab  (or  Refraction through  a
number  of  parallel  media, as  seen  from  a  medium  of  R. I.  n
0
)
Apparent  depth (distance of final image from final surface)
=  
t
n
rel
1
1
 + 
t
n
rel
2
2
 + 
t
n
rel
3
3
 
+......... + 
rel n
n
n
t
      
Apparent  shift =  t
1
 
?
?
?
?
?
?
?
?
?
rel 1
n
1
1 + 
 
t
2 
?
?
?
?
?
?
?
?
?
rel 2
n
1
1 +........+
?
?
?
?
?
?
?
?
?
rel n
n
n
1
8. Critical  Angle  and  Total  Internal  Reflection ( T. I. R.)
?C = sin
 ?1 
n
n
r
d
(i) Conditions  of   T. I. R.
(a) light is incident on the interface from denser medium.
(b) Angle of incidence should be greater than the critical
angle (i > c).
9. Refraction  Through  Prism
9.1 Characteristics of a prism
? = (i + e) ? (r
1
 + r
2
)  and  r
1
 + r
2
 = A
? ? ? ? ? ? = i + e ? A.
9.2 Variation of ? versus i
(1) There is one and only one angle of incidence for which  the angle
of deviation is minimum.
(2) When ? = ?
min 
,  the angle of minimum deviation, then i = e 
 
and
r
1
 = r
2
, the ray passes symmetrically w.r.t. the refracting surfaces.
We can show by simple calculation that ?
min
  = 2i
min
 – A
where i
min
 = angle of incidence for minimum deviation and r = A/2.
?  n
rel
 =
? ?
? ?
2
A
2
A
sin
sin
m
? ?
,  where  n
rel
 = 
n
n
prism
surroundings
Also ? ? ? ? ? ?
min
 = (n ? 1) A (for small values of ? A)
(3) For a thin prism ( A ?10
o
) and for small value of i, all values of
? ? ? = ( n
rel
 ? 1 ) A where n
rel
 = 
g surroundin
prism
n
n
10. Dispersion  Of  Light
The  angular splitting of a ray of white light into a number of components
and spreading in different directions is called  Dispersion  of  Light. This
phenomenon is because waves of different wavelength move with same
speed in vacuum but with different speeds in a medium.
The refractive index of a medium depends slightly on wavelength also.
This variation of refractive index with wavelength is given by Cauchy’s
formula.
Cauchy's formula   n
 
( ?) =a
b
?
?
2
   where  a and b are positive constants
of a medium.
Angle between the rays of the extreme colours in the refracted (dispersed) light is
called angle of dispersion.
For prism of small ‘A’ and with small ‘i’ : ? = (n
v
 – n
r
)A
Deviation of beam(also called mean deviation) ? = ?
y
 = (n
y
 – 1)A
Dispersive power ( ?) of the medium of the material of prism is given by:
? = 
1 n
n n
y
r v
?
?
For small angled prism ( A ?10
o 
) with light incident at small angle i :
1 n
n n
y
r v
?
?
 =
y
r v
?
? ? ?
 = 
y
?
?
= 
angular dispersion
deviation of mean ray yellow ( )