NCERT Textbook: Ray Optics & Optical Instruments | Physics Class 12 - NEET PDF Download

 Page 1


Chapter Nine
RAY OPTICS
AND OPTICAL
INSTRUMENTS
9.1  INTRODUCTION
Nature has endowed the human eye (retina) with the sensitivity to detect
electromagnetic waves within a small range of the electromagnetic
spectrum. Electromagnetic radiation belonging to this region of the
spectrum (wavelength of about 400 nm to 750 nm) is called light. It is
mainly through light and the sense of vision that we know and interpret
the world around us.
There are two things that we can intuitively mention about light from
common experience. First, that it travels with enormous speed and second,
that it travels in a straight line. It took some time for people to realise that
the speed of light is finite and measurable. Its presently accepted value
in vacuum is c = 2.99792458 × 10
8
 m s
–1
.  For many purposes, it suffices
to take c = 3 × 10
8
 m s
–1
. The speed of light in vacuum is the highest
speed attainable in nature.
The intuitive notion that light travels in a straight line seems to
contradict what we have learnt in Chapter 8, that light is an
electromagnetic wave of wavelength belonging to the visible part of the
spectrum. How to reconcile the two facts? The answer is that the
wavelength of light is very small compared to the size of ordinary objects
that we encounter commonly (generally of the order of a few cm or larger).
In this situation, as you will learn in Chapter 10, a light wave can be
considered to travel from one point to another, along a straight line joining
2024-25
Page 2


Chapter Nine
RAY OPTICS
AND OPTICAL
INSTRUMENTS
9.1  INTRODUCTION
Nature has endowed the human eye (retina) with the sensitivity to detect
electromagnetic waves within a small range of the electromagnetic
spectrum. Electromagnetic radiation belonging to this region of the
spectrum (wavelength of about 400 nm to 750 nm) is called light. It is
mainly through light and the sense of vision that we know and interpret
the world around us.
There are two things that we can intuitively mention about light from
common experience. First, that it travels with enormous speed and second,
that it travels in a straight line. It took some time for people to realise that
the speed of light is finite and measurable. Its presently accepted value
in vacuum is c = 2.99792458 × 10
8
 m s
–1
.  For many purposes, it suffices
to take c = 3 × 10
8
 m s
–1
. The speed of light in vacuum is the highest
speed attainable in nature.
The intuitive notion that light travels in a straight line seems to
contradict what we have learnt in Chapter 8, that light is an
electromagnetic wave of wavelength belonging to the visible part of the
spectrum. How to reconcile the two facts? The answer is that the
wavelength of light is very small compared to the size of ordinary objects
that we encounter commonly (generally of the order of a few cm or larger).
In this situation, as you will learn in Chapter 10, a light wave can be
considered to travel from one point to another, along a straight line joining
2024-25
Physics
222
FIGURE 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
them. The path is called a ray of light, and a bundle of such rays
constitutes a beam of light.
In this chapter, we consider the phenomena of reflection, refraction
and dispersion of light, using the ray picture of light. Using the basic
laws of reflection and refraction, we shall study the image formation by
plane and spherical reflecting and refracting surfaces. We then go on to
describe the construction and working of some important optical
instruments, including the human eye.
9.2  REFLECTION OF LIGHT BY SPHERICAL MIRRORS
We are familiar with the laws of reflection. The
angle of reflection (i.e., the angle between reflected
ray and the normal to the reflecting surface or
the mirror) equals the angle of incidence (angle
between incident ray and the normal). Also that
the incident ray, reflected ray and the normal to
the reflecting surface at the point of incidence lie
in the same plane (Fig. 9.1). These laws are valid
at each point on any reflecting surface whether
plane or curved. However, we shall restrict our
discussion to the special case of curved surfaces,
that is, spherical surfaces. The normal in this case
is to be taken as normal to the tangent to surface
at the point of incidence. That is, the normal is
along the radius, the line joining the centre of curvature of the mirror to
the point of incidence.
We have already studied that the geometric centre of a spherical mirror
is called its pole while that of a spherical lens is called its optical centre.
The line joining the pole and the centre of curvature of the spherical
mirror is known as the principal axis. In the case of spherical lenses, the
principal axis is the line joining the optical centre with its principal focus
as you will see later.
9.2.1  Sign convention
To derive the relevant formulae for
reflection by spherical mirrors and
refraction by spherical lenses, we must
first adopt a sign convention for
measuring distances. In this book, we
shall follow the Cartesian sign
convention. According to this
convention, all distances are measured
from the pole of the mirror or the optical
centre of the lens. The distances
measured in the same direction as the
incident light are taken as positive and
those measured in the direction
opposite to the direction of incident light are taken as negative (Fig. 9.2).
The heights measured upwards with respect to x-axis and normal to the
2024-25
Page 3


Chapter Nine
RAY OPTICS
AND OPTICAL
INSTRUMENTS
9.1  INTRODUCTION
Nature has endowed the human eye (retina) with the sensitivity to detect
electromagnetic waves within a small range of the electromagnetic
spectrum. Electromagnetic radiation belonging to this region of the
spectrum (wavelength of about 400 nm to 750 nm) is called light. It is
mainly through light and the sense of vision that we know and interpret
the world around us.
There are two things that we can intuitively mention about light from
common experience. First, that it travels with enormous speed and second,
that it travels in a straight line. It took some time for people to realise that
the speed of light is finite and measurable. Its presently accepted value
in vacuum is c = 2.99792458 × 10
8
 m s
–1
.  For many purposes, it suffices
to take c = 3 × 10
8
 m s
–1
. The speed of light in vacuum is the highest
speed attainable in nature.
The intuitive notion that light travels in a straight line seems to
contradict what we have learnt in Chapter 8, that light is an
electromagnetic wave of wavelength belonging to the visible part of the
spectrum. How to reconcile the two facts? The answer is that the
wavelength of light is very small compared to the size of ordinary objects
that we encounter commonly (generally of the order of a few cm or larger).
In this situation, as you will learn in Chapter 10, a light wave can be
considered to travel from one point to another, along a straight line joining
2024-25
Physics
222
FIGURE 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
them. The path is called a ray of light, and a bundle of such rays
constitutes a beam of light.
In this chapter, we consider the phenomena of reflection, refraction
and dispersion of light, using the ray picture of light. Using the basic
laws of reflection and refraction, we shall study the image formation by
plane and spherical reflecting and refracting surfaces. We then go on to
describe the construction and working of some important optical
instruments, including the human eye.
9.2  REFLECTION OF LIGHT BY SPHERICAL MIRRORS
We are familiar with the laws of reflection. The
angle of reflection (i.e., the angle between reflected
ray and the normal to the reflecting surface or
the mirror) equals the angle of incidence (angle
between incident ray and the normal). Also that
the incident ray, reflected ray and the normal to
the reflecting surface at the point of incidence lie
in the same plane (Fig. 9.1). These laws are valid
at each point on any reflecting surface whether
plane or curved. However, we shall restrict our
discussion to the special case of curved surfaces,
that is, spherical surfaces. The normal in this case
is to be taken as normal to the tangent to surface
at the point of incidence. That is, the normal is
along the radius, the line joining the centre of curvature of the mirror to
the point of incidence.
We have already studied that the geometric centre of a spherical mirror
is called its pole while that of a spherical lens is called its optical centre.
The line joining the pole and the centre of curvature of the spherical
mirror is known as the principal axis. In the case of spherical lenses, the
principal axis is the line joining the optical centre with its principal focus
as you will see later.
9.2.1  Sign convention
To derive the relevant formulae for
reflection by spherical mirrors and
refraction by spherical lenses, we must
first adopt a sign convention for
measuring distances. In this book, we
shall follow the Cartesian sign
convention. According to this
convention, all distances are measured
from the pole of the mirror or the optical
centre of the lens. The distances
measured in the same direction as the
incident light are taken as positive and
those measured in the direction
opposite to the direction of incident light are taken as negative (Fig. 9.2).
The heights measured upwards with respect to x-axis and normal to the
2024-25
Ray Optics and
Optical Instruments
223
principal axis (x-axis) of the mirror/lens are taken as positive (Fig. 9.2).
The heights measured downwards are taken as negative.
With a common accepted convention, it turns out that a single formula
for spherical mirrors and a single formula for spherical lenses can handle
all different cases.
9.2.2  Focal length of spherical mirrors
Figure 9.3 shows what happens when a parallel beam of light is incident
on (a) a concave mirror, and (b) a convex mirror. We assume that the rays
are paraxial, i.e., they are incident at points close to the pole P of the mirror
and make small angles with the principal axis. The reflected rays converge
at a point F on the principal axis of a concave mirror [Fig. 9.3(a)].
For a convex mirror, the reflected rays appear to diverge from a point F
on its principal axis [Fig. 9.3(b)]. The point F is called the principal focus
of the mirror. If the parallel paraxial beam of light were incident, making
some angle with the principal axis, the reflected rays would converge (or
appear to diverge) from a point in a plane through F normal to the principal
axis. This is called the focal plane of the mirror [Fig. 9.3(c)].
FIGURE 9.3  Focus of a concave and convex mirror.
The distance between the focus F and the pole P of the mirror is called
the focal length of the mirror, denoted by f. We now show that f = R/2,
where R is the radius of curvature of the mirror. The geometry of reflection
of an incident ray is shown in Fig. 9.4.
Let C be the centre of curvature of the mirror. Consider a ray parallel
to the principal axis striking the mirror at M. Then CM will be
perpendicular to the mirror at M. Let q be the angle of incidence, and MD
2024-25
Page 4


Chapter Nine
RAY OPTICS
AND OPTICAL
INSTRUMENTS
9.1  INTRODUCTION
Nature has endowed the human eye (retina) with the sensitivity to detect
electromagnetic waves within a small range of the electromagnetic
spectrum. Electromagnetic radiation belonging to this region of the
spectrum (wavelength of about 400 nm to 750 nm) is called light. It is
mainly through light and the sense of vision that we know and interpret
the world around us.
There are two things that we can intuitively mention about light from
common experience. First, that it travels with enormous speed and second,
that it travels in a straight line. It took some time for people to realise that
the speed of light is finite and measurable. Its presently accepted value
in vacuum is c = 2.99792458 × 10
8
 m s
–1
.  For many purposes, it suffices
to take c = 3 × 10
8
 m s
–1
. The speed of light in vacuum is the highest
speed attainable in nature.
The intuitive notion that light travels in a straight line seems to
contradict what we have learnt in Chapter 8, that light is an
electromagnetic wave of wavelength belonging to the visible part of the
spectrum. How to reconcile the two facts? The answer is that the
wavelength of light is very small compared to the size of ordinary objects
that we encounter commonly (generally of the order of a few cm or larger).
In this situation, as you will learn in Chapter 10, a light wave can be
considered to travel from one point to another, along a straight line joining
2024-25
Physics
222
FIGURE 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
them. The path is called a ray of light, and a bundle of such rays
constitutes a beam of light.
In this chapter, we consider the phenomena of reflection, refraction
and dispersion of light, using the ray picture of light. Using the basic
laws of reflection and refraction, we shall study the image formation by
plane and spherical reflecting and refracting surfaces. We then go on to
describe the construction and working of some important optical
instruments, including the human eye.
9.2  REFLECTION OF LIGHT BY SPHERICAL MIRRORS
We are familiar with the laws of reflection. The
angle of reflection (i.e., the angle between reflected
ray and the normal to the reflecting surface or
the mirror) equals the angle of incidence (angle
between incident ray and the normal). Also that
the incident ray, reflected ray and the normal to
the reflecting surface at the point of incidence lie
in the same plane (Fig. 9.1). These laws are valid
at each point on any reflecting surface whether
plane or curved. However, we shall restrict our
discussion to the special case of curved surfaces,
that is, spherical surfaces. The normal in this case
is to be taken as normal to the tangent to surface
at the point of incidence. That is, the normal is
along the radius, the line joining the centre of curvature of the mirror to
the point of incidence.
We have already studied that the geometric centre of a spherical mirror
is called its pole while that of a spherical lens is called its optical centre.
The line joining the pole and the centre of curvature of the spherical
mirror is known as the principal axis. In the case of spherical lenses, the
principal axis is the line joining the optical centre with its principal focus
as you will see later.
9.2.1  Sign convention
To derive the relevant formulae for
reflection by spherical mirrors and
refraction by spherical lenses, we must
first adopt a sign convention for
measuring distances. In this book, we
shall follow the Cartesian sign
convention. According to this
convention, all distances are measured
from the pole of the mirror or the optical
centre of the lens. The distances
measured in the same direction as the
incident light are taken as positive and
those measured in the direction
opposite to the direction of incident light are taken as negative (Fig. 9.2).
The heights measured upwards with respect to x-axis and normal to the
2024-25
Ray Optics and
Optical Instruments
223
principal axis (x-axis) of the mirror/lens are taken as positive (Fig. 9.2).
The heights measured downwards are taken as negative.
With a common accepted convention, it turns out that a single formula
for spherical mirrors and a single formula for spherical lenses can handle
all different cases.
9.2.2  Focal length of spherical mirrors
Figure 9.3 shows what happens when a parallel beam of light is incident
on (a) a concave mirror, and (b) a convex mirror. We assume that the rays
are paraxial, i.e., they are incident at points close to the pole P of the mirror
and make small angles with the principal axis. The reflected rays converge
at a point F on the principal axis of a concave mirror [Fig. 9.3(a)].
For a convex mirror, the reflected rays appear to diverge from a point F
on its principal axis [Fig. 9.3(b)]. The point F is called the principal focus
of the mirror. If the parallel paraxial beam of light were incident, making
some angle with the principal axis, the reflected rays would converge (or
appear to diverge) from a point in a plane through F normal to the principal
axis. This is called the focal plane of the mirror [Fig. 9.3(c)].
FIGURE 9.3  Focus of a concave and convex mirror.
The distance between the focus F and the pole P of the mirror is called
the focal length of the mirror, denoted by f. We now show that f = R/2,
where R is the radius of curvature of the mirror. The geometry of reflection
of an incident ray is shown in Fig. 9.4.
Let C be the centre of curvature of the mirror. Consider a ray parallel
to the principal axis striking the mirror at M. Then CM will be
perpendicular to the mirror at M. Let q be the angle of incidence, and MD
2024-25
Physics
224
be the perpendicular from M on the principal axis. Then,
ÐMCP = q and ÐMFP = 2q
Now,
tanq =
MD
CD
 and tan 2q = 
MD
FD
(9.1)
For small q, which is true for paraxial rays, tanq  » q,
tan 2q » 2q.  Therefore, Eq. (9.1) gives
MD
FD
 = 2 
MD
CD
or, FD = 
CD
2
(9.2)
Now, for small q, the point D is very close to the point P.
Therefore, FD = f and CD = R. Equation (9.2) then gives
f = R/2 (9.3)
9.2.3  The mirror equation
If rays emanating from a point actually meet at another point
after reflection and/or refraction, that point is called the image
of the first point. The image is real if the rays actually converge
to the point; it is virtual if the rays do not actually meet but
appear to diverge from the point when produced
backwards. An image is thus a point-to-point
correspondence with the object established through
reflection and/or refraction.
In principle, we can take any two rays emanating
from a point on an object, trace their paths, find their
point of intersection and thus, obtain the image of
the point due to reflection at a spherical mirror. In
practice, however, it is convenient to choose any two
of the following rays:
(i) The ray from the point which is parallel to the
principal axis. The reflected ray goes through
the focus of the mirror.
(ii) The ray passing through the centre of
curvature of a concave mirror or appearing to
pass through it for a convex mirror. The
reflected ray simply retraces the path.
(iii) The ray passing through (or directed towards) the focus of the concave
mirror or appearing to pass through (or directed towards) the focus
of a convex mirror. The reflected ray is parallel to the principal axis.
(iv) The ray incident at any angle at the pole. The reflected ray follows
laws of reflection.
Figure 9.5 shows the ray diagram considering three rays. It shows
the image A¢B¢ (in this case, real) of an object AB formed by a concave
mirror. It does not mean that only three rays emanate from the point A.
An infinite number of rays emanate from any source, in all directions.
Thus, point A¢ is image point of A if every ray originating at point A and
falling on the concave mirror after reflection passes through the point A¢.
FIGURE 9.4 Geometry of
reflection of an incident ray on
(a) concave spherical mirror,
and (b) convex spherical mirror.
FIGURE 9.5 Ray diagram for image
formation by a concave mirror.
2024-25
Page 5


Chapter Nine
RAY OPTICS
AND OPTICAL
INSTRUMENTS
9.1  INTRODUCTION
Nature has endowed the human eye (retina) with the sensitivity to detect
electromagnetic waves within a small range of the electromagnetic
spectrum. Electromagnetic radiation belonging to this region of the
spectrum (wavelength of about 400 nm to 750 nm) is called light. It is
mainly through light and the sense of vision that we know and interpret
the world around us.
There are two things that we can intuitively mention about light from
common experience. First, that it travels with enormous speed and second,
that it travels in a straight line. It took some time for people to realise that
the speed of light is finite and measurable. Its presently accepted value
in vacuum is c = 2.99792458 × 10
8
 m s
–1
.  For many purposes, it suffices
to take c = 3 × 10
8
 m s
–1
. The speed of light in vacuum is the highest
speed attainable in nature.
The intuitive notion that light travels in a straight line seems to
contradict what we have learnt in Chapter 8, that light is an
electromagnetic wave of wavelength belonging to the visible part of the
spectrum. How to reconcile the two facts? The answer is that the
wavelength of light is very small compared to the size of ordinary objects
that we encounter commonly (generally of the order of a few cm or larger).
In this situation, as you will learn in Chapter 10, a light wave can be
considered to travel from one point to another, along a straight line joining
2024-25
Physics
222
FIGURE 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
them. The path is called a ray of light, and a bundle of such rays
constitutes a beam of light.
In this chapter, we consider the phenomena of reflection, refraction
and dispersion of light, using the ray picture of light. Using the basic
laws of reflection and refraction, we shall study the image formation by
plane and spherical reflecting and refracting surfaces. We then go on to
describe the construction and working of some important optical
instruments, including the human eye.
9.2  REFLECTION OF LIGHT BY SPHERICAL MIRRORS
We are familiar with the laws of reflection. The
angle of reflection (i.e., the angle between reflected
ray and the normal to the reflecting surface or
the mirror) equals the angle of incidence (angle
between incident ray and the normal). Also that
the incident ray, reflected ray and the normal to
the reflecting surface at the point of incidence lie
in the same plane (Fig. 9.1). These laws are valid
at each point on any reflecting surface whether
plane or curved. However, we shall restrict our
discussion to the special case of curved surfaces,
that is, spherical surfaces. The normal in this case
is to be taken as normal to the tangent to surface
at the point of incidence. That is, the normal is
along the radius, the line joining the centre of curvature of the mirror to
the point of incidence.
We have already studied that the geometric centre of a spherical mirror
is called its pole while that of a spherical lens is called its optical centre.
The line joining the pole and the centre of curvature of the spherical
mirror is known as the principal axis. In the case of spherical lenses, the
principal axis is the line joining the optical centre with its principal focus
as you will see later.
9.2.1  Sign convention
To derive the relevant formulae for
reflection by spherical mirrors and
refraction by spherical lenses, we must
first adopt a sign convention for
measuring distances. In this book, we
shall follow the Cartesian sign
convention. According to this
convention, all distances are measured
from the pole of the mirror or the optical
centre of the lens. The distances
measured in the same direction as the
incident light are taken as positive and
those measured in the direction
opposite to the direction of incident light are taken as negative (Fig. 9.2).
The heights measured upwards with respect to x-axis and normal to the
2024-25
Ray Optics and
Optical Instruments
223
principal axis (x-axis) of the mirror/lens are taken as positive (Fig. 9.2).
The heights measured downwards are taken as negative.
With a common accepted convention, it turns out that a single formula
for spherical mirrors and a single formula for spherical lenses can handle
all different cases.
9.2.2  Focal length of spherical mirrors
Figure 9.3 shows what happens when a parallel beam of light is incident
on (a) a concave mirror, and (b) a convex mirror. We assume that the rays
are paraxial, i.e., they are incident at points close to the pole P of the mirror
and make small angles with the principal axis. The reflected rays converge
at a point F on the principal axis of a concave mirror [Fig. 9.3(a)].
For a convex mirror, the reflected rays appear to diverge from a point F
on its principal axis [Fig. 9.3(b)]. The point F is called the principal focus
of the mirror. If the parallel paraxial beam of light were incident, making
some angle with the principal axis, the reflected rays would converge (or
appear to diverge) from a point in a plane through F normal to the principal
axis. This is called the focal plane of the mirror [Fig. 9.3(c)].
FIGURE 9.3  Focus of a concave and convex mirror.
The distance between the focus F and the pole P of the mirror is called
the focal length of the mirror, denoted by f. We now show that f = R/2,
where R is the radius of curvature of the mirror. The geometry of reflection
of an incident ray is shown in Fig. 9.4.
Let C be the centre of curvature of the mirror. Consider a ray parallel
to the principal axis striking the mirror at M. Then CM will be
perpendicular to the mirror at M. Let q be the angle of incidence, and MD
2024-25
Physics
224
be the perpendicular from M on the principal axis. Then,
ÐMCP = q and ÐMFP = 2q
Now,
tanq =
MD
CD
 and tan 2q = 
MD
FD
(9.1)
For small q, which is true for paraxial rays, tanq  » q,
tan 2q » 2q.  Therefore, Eq. (9.1) gives
MD
FD
 = 2 
MD
CD
or, FD = 
CD
2
(9.2)
Now, for small q, the point D is very close to the point P.
Therefore, FD = f and CD = R. Equation (9.2) then gives
f = R/2 (9.3)
9.2.3  The mirror equation
If rays emanating from a point actually meet at another point
after reflection and/or refraction, that point is called the image
of the first point. The image is real if the rays actually converge
to the point; it is virtual if the rays do not actually meet but
appear to diverge from the point when produced
backwards. An image is thus a point-to-point
correspondence with the object established through
reflection and/or refraction.
In principle, we can take any two rays emanating
from a point on an object, trace their paths, find their
point of intersection and thus, obtain the image of
the point due to reflection at a spherical mirror. In
practice, however, it is convenient to choose any two
of the following rays:
(i) The ray from the point which is parallel to the
principal axis. The reflected ray goes through
the focus of the mirror.
(ii) The ray passing through the centre of
curvature of a concave mirror or appearing to
pass through it for a convex mirror. The
reflected ray simply retraces the path.
(iii) The ray passing through (or directed towards) the focus of the concave
mirror or appearing to pass through (or directed towards) the focus
of a convex mirror. The reflected ray is parallel to the principal axis.
(iv) The ray incident at any angle at the pole. The reflected ray follows
laws of reflection.
Figure 9.5 shows the ray diagram considering three rays. It shows
the image A¢B¢ (in this case, real) of an object AB formed by a concave
mirror. It does not mean that only three rays emanate from the point A.
An infinite number of rays emanate from any source, in all directions.
Thus, point A¢ is image point of A if every ray originating at point A and
falling on the concave mirror after reflection passes through the point A¢.
FIGURE 9.4 Geometry of
reflection of an incident ray on
(a) concave spherical mirror,
and (b) convex spherical mirror.
FIGURE 9.5 Ray diagram for image
formation by a concave mirror.
2024-25
Ray Optics and
Optical Instruments
225
We now derive the mirror equation or the relation between the object
distance (u), image distance (v) and the focal length ( f ).
From Fig. 9.5, the two right-angled triangles A¢B¢F and MPF are
similar. (For paraxial rays, MP can be considered to be a straight line
perpendicular to CP.) Therefore,
B A B F
PM FP
' ' '
=
or 
B A B F
BA FP
' ' '
=
 (
?
PM = AB) (9.4)
Since Ð APB = Ð A¢PB¢, the right angled triangles A¢B¢P and ABP are
also similar. Therefore,
B A B P
B A BP
' ' '
=
(9.5)
Comparing Eqs. (9.4) and (9.5), we get
B P – FP B F B P
FP FP BP
' ' '
= =
(9.6)
Equation (9.6) is a relation involving magnitude of distances. We now
apply the sign convention. We note that light travels from the object to
the mirror MPN. Hence this is taken as the positive direction. To reach
the object AB, image A¢B¢ as well as the focus F from the pole P, we have
to travel opposite to the direction of incident light. Hence, all the three
will have negative signs. Thus,
B¢ P = –v, FP = –f, BP = –u
Using these in Eq. (9.6), we get
– –
–
v f v
f u
+
=
–
or
– v f v
f u
=
      
v
f
v
u
= + 1
Dividing it by v, we get
      
1 1 1
v u f
+ =
(9.7)
This relation is known as the mirror equation.
The size of the image relative to the size of the object is another
important quantity to consider. We define linear magnification (m) as the
ratio of the height of the image (h¢) to the height of the object (h):
m = 
h
h
'
(9.8)
h and h¢ will be taken positive or negative in accordance with the accepted
sign convention. In triangles A¢B¢P and ABP, we have,
B A B P
BA BP
' ' '
=
With the sign convention, this becomes
2024-25