NCERT Textbook - Ray Optics and Optical Instruments Class 12 Notes | EduRev

Physics Class 12

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Class 12 : NCERT Textbook - Ray Optics and Optical Instruments Class 12 Notes | EduRev

 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
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
Physics
310
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.
PARTICLE MODEL OF LIGHT
Newton’s fundamental contributions to mathematics, mechanics, and gravitation often blind
us to his deep experimental and theoretical study of light. He made pioneering contributions
in the field of optics. He further developed the corpuscular model of light proposed by
Descartes. It presumes that light energy is concentrated in tiny particles called corpuscles.
He further assumed that corpuscles of light were massless elastic particles. With his
understanding of mechanics, he could come up with a simple model of reflection and
refraction. It is a common observation that a ball bouncing from a smooth plane surface
obeys the laws of reflection. When this is an elastic collision, the magnitude of the velocity
remains the same. As the surface is smooth, there is no force acting parallel to the surface,
so the component of momentum in this direction also remains the same. Only the component
perpendicular to the surface, i.e., the normal component of the momentum, gets reversed
in reflection. Newton argued that smooth surfaces like mirrors reflect the corpuscles in a
similar manner.
In order to explain the phenomena of refraction, Newton postulated that the speed of
the corpuscles was greater in water or glass than in air. However, later on it was discovered
that the speed of light is less in water or glass than in air.
In the field of optics, Newton – the experimenter, was greater than Newton – the theorist.
He himself observed many phenomena, which were difficult to understand in terms of
particle nature of light. For example, the colours observed due to a thin film of oil on water.
Property of partial reflection of light is yet another such example. Everyone who has looked
into the water in a pond sees image of the face in it, but also sees the bottom of the pond.
Newton argued that some of the corpuscles, which fall on the water, get reflected and some
get transmitted. But what property could distinguish these two kinds of corpuscles? Newton
had to postulate some kind of unpredictable, chance phenomenon, which decided whether
an individual corpuscle would be reflected or not. In explaining other phenomena, however,
the corpuscles were presumed to behave as if they are identical. Such a dilemma does not
occur in the wave picture of light. An incoming wave can be divided into two weaker waves
at the boundary between air and water.
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
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
Physics
310
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.
PARTICLE MODEL OF LIGHT
Newton’s fundamental contributions to mathematics, mechanics, and gravitation often blind
us to his deep experimental and theoretical study of light. He made pioneering contributions
in the field of optics. He further developed the corpuscular model of light proposed by
Descartes. It presumes that light energy is concentrated in tiny particles called corpuscles.
He further assumed that corpuscles of light were massless elastic particles. With his
understanding of mechanics, he could come up with a simple model of reflection and
refraction. It is a common observation that a ball bouncing from a smooth plane surface
obeys the laws of reflection. When this is an elastic collision, the magnitude of the velocity
remains the same. As the surface is smooth, there is no force acting parallel to the surface,
so the component of momentum in this direction also remains the same. Only the component
perpendicular to the surface, i.e., the normal component of the momentum, gets reversed
in reflection. Newton argued that smooth surfaces like mirrors reflect the corpuscles in a
similar manner.
In order to explain the phenomena of refraction, Newton postulated that the speed of
the corpuscles was greater in water or glass than in air. However, later on it was discovered
that the speed of light is less in water or glass than in air.
In the field of optics, Newton – the experimenter, was greater than Newton – the theorist.
He himself observed many phenomena, which were difficult to understand in terms of
particle nature of light. For example, the colours observed due to a thin film of oil on water.
Property of partial reflection of light is yet another such example. Everyone who has looked
into the water in a pond sees image of the face in it, but also sees the bottom of the pond.
Newton argued that some of the corpuscles, which fall on the water, get reflected and some
get transmitted. But what property could distinguish these two kinds of corpuscles? Newton
had to postulate some kind of unpredictable, chance phenomenon, which decided whether
an individual corpuscle would be reflected or not. In explaining other phenomena, however,
the corpuscles were presumed to behave as if they are identical. Such a dilemma does not
occur in the wave picture of light. An incoming wave can be divided into two weaker waves
at the boundary between air and water.
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
Ray Optics and
Optical Instruments
311
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
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 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
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
Physics
310
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.
PARTICLE MODEL OF LIGHT
Newton’s fundamental contributions to mathematics, mechanics, and gravitation often blind
us to his deep experimental and theoretical study of light. He made pioneering contributions
in the field of optics. He further developed the corpuscular model of light proposed by
Descartes. It presumes that light energy is concentrated in tiny particles called corpuscles.
He further assumed that corpuscles of light were massless elastic particles. With his
understanding of mechanics, he could come up with a simple model of reflection and
refraction. It is a common observation that a ball bouncing from a smooth plane surface
obeys the laws of reflection. When this is an elastic collision, the magnitude of the velocity
remains the same. As the surface is smooth, there is no force acting parallel to the surface,
so the component of momentum in this direction also remains the same. Only the component
perpendicular to the surface, i.e., the normal component of the momentum, gets reversed
in reflection. Newton argued that smooth surfaces like mirrors reflect the corpuscles in a
similar manner.
In order to explain the phenomena of refraction, Newton postulated that the speed of
the corpuscles was greater in water or glass than in air. However, later on it was discovered
that the speed of light is less in water or glass than in air.
In the field of optics, Newton – the experimenter, was greater than Newton – the theorist.
He himself observed many phenomena, which were difficult to understand in terms of
particle nature of light. For example, the colours observed due to a thin film of oil on water.
Property of partial reflection of light is yet another such example. Everyone who has looked
into the water in a pond sees image of the face in it, but also sees the bottom of the pond.
Newton argued that some of the corpuscles, which fall on the water, get reflected and some
get transmitted. But what property could distinguish these two kinds of corpuscles? Newton
had to postulate some kind of unpredictable, chance phenomenon, which decided whether
an individual corpuscle would be reflected or not. In explaining other phenomena, however,
the corpuscles were presumed to behave as if they are identical. Such a dilemma does not
occur in the wave picture of light. An incoming wave can be divided into two weaker waves
at the boundary between air and water.
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
Ray Optics and
Optical Instruments
311
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
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 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
Physics
312
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 ? be the angle
of incidence, and MD be the perpendicular from M on the
principal axis. Then,
?MCP = ? and ?MFP = 2?
Now,
tan? =
MD
CD
 and tan 2? = 
MD
FD
(9.1)
For small ?, which is true for paraxial rays, tan?  ˜ ?,
tan 2? ˜ 2?.  Therefore, Eq. (9.1) gives
MD
FD
 = 2 
MD
CD
or, FD = 
CD
2
(9.2)
Now, for small ?, 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
FIGURE 9.3  Focus of a concave and convex mirror.
FIGURE 9.4 Geometry of
reflection of an incident ray on
(a) concave spherical mirror,
and (b) convex spherical mirror.
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
Physics
310
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.
PARTICLE MODEL OF LIGHT
Newton’s fundamental contributions to mathematics, mechanics, and gravitation often blind
us to his deep experimental and theoretical study of light. He made pioneering contributions
in the field of optics. He further developed the corpuscular model of light proposed by
Descartes. It presumes that light energy is concentrated in tiny particles called corpuscles.
He further assumed that corpuscles of light were massless elastic particles. With his
understanding of mechanics, he could come up with a simple model of reflection and
refraction. It is a common observation that a ball bouncing from a smooth plane surface
obeys the laws of reflection. When this is an elastic collision, the magnitude of the velocity
remains the same. As the surface is smooth, there is no force acting parallel to the surface,
so the component of momentum in this direction also remains the same. Only the component
perpendicular to the surface, i.e., the normal component of the momentum, gets reversed
in reflection. Newton argued that smooth surfaces like mirrors reflect the corpuscles in a
similar manner.
In order to explain the phenomena of refraction, Newton postulated that the speed of
the corpuscles was greater in water or glass than in air. However, later on it was discovered
that the speed of light is less in water or glass than in air.
In the field of optics, Newton – the experimenter, was greater than Newton – the theorist.
He himself observed many phenomena, which were difficult to understand in terms of
particle nature of light. For example, the colours observed due to a thin film of oil on water.
Property of partial reflection of light is yet another such example. Everyone who has looked
into the water in a pond sees image of the face in it, but also sees the bottom of the pond.
Newton argued that some of the corpuscles, which fall on the water, get reflected and some
get transmitted. But what property could distinguish these two kinds of corpuscles? Newton
had to postulate some kind of unpredictable, chance phenomenon, which decided whether
an individual corpuscle would be reflected or not. In explaining other phenomena, however,
the corpuscles were presumed to behave as if they are identical. Such a dilemma does not
occur in the wave picture of light. An incoming wave can be divided into two weaker waves
at the boundary between air and water.
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
Ray Optics and
Optical Instruments
311
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
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 9.1 The incident ray, reflected ray
and the normal to the reflecting surface lie
in the same plane.
FIGURE 9.2  The Cartesian Sign Convention.
Physics
312
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 ? be the angle
of incidence, and MD be the perpendicular from M on the
principal axis. Then,
?MCP = ? and ?MFP = 2?
Now,
tan? =
MD
CD
 and tan 2? = 
MD
FD
(9.1)
For small ?, which is true for paraxial rays, tan?  ˜ ?,
tan 2? ˜ 2?.  Therefore, Eq. (9.1) gives
MD
FD
 = 2 
MD
CD
or, FD = 
CD
2
(9.2)
Now, for small ?, 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
FIGURE 9.3  Focus of a concave and convex mirror.
FIGURE 9.4 Geometry of
reflection of an incident ray on
(a) concave spherical mirror,
and (b) convex spherical mirror.
Ray Optics and
Optical Instruments
313
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'.
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,
BA BF
PM FP
   
 
or 
BA BF
BA FP
   
 
 ( ? PM = AB) (9.4)
Since ? APB = ? A'PB', the right angled triangles A'B'P and ABP are
also similar. Therefore,
BA B P
BA BP
   
 
(9.5)
Comparing Eqs. (9.4) and (9.5), we get
BP–FP BF 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
FIGURE 9.5 Ray diagram for image
formation by a concave mirror.
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