Download Jacaranda Physics 1 2E Chapter 1

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AREA OF STUDY 1
Reflecting light
Chapter 2
Refracting light
Chapter 3
Seeing colours
1
Wave-like
properties of
light
UNIT
Chapter 1
Jac Phys 1 2E - 01 Page 2 Tuesday, October 21, 2003 1:41 PM
Chapter
Reflecting
light
1
Remember
Before beginning this chapter, you should be able to:
• use ray diagrams to show how light is reflected from
smooth surfaces.
Key
ideas
After completing this chapter, you should be able to:
• use the speed of light to calculate the time light
takes to travel a stated distance
• appreciate that light travels in straight lines and be
able to give evidence for this
• use the ray model to explain the reflection of light
and the formation of images in plane and curved
mirrors
• describe images fully, referring to their location,
size, orientation and nature
• describe various applications of plane and curved
mirrors.
Figure 1.1
S
traight rays of sunlight coming through
the forest are seen because the light is
scattered by airborne dust particles and
light fog into our eyes.
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Sight is the sense by which humans and most other mammals get most of
their information about the world. This sense responds to light. Questions about light naturally arise. Where does light come from? What can
it do? How can its properties be explained?
LIGHT AND ITS PROPERTIES
Some obvious observations of light are:
• Sources of light are needed to see.
• Light travels very fast.
• Light produces shadows.
Sources of light
Objects that give off their own
light are described as luminous.
Luminous objects that produce
light as a result of being hot are
described as incandescent.
When we experience darkness at night or in an enclosed room, we know
that a source of light, such as the Sun or a lamp, is needed to light up the
darkness. Once a lamp is turned on, we can see features in the room
because the light from the lamp shines on them and is then reflected
into our eyes.
This means that objects can be classified into two groups. Objects seen
because they give off their own light are called luminous objects; those
seen because they reflect light are called non-luminous objects. The Sun,
torches and candles are luminous objects. Tables, chairs, cats and dogs
are non-luminous objects.
Some luminous objects produce light because they are hot. The Sun is
one example. The higher the temperature, the brighter the light, and
the colour also changes. These objects are called incandescent.
Figure 1.2
The Pleiades open star cluster in the constellation Taurus.
All stars are incandescent sources of light.
CHAPTER 1 REFLECTING LIGHT
3
Inve
Jac Phys 1 2E - 01 Page 4 Tuesday, October 21, 2003 1:41 PM
ations
stig
Inve
Investigation 1.1
Luminous or not?
ations
stig
Investigation 1.2
Luminosity and
temperature
Other objects are cold and produce light in another way. It involves
changes in the energy of electrons in the material brought about by
either chemical or electrical processes.
Speed of light
The gap we experience between seeing lightning and hearing thunder
shows that sound travels relatively slowly. Light seems to travel so fast that
to our experience its speed seems infinite; that is, we seem to observe
events at the instant they happen.
Galileo Galilei (1564–1642) was not convinced of this. He attempted to
determine the speed of light by measuring the time delay between the
flash of his lamp to an assistant on a distant mountain and the return
flash from his assistant’s lamp (see figure 1.3). No detectable delay was
observed and Galileo concluded that the speed of light was very high. A
longer distance was needed.
Figure 1.3
Web
Galileo used this
method to measure the speed of light. He
attempted to time, with his pulse, the delay
between uncovering his lantern and seeing
the light from his partner’s lantern, which
his partner uncovered at the moment when
he saw the light from Galileo’s lantern.
s
link
Speed of light
Skill
cks
che
Significant figures
(p. 493)
4
Olaus Roemer was a Danish astronomer born two years after
Galileo’s death. He observed that the time between eclipses of Jupiter’s
moons by Jupiter decreased as the Earth moved closer to Jupiter and
increased as the Earth moved away. Roemer reasoned that this was
because the distance the light travelled from Jupiter to Earth became
greater as the Earth’s orbit took it further from Jupiter (see figure 1.4).
Roemer used this time and the known diameter of the Earth’s orbit
about the Sun to estimate the speed of light. The value he obtained
was 2.7 × 108 m s−1.
Eventually, in the nineteenth century, with stronger light sources and
more precise timing devices, Galileo’s method could be used, but the
assistant was replaced by a mirror. The values obtained then were about
3.0 × 108 m s−1.
Early in the twentieth century, the American scientist Albert A. Michelson
(1852–1931) used a rapidly rotating eight-sided mirror (see figure 1.5). The
light was reflected to a distant mirror about 35 kilometres away then
WAVE-LIKE PROPERTIES OF LIGHT
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Skill
cks
che
SI units
(p. 490)
reflected back to the rotating mirror. For some particular rotation rates, this
light is reflected by one of the sides of the rotating mirror directly to the
observer. The rotation rate can be used to calculate the speed of light.
The value Michelson obtained was 2.997 96 × 108 m s−1. He actually
measured the distance of 35 km to an accuracy of 2.5 cm. The speed of
light is currently measured at 2.997 924 58 × 108 m s−1. It is rounded off
to 300 000 km s−1 for calculation purposes.
observer
B
fixed
mirror
Io (moon)
rotating mirror
with eight sides
A
Sun
Jupiter
Earth’s orbit
source of light
Figure 1.4
The time,
as seen from the Earth, for
Jupiter’s moon, Io, to orbit Jupiter
increases as the Earth moves
from A to B. (The diagram is not
to scale.)
SAMPLE
PROBLEM
1.1
Solution:
Figure 1.5 Light from the source reflects off one of the sides of the
rotating mirror towards a mirror 35 kilometres away. The returning beam hits
the rotating mirror. If one of the sides of the mirror is in the right position, the
light enters the eyepiece and can be seen by the observer. By measuring the
speed of rotation when the beam enters the eyepiece, the speed of light can be
calculated.
How long does light take to travel from the Sun to the Earth?
speed of light = 3.00 × 108 m s−1
distance from Sun to Earth = 1.49 × 1011 m
distance travelled
speed = --------------------------------------------time taken
distance travelled
⇒ time taken = --------------------------------------------speed
11
1.49 × 10 m
time = ---------------------------------------8
–1
3.00 × 10 m s
= 0.497 × 103 s
= 497 s
= 8 minutes 17 seconds.
Skill
cks
che
Scientific notation
(p. 492)
SAMPLE
PROBLEM
1.2
Solution:
35 km
How far does light travel in one year (one light-year)?
distance travelled = speed × time taken
distance = 3.00 × 108 m s−1 × (365.25 × 24 × 60 × 60) s
= 9.47 × 1015 m
= 9.47 × 1012 km.
Shadows
The bright Sun produces sharp shadows on the ground. The shape of the
shadow is the same shape as the object blocking the light (see figure 1.6).
This could happen only if light travels in a straight line.
CHAPTER 1 REFLECTING LIGHT
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Figure 1.6 The
straight rays passing the edge of
a skateboarder leave a sharp
shadow on the wall.
Ray model
A ray of light is a very narrow
pencil-like beam of light.
The need for sources of light, the great speed of light and the existence
of sharp shadows can be described by a ray model. The model assumes
that light travels in a straight line path called a light ray. A light ray can
be considered as an infinitely narrow beam of light and can be
represented as a straight line (see figure 1.7).
Figure 1.7 Light rays leave
a point on this pencil and travel in
straight lines in all directions. The pencil
is seen because of the ‘bundle’ of rays
that enter the eye.
PLANE MIRROR REFLECTION
When you look at yourself in a plane mirror, some of the light rays from
your nose, for example, travel in the direction of the mirror and reflect
off in the direction of your eye (see figure 1.8). What is happening at the
surface of the mirror to produce such a perfect image?
mirror
Figure 1.8 Light rays from the
tip of the nose reflect off the mirror and
enter the eye.
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WAVE-LIKE PROPERTIES OF LIGHT
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The angle of incidence is the
angle between the incident ray
and the normal.
The angle of reflection is the
angle between the reflected ray
and the normal.
To investigate the reflection of light, the angles made by the rays need
to be measured. Measurements of these angles show that, like a ball
bouncing off a flat wall, the angle of incidence equals the angle of
reflection (see figure 1.9).
normal
incident
ray
reflected
ray
angle of
angle of
incidence reflection
mirror
Figure 1.9
The ray approaching the mirror is called the incident ray.
The ray leaving the mirror is called the reflected ray. The normal is a line at right
angles to the mirror. The angles are measured between each ray and the normal.
When the path of a light ray is traced, it is found that the angle of incidence
always equals the angle of reflection.
The normal is a line that is
perpendicular to a surface or a
boundary between two surfaces.
The other seemingly trivial conclusion that can be drawn from the
investigation is that the incident ray, the normal and the reflected ray all
lie in the same plane (see figure 1.10).
normal
reflected ray
Figure 1.10
The incident
ray, the ‘normal’ to the surface of the
mirror and the reflected ray all lie in
the same plane, which is at right
angles to the plane of the mirror.
mirror
incident ray
REGULAR AND DIFFUSE
REFLECTION
Regular reflection, also
referred to as specular
reflection, is reflection from a
smooth surface.
Diffuse reflection is reflection
from a rough or irregular
surface.
Reflection from a smooth surface is called regular or specular reflection.
But what happens with an ordinary surface, such as this page? A page is
not smooth like a mirror. At the microscopic level, there are ‘hills and
valleys’. As the light rays come down into these hills and valleys, they still
reflect with the two angles the same but, because the surface is irregular,
the reflected rays emerge in all directions (see figure 1.11). This is called
diffuse reflection. Light rays from diffuse reflections — from the ground,
trees and other objects — enter the eye and enable the brain to make
sense of the world.
observer A
Figure 1.11
This is diffuse
reflection. Each of the incoming parallel
rays meets the irregular surface at a
different angle of incidence. The reflected
rays will therefore go off in different
directions, enabling observers in all
directions to receive light from the surface;
in other words, to see the surface.
observer B
irregular surface
CHAPTER 1 REFLECTING LIGHT
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FORMING IMAGES WITH A
PLANE MIRROR
Images are likenesses of
objects.
Web
A virtual image is seen because
light appears to be coming from
it. Light doesn’t actually pass
through it. Therefore a virtual
image cannot be ‘captured’ on a
screen.
s
link
The law of reflection can now be used to explain how images are formed
in plane mirrors and how to describe images. To describe an image, four
aspects need to be determined:
1. its location — where it is in relation to the mirror
2. its size — how big it is compared with the object
3. its orientation — whether it is upright or upside down
4. its nature — whether or not the image can be captured on a screen.
Figure 1.12 shows how the image that we see of a cat is formed. All the
rays leaving the cat’s ear that reach the mirror are reflected so that they
appear to be coming from the same region behind the screen. Some of
these rays enter your eye. Someone else standing beside you will receive
different rays even though the rays will appear to be coming from the
same region. The region from which the rays appear to be coming is the
location of the image of the cat’s ear.
The image is located as far behind the mirror as the object is in front
of the mirror. It is the same size and upright. The light rays entering the
eye only appear to come from the location of the image. The image is
described as a virtual image. A virtual image cannot be ‘captured’ on a
screen.
mirror
A1
A
B1
B
Image in a plane
mirror applets
Figure 1.12 When the rays of
light from the cat’s ear reflect off the mirror,
they enter a person’s eye as if they had
come from a region behind the mirror. This
region is the location of the virtual image. It
is located as far behind the mirror as the
cat’s ear is in front of the mirror.
Lateral inversion
Lateral inversion is the
apparent sideways reversal of
an image in a mirror when
compared to the object in front
of the mirror.
When you look into a mirror, you do not see yourself as others see you.
Your left and right sides appear reversed. This is called lateral inversion.
Some safety vehicles, such as ambulances, have their front labels adjusted
for lateral inversion so that drivers looking in their rear vision mirrors
will see the letters the right way round (see figures 1.13 and 1.14).
mirror
A
B
Figure 1.13
A1
B1
The rays from the letter ‘J’ reflect off the mirror and into
the eye. The loop of the ‘J’, from the eye’s point of view, is on the left, whereas in
the image the loop is on the right. This situation depends on where the eye is.
How would this situation look if the eye moved to the far left of the diagram,
behind the ‘J’?
8
WAVE-LIKE PROPERTIES OF LIGHT
Inve
Jac Phys 1 2E - 01 Page 9 Friday, October 24, 2003 1:19 PM
ations
stig
Figure 1.14
Some
vehicles, such as ambulances, have
their name laterally inverted on the
front of the vehicle. When the word
is seen through the rear vision
mirror of the car in front, it appears
the right way around and triggers a
more immediate response on the
part of the car’s driver.
Inve
Investigation 1.3
Lateral inversion
ations
stig
Inve
Investigation 1.4
Up periscope!
ations
stig
Investigation 1.5
Multiple images
Rotating mirrors
Another use of plane mirrors involves the effect on the reflected ray
when a mirror is turned. When a mirror rotates through a small angle,
the reflected ray is moved through twice that angle. This effect is put
to good use in sensitive scientific instruments. Sometimes scientific
instruments are used to measure very small values of quantities, such
as electric currents. A very small electrical input might turn a mirror
slightly. But when a beam of light is shone onto the mirror and
reflected onto a wall across the room, a more measurable deflection
can be observed (see figure 1.15).
mirror
g
r tiltin
y afte
d ra
flecte
re
reflected ray
before tilting
2θ
Web
incoming ray
Figure 1.15
s
link
θ
One-way mirror
The principle of the mirror or spot
mirror
galvanometer. A ray from a light source, for example, can be
tilted
reflected across a large distance. A very small rotation of the
mirror can move a light spot on a distant wall several centimetres.
AS A MATTER OF FACT
A
normal mirror has a thick coating of silver on
the rear side of the glass. The silver reflects in
a regular way because the glass surface is optically
flat. The outer surface of the silver cannot be used
because it does not reflect evenly and it corrodes,
so the silver is covered with a protective paint.
A two-way mirror, on the other hand, has a
thinner coating of silver. Some light can pass
through this, but most is reflected. From the
front, the mirror produces a normal mirror
image. From the rear, sufficient light passes
through for the sensitive human eye to see clearly
beyond it. However, a ‘secret observer’, behind
the mirror, can be detected because light can
travel along a light path in both directions. If a
light globe is turned on behind the mirror, some
light reflecting off the secret observer will pass
through the mirror to the front. The viewer in
front of the mirror will be able to see the faint
appearance of the secret observer. The secret
observer cannot be seen if there is no light
source behind the mirror.
Figure 1.16
CHAPTER 1 REFLECTING LIGHT
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CONCAVE MIRRORS
A concave mirror is a curved
mirror with the reflecting
surface on the inside of the
curve.
When a light source is placed in front of a plane mirror, the reflected
rays continue to move apart or diverge. If the mirror is curved into a
concave shape, the reflected rays would spread less widely. With just the
right shape, the reflected rays form in parallel lines. The shape to achieve
this is a parabola (see figure 1.17(c)). The parabolic shape is used in the
design of torches, car headlights and searchlights.
(a) Plane mirror
(b) Circular concave mirror
Figure 1.17
(a) Rays
from a light source in front of a
plane mirror are reflected in all
directions. (b) The curved circular
mirror brings the reflected rays
inwards. (c) When the curved
mirror is in the shape of a
parabola, the reflected rays
become parallel to each other
and the light source is said to be
at the focus of the curved mirror.
The focus of a concave mirror
is the point through which a set
of rays passes that is parallel to
the principal axis and reflected
from the mirror.
The principal axis of a curved
mirror is a line passing through
the centre of the mirror and
which is perpendicular to the
plane of the mirror.
The focal length is the distance
from the centre of a curved
mirror to its focus.
10
(c) Parabolic concave mirror
Figure 1.18
In a car headlight, the filament of the globe is placed at
the focus of the curved reflective surface. This produces a narrow beam of light.
Light can travel either way along a light path, so the arrows of the light
rays in figure 1.17 can be reversed. This means that all the parallel rays of
light that hit the concave mirror will pass through a point. This point
therefore has special significance. It is called the focus of the concave
mirror. The line through the centre of the mirror and the focus is called
the principal axis and the distance from the centre of the mirror to the
focus is called the focal length.
WAVE-LIKE PROPERTIES OF LIGHT
Jac Phys 1 2E - 01 Page 11 Tuesday, October 21, 2003 1:41 PM
The pole of a curved mirror is
at the centre of its reflecting
surface.
The centre of curvature of a
concave mirror is a point on the
principal axis at twice the focal
length from the pole of the
mirror.
A real image is one through
which light passes. A real image
can be seen on a screen placed
at the location of the image.
Ray tracing can be used to investiconcave
gate further the properties of the
mirror
principal axis
image formed by a concave mirror.
Ray tracing is a graphical technique
for finding the position and size of
images formed by mirrors.
centre of
focus (F)
As shown in figure 1.19, the centre curvature (C)
pole
of a concave mirror is called the pole.
The centre of curvature (see figure
1.20) is a point twice as far from the
pole as the focus. This statement is
true as long as the curved mirror is Figure 1.19 For small
small. For small mirrors, the circular mirrors, the focus is halfway
or spherical curved shape is easier to between the centre of curvature
manufacture than the ideal parabolic and the pole of the mirror.
shape and the difference between the
centre of
two shapes — spherical and parabolic
curvature
— would be minor. For specialist
mirrors, such as in the Hubble telescope, the parabolic shape is used.
The ray model can now be applied to
the concave mirror. In figure 1.21, rays
of light spread out from the head of the
object, hit the mirror and pass through
the image. This is called a real image.
(The concave mirror has been represented as a straight line because the Figure 1.20 A curved
rays are close to the principal axis.) or spherical mirror can be
This is called a real image because the imagined as being part of a
light passes through this location and complete sphere. The centre of
the image can be captured on a screen. curvature is at the point that would
Four rays can be drawn to locate the be at the centre of that sphere.
image.
• Ray 1 leaves the head of the object parallel to the principal axis,
reaches the mirror and is reflected back through the focus.
• Ray 2 passes through the focus before reaching the mirror. It is then
reflected in a direction parallel to the principal axis.
• Ray 3 travels towards the pole of the mirror. At this point, the section of
the concave mirror is at right angles to the principal axis, so the ray is
reflected at an equal angle below the axis. Thus the reflected ray passes
the object at the same distance below the axis that the object was above it.
• Ray 4 passes through the centre of curvature before reaching the mirror.
Because the centre of curvature is the centre of a sphere, this ray is travelling along the radius. The ray therefore meets the spherical mirror at
zero angle of incidence and thus is reflected back along the same line.
In figure 1.22, light rays drawn from the head of the object are
reflected by the concave mirror and all appear to come from another
point behind the mirror. This is the image of the head of the object. It is
called a virtual image because no light passes through it and it cannot be
captured on a screen. The four rays have been used to locate the image,
but every single ray from the head of the object that reaches the mirror
appears to come from the image.
In figures 1.21 and 1.22, the rays that leave the base of the object along
the principal axis towards C and F all come back along the axis. This
means that the image of the base of the object is also on the axis.
CHAPTER 1 REFLECTING LIGHT
11
eMo
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1
ling
del
object
Model of a
concave mirror
C
F
2
image
Inve
3
ations
stig
1
4
Figure 1.21
Web
Investigation 1.6
Concave mirrors —
and observation
exercise
s
link
Concave mirror
applet
The magnification of a mirror is
the ratio of the height of the
image to the height of the
object.
Light rays from the head of the object are reflected by
the concave mirror and all pass through another point, which is the image of the
head of the object. This is called a real image because the light passes through
this location and the image can be captured on a screen. The diagram includes
only four rays for the purpose of locating the image, but every single ray from the
head of the object that hits the mirror passes through the head of the image.
Note that the shaded section of figures 1.21 and 1.22 indicates that all
the rays from the head of the object that reach the mirror, not only the
four rays described on page 11, will pass through the head of the image.
The image can now be described as follows.
• Location: The scale of the diagram
can be used to work out the dis2
tance of the image from the mirror.
• Size: Similarly, the vertical scale —
or the comparison of the size of the
image with the size of the object —
object
image
can be used to calculate the height
of the image. The magnification is
C
F
the height of the image divided by
4
the height of the object.
• Orientation: The appearance of the
1
image immediately tells you whether
3
it is the same way up as the object or
upside down. If it is on the same
side of the principal axis, then the Figure 1.22 All four
rays appear to come from the same
image is upright.
• Nature: The image is real if the rays point. This is where the image of
actually pass through its location. the head of the object is located
This means that a screen can be put and it is a virtual image. Note that
at this point and the image will we could have located the image
appear on it. If the rays only look as with any two of the rays, but either
if they are coming from the image’s or both of the others could be used
location, as in a plane mirror, then to confirm the original location of
the image.
the image is a virtual image.
Looking at stars
The focal plane is at right
angles to the principal axis and
passes through the focus.
12
Parallel rays reaching a concave mirror could be coming from a very distant object, such as a star, so all the light hitting the mirror passes
through the focus. This same design can be applied to a telescope in
which a larger mirror can collect more light and enable astronomers to
see fainter stars. The light from stars at an angle to the principal axis is
still parallel and is brought to a focus at a point above or below the focus,
but in the same focal plane (see figure 1.23).
WAVE-LIKE PROPERTIES OF LIGHT
Jac Phys 1 2E - 01 Page 13 Tuesday, October 21, 2003 1:41 PM
Figure 1.23
parallel light
from star off
principal axis
principal axis
•
F
concave
focal mirror
plane
If a
concave mirror is oriented so that
its principal axis is pointing
directly at a star, then light from
that star will be collected at the
focus. For stars off to the side, at a
slight angle to the axis, their light
is collected at a point to the side of
the focus, but still in the same plane
at right angles to the axis. This
plane is called the focal plane.
Figure 1.24
Inve
Radio
telescopes use parabolic concave
mirrors to collect radio waves from
distant stars.
ations
stig
Investigation 1.7
Locating images in
concave mirrors
However, if the light comes from a light source close to the concave
mirror so that the incident rays are not parallel, the paths of the reflected
rays are not so easy to predict.
Using a formula to model
images in concave mirrors
Ray tracing diagrams can be used to develop a mathematical relationship
between the positions of the object and the image relative to the mirror.
As shown in figure 1.25:
u = distance from object to mirror
v = distance of image from mirror
f = focal length
u
Ho = height of object
B
D
Hi = height of image.
object
Ho
Figure 1.25
C
A
The description
of an image can be determined by use
of a formula.
G
F
O
Hi
image
E
v
positive distances
negative
distances
CHAPTER 1 REFLECTING LIGHT
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Jac Phys 1 2E - 01 Page 14 Friday, October 24, 2003 9:09 AM
In figure 1.25 the shaded triangles BAO and EGO are similar.
Hi
v
Therefore: ------- = --u
Ho
where
H
-------i is equal to the magnification, M, of the mirror.
Ho
[1]
The triangles DOF and EGF are also similar.
Hi
v–f
Therefore: ------- = ---------Ho
f
⇒
⇒
⇒
⇒
v
v–f
--- = ---------- (from [1])
f
u
vf = uv − uf
vf + uf = uv
1
1 1
--- + --- = --- (dividing by uvf )
u v
f
[2]
Because the image can be either in front of the mirror and therefore a real image or behind the mirror and therefore a virtual image,
a sign convention needs to be employed so as to distinguish these two
possibilities.
For real images that are in front of the mirror, the image distance is
positive. For virtual images that are behind the mirror, the image distance is negative.
Virtual images have a negative image distance, giving a negative value
v
to the ratio ---u- . However, the magnification is the ratio of two heights and
therefore cannot be negative. In such cases, only the size of the
numerical answer is taken.
SAMPLE
PROBLEM
1.3
Locate and describe the image formed by a 3.0 cm high object 25 cm in
front of a concave mirror with a focal length of 10 cm.
Ho = 3.0 cm; Hi = ?; u = 25 cm; v = ?; f = 10 cm
Solution:
1 1 1
Using --- + --- = --u v
f
1
1
1
------ + --- = -----25 v 10
1
1
1
--- = ------ − -----⇒
v 10 25
(5 – 2)
= ----------------50
⇒
⇒
14
WAVE-LIKE PROPERTIES OF LIGHT
3
= -----50
v 50
--- = -----1
3
50
v = -----3
= 16.67
= 17
(substituting data into mirror formula)
(combining fractions using lowest
common denominator)
(inverting each side of the equation)
(rounding the answer off to two
significant figures)
Jac Phys 1 2E - 01 Page 15 Tuesday, October 21, 2003 1:41 PM
Skill
cks
che
Significant figures
(p. 493)
SAMPLE
PROBLEM
1.4
The image is 17.0 cm in front of the mirror. It is real and inverted.
Hi
v
Magnification = ------- = --Ho u
H
17
-------i = -----3.0 25
2
= --3
2
⇒
Hi = --- × 3.0
3
= 2.0
The image is 2.0 cm high.
Locate and describe the image formed by a 5.0 cm high object 10 cm in
front of a concave mirror with a focal length of 15 cm.
Solution:
Ho = 5.0 cm; Hi = ?; u = 10 cm; v = ?; f = 15 cm
Inve
Using
ations
stig
⇒
eMo
Investigation 1.8
Soup spoon
ling
del
Exploring a concave
mirror with a
spreadsheet
1 1 1
--- + --- = --u v
f
1
1
1
------ + --- = -----10 v 15
1
1
1
--- = ------ − -----v 15 10
(2 – 3)
= ----------------30
(substituting data into mirror formula)
(combining fractions using lowest common
denominator)
–1
= -----30
v 30
--- = -----⇒
(inverting each side of the equation)
1 –1
⇒
v = −30
The image is 30 cm behind the mirror. It is virtual and upright.
Hi
v
Magnification = ------- = --Ho u
H
30
-------i = ------ = 3
5.0 10
Hi = 3 × 5.0 = 15
The image is 15 cm high.
CONVEX MIRRORS
A convex mirror is a curved
mirror with the reflecting
surface on the outside of the
curve.
Convex mirrors curve outwards rather than inwards. If you look through
a convex mirror, you have a wider angle of view. Convex mirrors are used
as security mirrors in supermarkets and for safety purposes to assist
drivers leaving driveways with restricted views of the traffic (see figures
1.26 and 1.27).
Like the concave mirror, the convex mirror has a focus, but it is a virtual focus because it is behind the mirror (see figure 1.28). Incident light
rays parallel to the principal axis are reflected away as if they had come
from the focus behind the mirror. However, the rays used in the ray
tracing for the concave mirror can still be used if you keep in mind that
the focus and the centre of curvature are on the other side of the mirror.
CHAPTER 1 REFLECTING LIGHT
15
Jac Phys 1 2E - 01 Page 16 Tuesday, October 21, 2003 1:41 PM
The rays from the object spread out or diverge after hitting
the mirror, so it is impossible to form a real image with a
convex mirror.
Figure 1.26
Web
Convex mirrors are used when a wide
view of an area is required.
eye
s
link
3
object
1
Convex mirror applet
image
F
2
convex
mirror
convex mirror
Figure 1.28
Figure 1.27
Inve
A convex
mirror is curved outwards. A person
standing in front of one will receive
light coming from a very wide angle.
Compare this with figure 1.17.
ations
stig
Investigation 1.9
Convex mirrors
SAMPLE
PROBLEM
Ray 1 heading
towards the focus is reflected parallel to
the principal axis. Ray 2 going towards
the pole is reflected at an equal angle
below the axis. Ray 3 parallel to the
principal axis is reflected up as if it came
from the focus.
1.5
Solution:
The same formula as was used for concave mirrors (see page 14) also
works for convex mirrors, but the focal length is represented by a
negative number because the focus is a virtual one. For this reason,
convex mirrors are sometimes called negative mirrors.
Locate and describe the image formed by a 5.0 cm high object 10 cm in
front of a convex mirror with a focal length of 15 cm.
Ho = 5.0 cm; Hi = ?; u = 10 cm; v = ?; f = −15 cm
1 1 1
Using --- + --- = --u v
f
1
1
1
------ + --- = -----10 v – 5
1
1
1
--- = − ------ − -----⇒
v
15 10
( −2 + −3 )
= ------------------------ (combining fractions using lowest common
30
denominator)
–5
= -----30
v
30
--- = -----⇒
(inverting each side of the equation)
1 –5
⇒
v = −6.0
The image is 6.0 cm behind the mirror. It is virtual and upright.
16
WAVE-LIKE PROPERTIES OF LIGHT
Jac Phys 1 2E - 01 Page 17 Tuesday, October 21, 2003 1:41 PM
eMo
Hi
v
Magnification = ------- = --Ho u
ling
del
Exploring a convex
mirror with a
spreadsheet
H
6.0
-------i = ------5.0
10
= 0.6
⇒
Hi = 0.6 × 5.0
= 3.0
The image is 3.0 cm high.
USING CONCAVE MIRRORS IN
THE REFLECTING TELESCOPE
The reflecting telescope (see figure 1.29) was designed by Isaac Newton
(1642–1727) to overcome problems with telescopes that used only lenses.
Lenses can produce distortion of the
image in that the light can spread into
colours when it passes through them
(see chapter 3) and large lenses sometimes sag when supported at their
edges. Also, large mirrors with one
curved surface are easier to make than
large lenses with two curved surfaces.
As shown in figure 1.30, light from a distant star
arrives as parallel rays, so the image of the star is
formed in a plane at the focus (F) of the concave
mirror. A small, tilted plane mirror is placed just
in front of the focus and reflects light out of the
side of the tube into a lens which reproduces the
original parallel rays. This means the image is
magnified. The actual magnification is the ratio
of the two focal lengths.
focal length of concave mirror
Magnification = -------------------------------------------------------------------------------------------------------------focal length of lens
Figure 1.29
The Newtonian telescope, with a
tube 11.43 cm in diameter
concave mirror
plane mirror
parallel
light
from
distant
star
F
Figure 1.30
convex lens
The
telescope designed by Isaac Newton
uses a concave mirror to bring the
light from a distant star to a focus but,
before the light reaches the focus, it is
reflected out to the side by a plane
mirror into a convex lens.
CHAPTER 1 REFLECTING LIGHT
17
Jac Phys 1 2E - 01 Page 18 Tuesday, October 21, 2003 1:41 PM
As
telescopes
became
more
concave
mirror
powerful, that is, able to collect more
light and hence see more stars and convex
more detail, the Newtonian design mirror
became cumbersome. The French
optician N. Cassegrain improved the
design by cutting a hole in the middle Figure 1.31 Diagram
of the concave mirror and reflecting of a Cassegrain telescope
the light through it with a very small
convex mirror placed in front of the large concave mirror (see figure 1.31).
AS A MATTER OF FACT
T
Web
he eyes of most animals contain transparent lenses. The scallop,
however, uses a concave mirror to focus light onto its retina.
The retina is the surface of receptor cells that send signals to the
brain.
s
link
Scallop eyes
concave
mirror
retina
Figure 1.32
A scallop’s eye. The scallop uses a set of reflective
cells in the shape of a concave mirror to produce a real image on its ‘screen’
or retina.
PROBLEMS WITH CURVED
MIRRORS
Spherical aberration is the
distortion of an image
produced by a concave mirror
that is spherical rather than
parabolic. In small mirrors
spherical aberration is less
noticeable than in larger
mirrors.
18
As discussed on page 11, most small concave mirrors are spherical rather
than parabolic. In small mirrors, differences between the circle and the
parabola are not great. For larger mirrors or when high quality images
are required, the distortion or spreading of the image produced by a
spherical mirror is a problem. This distortion is called spherical
aberration. It occurs because a spherical mirror does not reflect parallel
rays to a single point. The focus is ‘spread out’, as shown in figure 1.33.
WAVE-LIKE PROPERTIES OF LIGHT
Jac Phys 1 2E - 01 Page 19 Tuesday, October 21, 2003 1:41 PM
(a) Large spherical mirror
parallel rays
(b) Parabolic mirror
parallel rays
Figure 1.33
Inve
(a) A large spherical mirror reflects the outer parallel
rays to a different point from the inner rays. This produces a distorted image,
referred to as a spherical aberration. (b) A parabolic mirror brings all the rays to
a single point.
ations
stig
An example of spherical aberration is the pattern of light produced by
a ceiling light reflecting from the inside of the coffee cup onto the
surface of the coffee. The full circular shape of the mug spreads the
point focus out into a curve, called a caustic curve.
Investigation 1.10
Caustic curve
CHAPTER 1 REFLECTING LIGHT
19
Jac Phys 1 2E - 01 Page 20 Tuesday, October 21, 2003 1:41 PM
• Light sources are called luminous objects.
Some luminous objects give off light because
they are hot; these are called incandescent
objects.
Radius of Earth’s orbit about the Sun
= 1.49 × 1011 m
Radius of Mars’s orbit about the Sun
= 2.28 × 1011 m
Radius of Neptune’s orbit about the Sun
= 4.50 × 1012 m
• Light travels in straight lines through air at a
speed of 3.0 × 108 m s–1. Shadows provide evidence that light travels in straight lines.
5. Copy the following figure and draw the incident and reflected rays from the two ends of
the object to the eye. Locate the image.
SUMMARY
• Modelling light as a pencil-like ray helps
describe the reflection of light.
object
plane
mirror
• When light meets a surface the angle of incidence equals the angle of reflection. The incident ray, the normal to the surface and the
reflected ray all lie in the same plane.
• If the surface is smooth like that of a mirror,
the regular reflection enables images to be seen
in the mirror.
6. Calculate the angles, a, b and c in the following
figure.
• The image in a plane mirror is as far behind
the mirror as the object is in front of the
mirror.
• If a mirror is curved, the properties of the
images formed by the mirror are different from
the properties of the images in plane mirrors.
CHAPTER REVIEW
• The ray model explains the formation and
properties of images in plane and curved mirrors.
• The formation of images in concave and convex
mirrors can be mathematically modelled with
50°
c
a
v
f
Ho
A
A
u
QUESTIONS
Understanding
1. Draw a Venn diagram for the following words
which describe the different objects you can see:
luminous, non-luminous, and incandescent.
2. Describe the light path from a light source to
your eye in seeing an object.
3. Use the ray model and the sources of light to
rephrase the statements (a) ‘I looked at a
flower through the window’ and (b) ‘I
watched the TV’.
4. Calculate the longest and shortest time for a
radio signal travelling at the speed of light to go
from the Earth to a space probe when the space
probe is (a) near Mars and (b) near Neptune.
20
WAVE-LIKE PROPERTIES OF LIGHT
b
7. The two arrowed lines in the figures below
represent reflected rays. The line AB represents the plane mirror. Locate the image
and the light source in each of the two figures.
Hi
v
- = --- .
the equations --1- + 1--- = 1--- and M = ------u
mirror
B
B
8. What type of car does ATOYOT Car Sales sell?
9. A student argues that you cannot photograph
a virtual image because light rays do not pass
through the space where the image is
formed. How would you argue against this
statement?
10. Sketch the path of each of the rays entering
each of the pair of joined mirrors in the
following figure.
Jac Phys 1 2E - 01 Page 21 Tuesday, October 21, 2003 1:41 PM
19. (a) You are standing 2.0 m in front of a plane
mirror and you wish to take a sharp photograph of yourself in the mirror. At what
distance do you set the camera lens?
(b) Your friend is standing beside you, 1.0 m
away. At what distance do you set the camera
lens for a sharp photograph of your friend?
12. You are walking towards a plane mirror at a
speed of 1.0 m s−1. How fast is your image
walking? How quickly are you and your image
approaching each other?
20. Use two or more plane mirrors to produce an
image of you that is not laterally inverted; that
is, you see yourself as others see you. Use two
point sources, marked L and R, at the end of a
short line.
13. Use your understanding of the reflection of
light to explain how full gloss and matt paints
differ.
14. Produce a table summarising all the properties
of the image produced by a concave mirror for
the following object positions: beyond the
centre of curvature (C); at C; between C and
the focus (F); at F; inside F.
15. Use either a scaled drawing or the concave
mirror formula to determine the full description of the image (that is, size, location, orientation and nature) for the following situations:
(a) a 2.0 cm high object 12 cm from a concave
mirror with a focal length of 8.0 cm
(b) a 5.0 cm high object 1.0 m from a concave
mirror with a focal length of 10 cm
(c) a 4.0 cm high object 12 cm from a concave
mirror with a focal length of 20 cm.
16. The formula for the concave mirror can also
be used for convex mirrors with only one
change: making the focal length a negative
quantity. Determine the nature of the image of
a 4.0 cm object 15 cm in front of a convex
mirror with a focal length of 10 cm.
Application
17. Explain how early astronomers knew the Moon
must have a rough surface.
18. Assume that your eyes are located at the top of
your 1.60 m tall body.
(a) What size plane mirror would you need to
see your whole body if you were standing
1 m from it?
(b) How far from the floor should the bottom
edge of the mirror be placed?
(c) How would your answers to (a) and (b)
change if you assumed your eyes were
10 cm from the top of your head?
(d) How would your answers to (a) and (b)
change if you were 2 m from the mirror?
Use diagrams and words to explain your answers.
21. Design a system of plane mirrors to enable an
immobile, bedridden patient to read a book
placed on the patient’s chest.
22. Draw three ray diagrams to investigate how the
size of an image in a plane mirror depends on
the distance of the object from the mirror.
23. A plane mirror reflects a ray of light and
changes its direction. The angle of deviation is
the angle between the reflected ray and the
continuation of the incident ray. How is the
angle of deviation related to the glancing
angle (that is, the angle between the incident
ray and the mirror)?
24. The figure below shows an incident ray (R1)
reflected off a mirror (m1). The mirror is
rotated (m2) and the reflected ray moves (R2).
Use the angles in the diagram to prove that the
angle of deflection of the reflected ray equals
twice the angle of rotation of the mirror.
I
N2 N1
m1
m2 a
R2
R1
a
m2
mirror
m1 position
25. Prove that the distance that light travels from a
source, P, to another point, Q, via a plane
mirror is the shortest when the angle of incidence equals the angle of reflection. (Hint:
Draw a straight line from P to the image of Q.)
26. What is the focal length of a plane mirror?
27. A make-up or shaving mirror is a concave
mirror designed to produce a magnified
image. Your face is inside the focus. A comfortable distance from your face to the mirror
would be about 30 cm.
(a) If you want a magnification of 2.0, what
should be the focal length of the mirror?
(b) Where is your image located?
CHAPTER 1 REFLECTING LIGHT
21
CHAPTER REVIEW
11. Sketch the path of a ray emitted from a point
between two parallel mirrors (see the following
figure).
Jac Phys 1 2E - 01 Page 22 Tuesday, October 21, 2003 1:41 PM
(c) If the height of your face is 16 cm, what
diameter mirror would you need to see
your whole face?
28. A dentist asks you to design a concave mirror
to produce a magnified upright image of a
tooth. The magnification should be about 3
with a space of 2 cm between the tooth and
the mirror.
(a) What focal length mirror will you design?
(b) The dentist tests out your mirror but
rejects it. She says that it produced a distorted image of the tooth because teeth
are three-dimensional objects. Why was the
image distorted?
29. A concave mirror with a focal length of 20 cm
produces an image located an infinite distance
away of an object placed in front of the mirror.
Where is the object located?
30. Find the size of the image of the Moon formed
by a concave mirror with a focal length of
20 cm if it is 3.8 × 105 km away and its radius is
1740 km. How could the image be made
brighter? How could it be made bigger?
31. A real image is formed by a concave mirror.
How will the image change if the top half of
the mirror is removed?
CHAPTER REVIEW
More of a challenge
32. When you look into a plane mirror, your left
and right sides appear reversed. This is called
‘lateral inversion’. How could you place the
mirror so that your image is upside down?
22
WAVE-LIKE PROPERTIES OF LIGHT
33. How can you see raindrops if water is transparent?
34. A laser beam is sent from the surface of the
Earth to a corner reflector which was left on
the Moon by the Apollo astronauts. The
reflector sends the beam back to Earth in the
same direction. The time of the round trip for
the beam of light is about 2.479 seconds.
(a) Calculate the distance of the centre of the
Moon from the centre of the Earth, given
the following data:
speed of light = 2.998 × 108 m s−1
radius of Earth = 6380 km
radius of Moon = 1740 km.
(b) The Earth was continuing to spin while the
laser beam was in transit. How far did the
receiver on the Earth’s surface move in
that time?
(c) What are the implications of these results
for the design of this exercise?
35. A corner reflector is made of three plane mirrors each at right angles to each other, like the
corner of a room. How many images would
you see?
36. The smooth surface of a very large lake is
slightly curved because the Earth is round. In
fact, the surface is a convex mirror with a focal
length equal to half the Earth’s radius. If the
radius of the Earth is 6380 km and the moon,
which has a radius of 1740 km, is 3.8 × 105 km
away, calculate the location and size of the
Moon’s image formed by the lake’s surface.