Refracting light Wave-like - Jacaranda | Shop · Refracting light Chapter 3 Seeing colours Jac Phys 1 2E - 01 Page 1 Tuesday, October 21, 2003 1:41 PM. ... seen because they reﬂect

$Page 1: Refracting light Wave-like - Jacaranda | Shop · Refracting light Chapter 3 Seeing colours Jac Phys 1 2E - 01 Page 1 Tuesday, October 21, 2003 1:41 PM. ... seen because they reﬂect$
UNIT 1

Wave-likeproperties of

light

AREA OF STUDY 1

Chapter 1

Reflecting light

Chapter 2

Refracting light

Chapter 3

Seeing colours

Jac Phys 1 2E - 01 Page 1 Tuesday, October 21, 2003 1:41 PM

Chapter

1

Reflectinglight

Figure 1.1

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.

S

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.


CHAPTER 1 REFLECTING LIGHT 3

Sight is the sense by which humans and most other mammals get most oftheir information about the world. This sense responds to light. Ques-tions about light naturally arise. Where does light come from? What canit 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

When we experience darkness at night or in an enclosed room, we knowthat a source of light, such as the Sun or a lamp, is needed to light up thedarkness. Once a lamp is turned on, we can see features in the roombecause the light from the lamp shines on them and is then reflectedinto our eyes.

This means that objects can be classified into two groups. Objects seenbecause they give off their own light are called

luminous

objects; thoseseen because they reflect light are called non-luminous objects. The Sun,torches and candles are luminous objects. Tables, chairs, cats and dogsare non-luminous objects.

Some luminous objects produce light because they are hot. The Sun isone example. The higher the temperature, the brighter the light, andthe 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.

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.


WAVE-LIKE PROPERTIES OF LIGHT4

Other objects are cold and produce light in another way. It involveschanges in the energy of electrons in the material brought about byeither chemical or electrical processes.

Speed of light

The gap we experience between seeing lightning and hearing thundershows that sound travels relatively slowly. Light seems to travel so fast thatto our experience its speed seems infinite; that is, we seem to observeevents at the instant they happen.

Galileo Galilei (1564–1642) was not convinced of this. He attempted todetermine the speed of light by measuring the time delay between theflash of his lamp to an assistant on a distant mountain and the returnflash from his assistant’s lamp (see figure 1.3). No detectable delay wasobserved and Galileo concluded that the speed of light was very high. Alonger distance was needed.

Olaus Roemer was a Danish astronomer born two years afterGalileo’s death. He observed that the time between eclipses of Jupiter’smoons by Jupiter decreased as the Earth moved closer to Jupiter andincreased as the Earth moved away. Roemer reasoned that this wasbecause the distance the light travelled from Jupiter to Earth becamegreater 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 orbitabout the Sun to estimate the speed of light. The value he obtainedwas 2.7

×

10

8

m s

−

1

.Eventually, in the nineteenth century, with stronger light sources and

more precise timing devices, Galileo’s method could be used, but theassistant was replaced by a mirror. The values obtained then were about3.0

×

10

8

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). Thelight was reflected to a distant mirror about 35 kilometres away then

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Investigation 1.1Luminous or not?

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stigations

Investigation 1.2Luminosity and

temperature

Figure 1.3 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.

Web

links

Speed of light

Skillc

hecks

Significant figures(p. 493)



reflected back to the rotating mirror. For some particular rotation rates, thislight is reflected by one of the sides of the rotating mirror directly to theobserver. The rotation rate can be used to calculate the speed of light.

The value Michelson obtained was 2.997 96

×

10

8

m s

−

1

. He actuallymeasured the distance of 35 km to an accuracy of 2.5 cm. The speed oflight is currently measured at 2.997 924 58

×

10

8

m s

−

1

. It is rounded offto 300 000 km s

−

1

for calculation purposes.

How long does light take to travel from the Sun to the Earth?

speed of light

=

3.00

×

10

8

m s

−

1

distance from Sun to Earth

=

1.49

×

10

11

m

speed

=

⇒

time taken

=

time

=

=

0.497

×

10

3

s

=

497 s

=

8 minutes 17 seconds.

How far does light travel in one year (one light-year)?

distance travelled

=

speed

×

time takendistance

=

3.00

×

10

8

m s

−

1

×

(365.25

×

24

×

60

×

60) s

=

9.47

×

10

15

m

=

9.47

×

10

12

km.

ShadowsThe bright Sun produces sharp shadows on the ground. The shape of theshadow 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.

Skillc

hecks

SI units(p. 490)

Sun

B

Earth’s orbit

AIo (moon)

Jupiter

observer

source of light

fixedmirrorrotating mirror

with eight sides

35 km

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.)

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.

SAMPLEPROBLEM 1.1

Solution:

distance travelledtime taken

---------------------------------------------

distance travelledspeed

---------------------------------------------

Skillc

hecks

Scientific notation(p. 492)

1.49 1011 m×3.00 108 m s 1–×-----------------------------------------

SAMPLEPROBLEM 1.2

Solution:



Ray model

The need for sources of light, the great speed of light and the existenceof sharp shadows can be described by a ray model. The model assumesthat light travels in a straight line path called a light

ray

. A light ray canbe considered as an infinitely narrow beam of light and can berepresented as a straight line (see figure 1.7).

PLANE MIRROR REFLECTION

When you look at yourself in a plane mirror, some of the light rays fromyour nose, for example, travel in the direction of the mirror and reflectoff in the direction of your eye (see figure 1.8). What is happening at thesurface of the mirror to produce such a perfect image?

Figure 1.6 The straight rays passing the edge of a skateboarder leave a sharp shadow on the wall.

A ray of light is a very narrow pencil-like beam of light.

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.

mirror

Figure 1.8 Light rays from the tip of the nose reflect off the mirror and enter the eye.



To investigate the reflection of light, the angles made by the rays needto be measured. Measurements of these angles show that, like a ballbouncing off a flat wall, the

angle of incidence

equals the

angle ofreflection

(see figure 1.9).

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 other seemingly trivial conclusion that can be drawn from theinvestigation is that the incident ray, the

normal

and the reflected ray alllie in the same plane (see figure 1.10).

REGULAR AND DIFFUSE REFLECTION

Reflection from a smooth surface is called

regular

or

specular reflection

.But what happens with an ordinary surface, such as this page? A page isnot smooth like a mirror. At the microscopic level, there are ‘hills andvalleys’. As the light rays come down into these hills and valleys, they stillreflect 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 makesense of the world.

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.

normal

angle ofreflection

angle ofincidence

incidentray

reflectedray

mirror

The normal is a line that is perpendicular to a surface or a boundary between two surfaces.

normal

mirror

reflected ray

incident 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.

Regular reflection, also referred to as specular reflection, is reflection from a smooth surface.

Diffuse reflection is reflection from a rough or irregular surface.

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. irregular surface

observer A

observer B



FORMING IMAGES WITH A PLANE MIRRORThe law of reflection can now be used to explain how images are formedin plane mirrors and how to describe images. To describe an image, fouraspects need to be determined:1. its location — where it is in relation to the mirror2. its size — how big it is compared with the object3. its orientation — whether it is upright or upside down4. 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 therays leaving the cat’s ear that reach the mirror are reflected so that theyappear to be coming from the same region behind the screen. Some ofthese rays enter your eye. Someone else standing beside you will receivedifferent rays even though the rays will appear to be coming from thesame region. The region from which the rays appear to be coming is thelocation of the image of the cat’s ear.

The image is located as far behind the mirror as the object is in frontof the mirror. It is the same size and upright. The light rays entering theeye only appear to come from the location of the image. The image isdescribed as a virtual image. A virtual image cannot be ‘captured’ on ascreen.

Lateral inversionWhen 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 adjustedfor lateral inversion so that drivers looking in their rear vision mirrorswill see the letters the right way round (see figures 1.13 and 1.14).

Figure 1.13 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’?

Images are likenesses of objects.

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.

Web

links

Image in a planemirror applets

mirror

B

A

B1

A1

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 is the apparent sideways reversal of an image in a mirror when compared to the object in front of the mirror.

mirror

B

A

B1

A1



Rotating mirrorsAnother use of plane mirrors involves the effect on the reflected raywhen a mirror is turned. When a mirror rotates through a small angle,the reflected ray is moved through twice that angle. This effect is putto good use in sensitive scientific instruments. Sometimes scientificinstruments are used to measure very small values of quantities, suchas electric currents. A very small electrical input might turn a mirrorslightly. But when a beam of light is shone onto the mirror andreflected onto a wall across the room, a more measurable deflectioncan be observed (see figure 1.15).

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.

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stigations

Investigation 1.3Lateral inversion

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Investigation 1.4Up periscope!

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Investigation 1.5Multiple images

mirror

mirrortilted

2

reflected ray after tilting

incoming ray

reflected ray before tilting

θ

θ

Figure 1.15 The principle of the mirror or spot galvanometer. A ray from a light source, for example, can be reflected across a large distance. A very small rotation of the mirror can move a light spot on a distant wall several centimetres.

Web

links

One-way mirror

normal mirror has a thick coating of silver onthe rear side of the glass. The silver reflects in

a regular way because the glass surface is opticallyflat. The outer surface of the silver cannot be usedbecause 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 athinner coating of silver. Some light can passthrough this, but most is reflected. From thefront, the mirror produces a normal mirrorimage. From the rear, sufficient light passesthrough for the sensitive human eye to see clearlybeyond it. However, a ‘secret observer’, behindthe mirror, can be detected because light cantravel along a light path in both directions. If alight globe is turned on behind the mirror, somelight reflecting off the secret observer will passthrough the mirror to the front. The viewer in

front of the mirror will be able to see the faintappearance of the secret observer. The secretobserver cannot be seen if there is no lightsource behind the mirror.

AAS A MATTER OF FACT

Figure 1.16

Jac Phys 1 2E - 01 Page 9 Friday, October 24, 2003 1:19 PM


CONCAVE MIRRORSWhen a light source is placed in front of a plane mirror, the reflectedrays continue to move apart or diverge. If the mirror is curved into aconcave shape, the reflected rays would spread less widely. With just theright shape, the reflected rays form in parallel lines. The shape to achievethis is a parabola (see figure 1.17(c)). The parabolic shape is used in thedesign of torches, car headlights and searchlights.

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 lightrays in figure 1.17 can be reversed. This means that all the parallel rays oflight that hit the concave mirror will pass through a point. This pointtherefore has special significance. It is called the focus of the concavemirror. The line through the centre of the mirror and the focus is calledthe principal axis and the distance from the centre of the mirror to thefocus is called the focal length.

A concave mirror is a curved mirror with the reflecting surface on the inside of the curve.

(b) Circular concave mirror(a) Plane mirror

(c) Parabolic 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.



Ray tracing can be used to investi-gate further the properties of theimage formed by a concave mirror.Ray tracing is a graphical techniquefor finding the position and size ofimages formed by mirrors.

As shown in figure 1.19, the centreof a concave mirror is called the pole.The centre of curvature (see figure1.20) is a point twice as far from thepole as the focus. This statement istrue as long as the curved mirror issmall. For small mirrors, the circularor spherical curved shape is easier tomanufacture than the ideal parabolicshape and the difference between thetwo shapes — spherical and parabolic— would be minor. For specialistmirrors, such as in the Hubble tele-scope, the parabolic shape is used.

The ray model can now be applied tothe concave mirror. In figure 1.21, raysof light spread out from the head of theobject, hit the mirror and pass throughthe image. This is called a real image.(The concave mirror has been repre-sented as a straight line because therays are close to the principal axis.)This is called a real image because thelight passes through this location andthe image can be captured on a screen.

Four rays can be drawn to locate theimage.• 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 isreflected at an equal angle below the axis. Thus the reflected ray passesthe 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 trav-elling along the radius. The ray therefore meets the spherical mirror atzero 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 anotherpoint behind the mirror. This is the image of the head of the object. It iscalled a virtual image because no light passes through it and it cannot becaptured 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 mirrorappears to come from the image.

In figures 1.21 and 1.22, the rays that leave the base of the object alongthe principal axis towards C and F all come back along the axis. Thismeans that the image of the base of the object is also on the axis.

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.

centre ofcurvature (C)

focus (F)

concavemirror

principal axis

pole

Figure 1.19 For small mirrors, the focus is halfway between the centre of curvature and the pole of the mirror.

centre ofcurvature

Figure 1.20 A curved or spherical mirror can be imagined as being part of a complete sphere. The centre of curvature is at the point that would be at the centre of that sphere.

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.



Figure 1.21 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 allthe rays from the head of the object that reach the mirror, not only thefour 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 dis-tance of the image from the mirror.

• Size : Similarly, the vertical scale —or the comparison of the size of theimage with the size of the object —can be used to calculate the heightof the image. The magnification isthe height of the image divided bythe height of the object.

• Orientation : The appearance of theimage immediately tells you whetherit is the same way up as the object orupside down. If it is on the sameside of the principal axis, then theimage is upright.

• Nature : The image is real if the raysactually pass through its location.This means that a screen can be putat this point and the image willappear on it. If the rays only look asif they are coming from the image’slocation, as in a plane mirror, thenthe image is a virtual image.

Looking at starsParallel rays reaching a concave mirror could be coming from a very dis-tant object, such as a star, so all the light hitting the mirror passesthrough the focus. This same design can be applied to a telescope inwhich a larger mirror can collect more light and enable astronomers tosee fainter stars. The light from stars at an angle to the principal axis isstill 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).

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Investigation 1.6Concave mirrors —

and observationexercise

eMode

lling

Model of aconcave mirror

FC

image

object

1

2

3

14

Web

links

Concave mirrorapplet

FC

imageobject

1

2

3

4

Figure 1.22 All four rays appear to come from the same point. This is where the image of the head of the object is located and it is a virtual image. Note that we could have located the image with any two of the rays, but either or both of the others could be used to confirm the original location of the image.

The magnification of a mirror is the ratio of the height of the image to the height of the object.

The focal plane is at right angles to the principal axis and passes through the focus.



However, if the light comes from a light source close to the concavemirror so that the incident rays are not parallel, the paths of the reflectedrays are not so easy to predict.

Using a formula to model images in concave mirrorsRay tracing diagrams can be used to develop a mathematical relationshipbetween the positions of the object and the image relative to the mirror.As shown in figure 1.25:

u = distance from object to mirrorv = distance of image from mirrorf = focal length

Ho = height of objectHi = height of image.

parallel lightfrom star offprincipal axis

concavemirror

principal axis

focalplane

F •

Figure 1.23 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 Radio telescopes use parabolic concave mirrors to collect radio waves from distant stars.

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stigations

Investigation 1.7Locating images in

concave mirrors

F

positive distancesnegativedistances

Ho

Hi

C G

BD

OA

E

u

v

object

imageFigure 1.25 The description of an image can be determined by use of a formula.



In figure 1.25 the shaded triangles BAO and EGO are similar.

Therefore: = [1]

where

is equal to the magnification, M, of the mirror.

The triangles DOF and EGF are also similar.

Therefore: =

⇒ = (from [1])

⇒ vf = uv − uf⇒ vf + uf = uv

⇒ + = (dividing by uvf ) [2]

Because the image can be either in front of the mirror and there-fore a real image or behind the mirror and therefore a virtual image,a sign convention needs to be employed so as to distinguish these twopossibilities.

For real images that are in front of the mirror, the image distance ispositive. For virtual images that are behind the mirror, the image dis-tance is negative.

Virtual images have a negative image distance, giving a negative valueto the ratio . However, the magnification is the ratio of two heights andtherefore cannot be negative. In such cases, only the size of thenumerical answer is taken.

Locate and describe the image formed by a 3.0 cm high object 25 cm infront of a concave mirror with a focal length of 10 cm.

Ho = 3.0 cm; Hi = ?; u = 25 cm; v = ?; f = 10 cm

Using + =

+ = (substituting data into mirror formula)

⇒ = −

= (combining fractions using lowest common denominator)

=

⇒ = (inverting each side of the equation)

⇒ v =

= 16.67= 17 (rounding the answer off to two

significant figures)

HiHo-------

vu---

Hi

Ho-------

Hi

Ho-------

v f–f

----------

vu---

v f–f

----------

1u---

1v---

1f---

vu----

SAMPLEPROBLEM 1.3

Solution:

1u--- 1

v---

1f---

125------ 1

v--- 1

10------

1v--- 1

10------ 1

25------

5 2–( )50

-----------------

350------

v1--- 50

3------

503

------

Jac Phys 1 2E - 01 Page 14 Friday, October 24, 2003 9:09 AM


The image is 17.0 cm in front of the mirror. It is real and inverted.

Magnification = =

=

=

⇒ Hi = × 3.0

= 2.0The image is 2.0 cm high.

Locate and describe the image formed by a 5.0 cm high object 10 cm infront of a concave mirror with a focal length of 15 cm.

Ho = 5.0 cm; Hi = ?; u = 10 cm; v = ?; f = 15 cm

Using + =

+ = (substituting data into mirror formula)

⇒ = −

= (combining fractions using lowest commondenominator)

=


⇒ v = −30The image is 30 cm behind the mirror. It is virtual and upright.

Magnification = =

= = 3

Hi = 3 × 5.0 = 15The image is 15 cm high.

CONVEX MIRRORSConvex mirrors curve outwards rather than inwards. If you look througha convex mirror, you have a wider angle of view. Convex mirrors are usedas security mirrors in supermarkets and for safety purposes to assistdrivers leaving driveways with restricted views of the traffic (see figures1.26 and 1.27).

Like the concave mirror, the convex mirror has a focus, but it is a vir-tual focus because it is behind the mirror (see figure 1.28). Incident lightrays parallel to the principal axis are reflected away as if they had comefrom the focus behind the mirror. However, the rays used in the raytracing for the concave mirror can still be used if you keep in mind thatthe focus and the centre of curvature are on the other side of the mirror.

Skillc

hecks

Significant figures(p. 493)

Hi

Ho------- v

u---

Hi

3.0------- 17

25------

23---

23---

SAMPLEPROBLEM 1.4

Solution:

1u--- 1

v--- 1

f---

110------ 1

v--- 1

15------

1v--- 1

15------ 1

10------

2 3–( )30

-----------------

1–30------

v1--- 30

1–------

Inve

stigations

Investigation 1.8Soup spoon

Hi

Ho------- v

u---

Hi

5.0------- 30

10------

eMode

lling

Exploring a concavemirror with aspreadsheet

A convex mirror is a curved mirror with the reflecting surface on the outside of the curve.



The rays from the object spread out or diverge after hittingthe mirror, so it is impossible to form a real image with aconvex mirror.

Figure 1.26 Convex mirrors are used when a wide view of an area is required.

The same formula as was used for concave mirrors (see page 14) alsoworks for convex mirrors, but the focal length is represented by anegative 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 infront of a convex mirror with a focal length of 15 cm.

Ho = 5.0 cm; Hi = ?; u = 10 cm; v = ?; f = −15 cm

Using + =

+ =

⇒ = − −

= (combining fractions using lowest common denominator)

=


⇒ v = −6.0The image is 6.0 cm behind the mirror. It is virtual and upright.

Web

links

Convex mirror applet

eye

convexmirror

object

image

3

1

2convex mirror

F

Figure 1.27 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.

Figure 1.28 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 camefrom the focus.

Inve

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Investigation 1.9Convex mirrors

SAMPLEPROBLEM 1.5

Solution:

1u--- 1

v--- 1

f---

110------ 1

v--- 1

5–------

1v--- 1

15------ 1

10------

−2 −3+( )30

------------------------

5–30------

v1--- 30

5–------



Magnification = =

=

= 0.6⇒ Hi = 0.6 × 5.0

= 3.0The image is 3.0 cm high.

USING CONCAVE MIRRORS IN THE REFLECTING TELESCOPEThe 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 theimage in that the light can spread intocolours when it passes through them(see chapter 3) and large lenses some-times sag when supported at theiredges. Also, large mirrors with onecurved surface are easier to make thanlarge lenses with two curved surfaces.

As shown in figure 1.30, light from a distant stararrives as parallel rays, so the image of the star isformed in a plane at the focus (F) of the concavemirror. A small, tilted plane mirror is placed justin front of the focus and reflects light out of theside of the tube into a lens which reproduces theoriginal parallel rays. This means the image ismagnified. The actual magnification is the ratioof the two focal lengths.

Magnification =

Figure 1.29 The Newtonian telescope, with atube 11.43 cm in diameter

Hi

Ho------- v

u---

Hi

5.0------- 6.0

10-------eM

odelling

Exploring a convexmirror with aspreadsheet

focal length of concave mirrorfocal length of lens

--------------------------------------------------------------------------------------------------------------

Figure 1.30 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.

convex lens

plane mirror

concave mirror

F

parallellightfromdistantstar



As telescopes became morepowerful, that is, able to collect morelight and hence see more stars andmore detail, the Newtonian designbecame cumbersome. The Frenchoptician N. Cassegrain improved thedesign by cutting a hole in the middleof the concave mirror and reflectingthe light through it with a very smallconvex mirror placed in front of the large concave mirror (see figure 1.31).

PROBLEMS WITH CURVED MIRRORS

As discussed on page 11, most small concave mirrors are spherical ratherthan parabolic. In small mirrors, differences between the circle and theparabola are not great. For larger mirrors or when high quality imagesare required, the distortion or spreading of the image produced by aspherical mirror is a problem. This distortion is called

sphericalaberration

. It occurs because a spherical mirror does not reflect parallelrays to a single point. The focus is ‘spread out’, as shown in figure 1.33.

Web

links

Scallop eyes

convexmirror

concavemirror

Figure 1.31 Diagram of a Cassegrain telescope

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 thebrain.

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.

T

concavemirror

retina

AS A MATTER OF FACT

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.



Figure 1.33

(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.

An example of spherical aberration is the pattern of light produced bya ceiling light reflecting from the inside of the coffee cup onto thesurface of the coffee. The full circular shape of the mug spreads thepoint focus out into a curve, called a caustic curve.

parallel rays

parallel rays

(a) Large spherical mirror

(b) Parabolic mirror

Inve

stigations

Investigation 1.10Caustic curve



CHAPTER REVIEW

• Light sources are called luminous objects.Some luminous objects give off light becausethey are hot; these are called incandescentobjects.

• Light travels in straight lines through air at aspeed of 3.0

×

10

8

m s

–1

. Shadows provide evi-dence that light travels in straight lines.

• Modelling light as a pencil-like ray helpsdescribe the reflection of light.

• When light meets a surface the angle of inci-dence equals the angle of reflection. The inci-dent ray, the normal to the surface and thereflected ray all lie in the same plane.

• If the surface is smooth like that of a mirror,the regular reflection enables images to be seenin the mirror.

• The image in a plane mirror is as far behindthe mirror as the object is in front of themirror.

• If a mirror is curved, the properties of theimages formed by the mirror are different fromthe properties of the images in plane mirrors.

• The ray model explains the formation andproperties of images in plane and curved mir-rors.

• The formation of images in concave and convexmirrors can be mathematically modelled with

the equations + =

and

M

=

=

.

Understanding1. 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 toyour eye in seeing an object.

3. Use the ray model and the sources of light torephrase the statements (a) ‘I looked at aflower through the window’ and (b) ‘Iwatched the TV’.

4. Calculate the longest and shortest time for aradio signal travelling at the speed of light to gofrom the Earth to a space probe when the spaceprobe is (a) near Mars and (b) near Neptune.

Radius of Earth’s orbit about the Sun = 1.49 × 1011 mRadius of Mars’s orbit about the Sun = 2.28 × 1011 mRadius of Neptune’s orbit about the Sun = 4.50 × 1012 m

5. Copy the following figure and draw the inci-dent and reflected rays from the two ends ofthe object to the eye. Locate the image.

6. Calculate the angles, a, b and c in the followingfigure.

7. The two arrowed lines in the figures belowrepresent reflected rays. The line AB rep-resents the plane mirror. Locate the imageand the light source in each of the two figures.

8. What type of car does ATOYOT Car Sales sell?

9. A student argues that you cannot photographa virtual image because light rays do not passthrough the space where the image isformed. How would you argue against thisstatement?

10. Sketch the path of each of the rays enteringeach of the pair of joined mirrors in thefollowing figure.

SUMMARY

1u--- 1

v--- 1

f--- H i

Ho--------

vu---

QUESTIONS

object

planemirror

mirror50°

a bc

A

B

A

B



CHAPTER REVIEW

11. Sketch the path of a ray emitted from a pointbetween two parallel mirrors (see the followingfigure).

12. You are walking towards a plane mirror at aspeed of 1.0 m s−1. How fast is your imagewalking? How quickly are you and your imageapproaching each other?

13. Use your understanding of the reflection oflight to explain how full gloss and matt paintsdiffer.

14. Produce a table summarising all the propertiesof the image produced by a concave mirror forthe following object positions: beyond thecentre of curvature (C); at C; between C andthe focus (F); at F; inside F.

15. Use either a scaled drawing or the concavemirror formula to determine the full descrip-tion of the image (that is, size, location, orien-tation 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 alsobe used for convex mirrors with only onechange: making the focal length a negativequantity. Determine the nature of the image ofa 4.0 cm object 15 cm in front of a convexmirror with a focal length of 10 cm.

Application17. Explain how early astronomers knew the Moon

must have a rough surface.

18. Assume that your eyes are located at the top ofyour 1.60 m tall body.(a) What size plane mirror would you need to

see your whole body if you were standing1 m from it?

(b) How far from the floor should the bottomedge of the mirror be placed?

(c) How would your answers to (a) and (b)change if you assumed your eyes were10 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.

19. (a) You are standing 2.0 m in front of a planemirror and you wish to take a sharp photo-graph of yourself in the mirror. At whatdistance do you set the camera lens?

(b) Your friend is standing beside you, 1.0 maway. At what distance do you set the cameralens for a sharp photograph of your friend?

20. Use two or more plane mirrors to produce animage of you that is not laterally inverted; thatis, you see yourself as others see you. Use twopoint sources, marked L and R, at the end of ashort line.

21. Design a system of plane mirrors to enable animmobile, bedridden patient to read a bookplaced on the patient’s chest.

22. Draw three ray diagrams to investigate how thesize of an image in a plane mirror depends onthe distance of the object from the mirror.

23. A plane mirror reflects a ray of light andchanges its direction. The angle of deviation isthe angle between the reflected ray and thecontinuation of the incident ray. How is theangle of deviation related to the glancingangle (that is, the angle between the incidentray and the mirror)?

24. The figure below shows an incident ray (R1)reflected off a mirror (m1). The mirror isrotated (m2) and the reflected ray moves (R2).Use the angles in the diagram to prove that theangle of deflection of the reflected ray equalstwice the angle of rotation of the mirror.

25. Prove that the distance that light travels from asource, P, to another point, Q, via a planemirror is the shortest when the angle of inci-dence 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 concavemirror designed to produce a magnifiedimage. Your face is inside the focus. A comfort-able distance from your face to the mirrorwould 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?

mirrorposition

N2R2

R1

m2

m1m2

m1

N1I

aa


WAVE-LIKE PROPERTIES OF LIGHT22CHAPTER REVIEW

(c) If the height of your face is 16 cm, whatdiameter mirror would you need to seeyour whole face?

28. A dentist asks you to design a concave mirrorto produce a magnified upright image of atooth. The magnification should be about 3with a space of 2 cm between the tooth andthe 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 dis-torted image of the tooth because teethare three-dimensional objects. Why was theimage distorted?

29. A concave mirror with a focal length of 20 cmproduces an image located an infinite distanceaway of an object placed in front of the mirror.Where is the object located?

30. Find the size of the image of the Moon formedby a concave mirror with a focal length of20 cm if it is 3.8 × 105 km away and its radius is1740 km. How could the image be madebrighter? 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 ofthe mirror is removed?

More of a challenge32. When you look into a plane mirror, your left

and right sides appear reversed. This is called‘lateral inversion’. How could you place themirror so that your image is upside down?

33. How can you see raindrops if water is trans-parent?

34. A laser beam is sent from the surface of theEarth to a corner reflector which was left onthe Moon by the Apollo astronauts. Thereflector sends the beam back to Earth in thesame direction. The time of the round trip forthe beam of light is about 2.479 seconds.(a) Calculate the distance of the centre of the

Moon from the centre of the Earth, giventhe following data:speed of light = 2.998 × 108 m s−1

radius of Earth = 6380 kmradius of Moon = 1740 km.

(b) The Earth was continuing to spin while thelaser beam was in transit. How far did thereceiver on the Earth’s surface move inthat time?

(c) What are the implications of these resultsfor the design of this exercise?

35. A corner reflector is made of three plane mir-rors each at right angles to each other, like thecorner of a room. How many images wouldyou see?

36. The smooth surface of a very large lake isslightly curved because the Earth is round. Infact, the surface is a convex mirror with a focallength equal to half the Earth’s radius. If theradius of the Earth is 6380 km and the moon,which has a radius of 1740 km, is 3.8 × 105 kmaway, calculate the location and size of theMoon’s image formed by the lake’s surface.


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Refracting light Wave-like - Jacaranda | Shop · Refracting light Chapter 3 Seeing colours Jac Phys 1 2E - 01 Page 1 Tuesday, October 21, 2003 1:41 PM. ... seen because they reﬂect