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=
CHAPTER� 23� -� MIRRORS AND LENSES
= �� ( MORE GEOMETRIC OPTICS )

MATERIAL� FROM� SECTIONS 23.1, 23.2, 23.3, 23.6

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DEFINITIONS<= span style=3D'mso-spacerun:yes'>� AND� CONVENTIONS -

principal axis

h

h�

s

s�

OBJECT

IMAGE

=	=	=
s� -� object distance (text: notation, p)�� h � object height
s� = -� image distance (text notation, q)�� h= � � image height

n � index of re= fraction of
�� the element (only significant f= or lenses)

=	<= /td>
NOTE: text uses=
p �= object distance
q -= � image distance

=
R �= radius of curvature of a reflecting
�� or refracting surface

R

f� -� focal length of the element

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h

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CONVENTIONS:

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A spherical mir= ror with a radius of curvature, R has a focal length, f given by:

The front of a mirror is the reflecting side. The front side of a mirror is = +

R

C

<= /td>	=
= Concave Spherical Mirror �
The= center of curvature, C is
in front.=

<= /td>
front
surf= ace

R and f are bot= h > 0 ( + )

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=
Convex Spher= ical �Mirror �
The= center of curvature is in back.

R

principal axis

C

R and f are bot= h < 0 ( - )

<= /td>
front

surf= ace

or since R� =3D� 2f:

mirror equation<= /font>

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I� CONCAVE MIRROR� -� f > 0

=
several cases e= xist:

(i)= s > f� (object is outside the focal poi= nt)

and

=
so:= � s� > 0� -�� a real image (projectable) and on the same side as the object
��= �� (the + side for mirrors)

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=
= RAY DIAGRAM� -� Images are formed where multiple rays from the object
converge (real)= or from where they seem to emanate (virtual)

s

prin. axis

O

C

F

PRINCIPAL RA= YS -

1� through C�� straight back

2� parallel to principal axis�� through F (focal point)

3 through F<= span style=3D'mso-spacerun:yes'>��= �� parallel to principal axis

4� on center of mirror�� equal = angle below principal axis

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RAY DIAGRAM<= span style=3D'mso-spacerun:yes'>� -� CONCAVE MIRROR, s > f

s

2

3

F

prin. axis

O

C

I

s�

This shows R= eal Image formation

<= /td>
A p= oint where two rays from a point in the object intersect locate that point

in the image

<= /td>
s� = > 0 meaning the image is real, on the same side as the object and
inverted<= /font>

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(ii) s� <� f� ( object is inside the focal point )

and

s�� <� 0

a� virtual� image

RAY DIAGRAM -

s�

prin. axis

O

C

F

viewer

Image

<= /td>
Ima= ge is virtual, on same side as the object,

upright

<= /td>	<= /td>
BUT = THE RAYS
NEVE= R PASSED

THRO= UGH THE
IMAGE POINT

<= /td>	=
TO = A VIEWER IN FRONT OF THE MIRROR
THE= RAYS �SEEM� TO COME FROM THE
IMAGE POINT

s

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s

prin. axis

O

C

F

=
HOMEWORK - <= /u>�What is s� for

�� (iii)� s� =3D

(iv) s =3D� f

<= /td>
The= focal length can be thought of as the image distance when the object is at=
infinity<= /font>

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II� CONVEX SPHERICAL MIRROR<= br>

R

s

O

F

C<= br>

R < 0, there= fore, f < 0

=
now= : s >0� and f < 0,�� therefore:

s�� <� 0�� whether s > f= � or� s < f

<= /td>
The= image is always virtual for
a convex mirror<= /font>

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Diagram for s &g= t; f

R

Image

O

F

C<= br>

s

<= /td>	=
Vir= tual image, on opposite side of mirror from the object
The= far side of the mirror is the negative side for mirrors;
Upright image

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<= /td>	=
So = for a plane mirror, the image
equ= als the negative of the object
dis= tance. and since s� < 0, the image
is always virtua= l

=
III PLANE MI= RRORS �

�f� =3D� ∞� because� R� =3D� ∞

Now since:

s�� =3D� -s

<= /td>	<= /td>	<= /td>	<= /td>
For= a plane mirror,
reg= ardless the object�s
loc= ation relative to the
pri= nciple axis, there will be a
vir= tual image of it behind the
mirror at s� =3D= -s

s

s�

O

I

front

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MAGNIFICATIO= N� (MIRRORS AND LENSES) -

h�

h

object

image

M� -� linear magnification

=
M > 0� -� image is erect
M &= lt; 0� -� image is inverted

It can be shown= that (for a mirror) :

which can be co= nverted to :

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EXAMPLE = -�� Section 23.2� #19 Serway & Vuille

s

C

image

object

F

s�

<= /td>		=
(a= )	con= cave� ( convex mirrors only produce virtu= al images which can�t
�� be projected )=

<= /td>
(b)= M� =3D� -s� / s� =3D� -5� ( because for a real image s� > 0 )
s� =3D� 5s

s� =3D� s� / 5� =3D� 5 m / 5� =3D� 1 m

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s

C

object

F

s�

=
= CLASS EXERCISE - �For the previous problem= find the focal length
�� o= f the mirror

f� =3D� (5/6) m� =3D� 0.83 m

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THIN� LENSES -

=	=
Alm= ost all of the relationships and conventions used for spherical
mir= rors can be applied to thin lenses the major difference has to do
with location o= f image types

=	<= /td>	<= /td>
MIRRORS� -� REAL IMAGES: SAME SIDE AS OBJECT
�� VIRTUAL IMAGES: OPPOSITE SIDE FROM OBJECT
LEN= SES�� -�� REAL IMAGES: OPPOSITE SIDE FROM OBJECT
�� VIRTUAL IMAGES= : SAME SIDE AS OBJECT

=	=
�� A lens is composed of a refracting material of width, t. The two
sur= faces have radii of curvature, R₁ and R₂. the index of refraction
of = the lens is n. If the lens is surrounded by air ( n_air =3D 1.0) and t is
assumed to be n= egligible, then:

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R₂

image

object

F

F

R₁

The focal lengt= h, f of the lens is defined as:

then we have ( = just as for mirrors ):

lens makers� fo= rmula

lens equation

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CHAPTER 23 - MIRRORS AND LENSES

= ; &n= bsp; ( MORE GEOMETRIC OPTICS )

MATER= IAL FROM SECTIONS 23.1, 23.2, 23.3, 23.6

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DEFINITIONS&n= bsp; AND CONVENTIONS -

s

s’

h

h’

OBJECT

IMAGE

= OPTICAL ELEMENT

(MIRROR OR LENS)

= s - object distance (text: notation, p) = ; &n= bsp; = ; h – object height

s’ - image distance (text notat= ion, q) = ; &n= bsp; = ; h’ – image height

n – index of refraction of

the eleme= nt (only significant for lenses)

R

R – radius of curvature of a reflecting

or refrac= ting surface

f - focal length of the element

principal axis

NOTE: text uses

p – object distance

q - image distance

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= NOTE: THE TEXT EXPLAINS REFLECTION AND REFRACTION AT A

SURFACE AND USING LAWS OF REFELCTION AND REFRACTION AND

THE PARAXIAL APPROXIMATION, THE TEXT DERIVES THE EXPRESSIONS

FOR IMAGE LOCATION

s

s’

h

h’

IMAGE

OBJECT

= CONVENTIONS:

real image - light forming the image passes through = the image point

virtual image – light forming the image only seems to emanate from the

= ; &n= bsp; image point

= NOTE: IT IS POSSIBLE TO HAVE S <= ; 0 (VIRTUAL OBJECT) FOR SYSTEMS

OF TWO OR MORE ELEMENTS

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A spherical mirror with a radius of curvature, R has a focal length, f given= by:

The front of a mirror is the reflecting side. The front side of a mirror is <= span style=3D'font-size:111%'>+

front

surface

C

R

Concave Spherical Mirror –

The center of curvature, C is

in front.

R and f are both > 0 ( + )

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= Convex Spherical –Mirror –

The center of curvature is in back.

R

C

front

surface

R and f are both < 0 ( - )

principal axis

or since R =3D 2f:

mirror equation

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I CONCAVE MIRROR - f > 0

several cases exist:

(i) s > f (object is outsi= de the focal point)

and

= so: s’ > 0 - a real image (projec= table) and on the same side as the object

= ; &n= bsp; = ; &n= bsp; = ; &n= bsp; (the + side for mirrors)

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= RAY DIAGRAM - Images are formed where multiple = rays from the object

converge (real) or from where they seem to emanate (virtual)

C

F

s

O

PRINCIPAL RAYS -

Incident = ; &n= bsp; = ; &n= bsp; Reflected

1 through C = ; &n= bsp; = ; straight back

2 parallel to principal axis = ; &n= bsp; through F (focal point)

3 through F = ; &n= bsp; = ; &n= bsp; parallel to principal axis

4 on center of mirror = ; &n= bsp; equal angle below principal axis

prin. axis

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RAY DIAGRAM - CONCAVE MIRROR, s > f

C

F

s

O

prin. axis

2

I

3

s’

This shows Real Image formation

A point where two rays from a point in the object intersect locate that point

in the image

= s’ > 0 meaning the image is real, on the same side as the object and

inverted

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(ii) s < f ( object is inside the focal point )

and

s’ < 0

a virtual image

RAY DIAGRAM -

C

F

s’

O

prin. axis

s

Image

viewer

= Image is virtual, on same side as the object,

upright

TO A VIEWER IN FRONT OF THE MIRROR

THE RAYS “SEEM” TO COME FROM THE

IMAGE POINT

= BUT THE RAYS

NEVER PASSED

THROUGH THE

IMAGE POINT

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