Notation for Mirrors and Lenses
• The object distance is the distance from the
object to the mirror or lens
– Denoted by p
• The image distance is the distance from the image
to the mirror or lens
– Denoted by q
• The lateral magnification of the mirror or lens is
the ratio of the image height to the object height
– Denoted by M
Images
• Images are always located by extending
diverging rays back to a point at which they
intersect
• Images are located either at a point from
which the rays of light actually diverge or at
a point from which they appear to diverge
Types of Images
• A real image is formed when light rays pass
through and diverge from the image point
– Real images can be displayed on screens
• A virtual image is formed when light rays
do not pass through the image point but
only appear to diverge from that point
– Virtual images cannot be displayed on screens
Images Formed by Flat Mirrors
• Simplest possible
mirror
• Light rays leave the
source and are
reflected from the
mirror
• Point I is called the
image of the object at
point O
• The image is virtual
Images Formed by Flat Mirrors
• One ray starts at point P,
travels to Q and reflects
back on itself
• Another ray follows the
path PR and reflects
according to the law of
reflection
• The triangles PQR and
P’QR are congruent
Lateral Magnification
• Lateral magnification, M, is defined as
Image height h '
M

Object height h
– This is the general magnification for any type of mirror
– It is also valid for images formed by lenses
– Magnification does not always mean bigger, the size
can either increase or decrease
• M less than 1 -> image size decreased
• M greater than 1 -> image size increased
Reversals in a Flat Mirror
• A flat mirror produces
an image that has an
apparent left-right
reversal
– For example, if you
raise your right hand
the image you see
raises its left hand
Spherical Mirrors
• A spherical mirror has the shape of a section of a
sphere
• The mirror focuses incoming parallel rays to a
point
• A concave spherical mirror has the silvered
surface of the mirror on the inner, or concave, side
of the curve
• A convex spherical mirror has the silvered surface
of the mirror on the outer, or convex, side of the
curve
Convex vs Concave
Spherical aberation
No clear focusing of the incident rays
Spherical mirrors
R
R
R
f
2
Spherical Mirrors
    i
     r    i
    
i

S



C
r

P F
 
R
p
d d d
2  
R p q
2 1 1 1
  
R p q f
d
p
  sin   tan  
d
R
  sin   tan  
d
q
d
q
2    
  sin   tan  
Ray Tracing


C
F
• Parallel ray hit the mirror and heads to the focus
• Ray through focus hits the mirror and comes off parallel
• Ray through center of curvature returns to center of
curvature
Ray Tracing


F
C
• Parallel ray hits the mirror and reflects as if it started at the focus
• Ray heading to the focus hits the mirror and comes off parallel
• Ray heading to the center of curvature returns along the same path
Magnification
hi h
M 
ho h
Gain is a better
description since M
might be < 1
h
q

h
p
Or:
h q

h
p
h
q
M  
h
p
Sign Conventions
Example: Find the image
C
F
f = 20 cm
p = 32 cm
h = 5 cm
Images Formed by Refraction
n1 sin 1  n 2 sin 2  n11  n 22
1    
  2  
n1  n 2    n 2  n1  
  sin   tan  
d
p
  sin   tan  
d
R
  sin   tan  
d
q
n1 n 2  n 2  n1 


p q
R
Sign Conventions for Refracting Surfaces
Flat Refracting Surfaces
• If a refracting surface is
flat, then R is infinite
• Then q = -(n2 / n1)p
– The image formed by a
flat refracting surface is
on the same side of the
surface as the object
• A virtual image is
formed
Locating the Image Formed by a Lens
• The lens has an index of refraction
n and two spherical surfaces with
radii of R1 and R2
– R1 is the radius of curvature of the
lens surface that the light of the
object reaches first
– R2 is the radius of curvature of the
other surface
• The object is placed at point O at a
distance of p1 in front of the first
surface
n1 n2 n2  n1
1 n n 1


  
p q
R
p1 q1
R1
p2  q1  t
n1 n2 n2  n1
n
1 1 n





p q
R
p2 q2
R2
 1
1 1
1 

  n  1  

p1 q2
R
R
2 
 1
Small!!
Lens Makers’ Equation
• The focal length of a thin lens is the image
distance that corresponds to an infinite
object distance
– This is the same as for a mirror
• The lens makers’ equation is
 1
1 1
1  1
  (n  1)  

p q
 R1 R2  ƒ
Thin lenses - converging
Thin lenses - diverging
Ray Tracing Rules
• A ray parallel to the
axis exits through
the focal point
• A ray through the
focal point leaves
parallel to the axis
• A ray through the
center is undeviated
Ray Diagram for Converging
Lens, p < f
•
•
•
•
The image is virtual
The image is upright
The image is larger than the object
The image is on the front side of the lens
Ray tracing for a diverging lens
Sign Conventions
Example
A 2 cm tall object is placed 30 cm in front of a diverging
lens with a focal length of –20 cm. Find the location of the
image, its classification, the size of the image, and the
magnification. Construct a ray diagram.
Lateral (Transverse) magnification
h
F1
F2
do
h
m
h
h’
di
But:
h q

h
p
h
q
M 
h
p
Or:
h q
 
h p
Multiple Thin Lenses
• Find the image distance of the first lens.
• Use the image of the first lens as the object
of the second lens.
• Find the new object distance by correcting
with the separation of the lenses.
• Find the image of the second lens.
• Transverse Magnification is the product of
the magnifications of the individual lenses
Example
f1  20.0cm
f2  25.0cm
p  60cm
Find the location and magnification of the final image formed
9.
A spherical convex mirror has a radius of curvature with
a magnitude of 40.0 cm. Determine the position of the virtual
image and the magnification for object distances of (a) 30.0 cm
and (b) 60.0 cm. (c) Are the images upright or inverted?
11.
A concave mirror has a radius of curvature of 60.0 cm.
Calculate the image position and magnification of an object
placed in front of the mirror at distances of (a) 90.0 cm and (b)
20.0 cm. (c) Draw ray diagrams to obtain the image
characteristics in each case.
12.
A concave mirror has a focal length of 40.0 cm.
Determine the object position for which the resulting image is
upright and four times the size of the object.
29.
The left face of a biconvex lens has a radius of curvature of
magnitude 12.0 cm, and the right face has a radius of curvature of
magnitude 18.0 cm. The index of refraction of the glass is 1.44. (a)
Calculate the focal length of the lens. (b) What If? Calculate the
focal length the lens has after is turned around to interchange the
radii of curvature of the two faces.
31.
A thin lens has a focal length of 25.0 cm. Locate and
describe the image when the object is placed (a) 26.0 cm and (b)
24.0 cm in front of the lens.
34.
A person looks at a gem with a jeweler’s loupe—a
converging lens that has a focal length of 12.5 cm. The loupe
forms a virtual image 30.0 cm from the lens. (a) Determine the
magnification. Is the image upright or inverted? (b) Construct a
ray diagram for this arrangement.