NOTES
Locating Images
only using the
parallel, focal,
and/or radial
rays.
Real images form on the side of a mirror where the objects are,
and virtual images form on the opposite side.
Mirror Equation and Magnification
(f = r/2)
•  s is positive if the object is in front of the mirror (real object)
•  s is negative if it is in back of the mirror (virtual object)
•  s’ is positive if the image is in front of the mirror (real image)
•  s’ is negative if it is in back of the mirror (virtual image)
•  m is positive if image and object have the same orientation (upright)
•  m is negative if they have opposite orientation (inverted)
•  f and r are positive if center of curvature in front of mirror (concave)
•  f and r are negative if it is in back of the mirror (convex)
Summary of Lens (RED) and Mirror (Blue) Notations
1/s + 1/s’ = 1/f
(magnification m) m = - s’/ s inverted image
m = + s’/ s erect image
+s object front distance +s’ image back distance
-s’ image front distance
+s object front distance +s’ image front distance)
-s’ image back distance
f focal length
f >0
(convex Lens)
(concave mirror)
f<0
(concave Lens)
(convex mirror)
LENS REAL INVERTED IMAGE
1/2F + 1/2F =1/F
(1/s + 1/s’ = 1/F)
M= -s’/s = -1
M = -1
Refraction at a Single Surface
Derivation:
A point object O is placed on the central axis of a
convex refracting surface. The center of curvature of
the surface is at C. It is easy to see (for small angles)
But
s
S’
Setting
for parallel rays,
the focal length is
f (and r)>0 for convex surface,
f (and r)<0 for concave surface
•  Parallel ray refracts through the
focal point.
•  A ray through the focal point
refracts parallel to the central axis.
•  A ray through the center of
curvature refracts straight.
NOTES
Thin Lens Formulas
Relabeling,
LENS MAKERS FORMULA
1/f = [n2/n1-1] [1/r1 -1/r2]
n2/n1 = n in books notation (bad)!
n2 inside lens
n1 the medium
DEMONSTRATION
7A-31
UNDER WATER AIR LENS DEMONSTRATION
Derive the Lens Equation
(<0)
But
  True for thin lens and paraxial
rays.
  magnification m = h’/h = - q/p
Thin Lenses Nomenclature
•  A lens is a piece of transparent material with two
refracting surfaces whose central axes coincide. A
lens is thin if its thickness is small compared to all
other lengths (s, s’, radii of curvature).
f>0
Convergent lens
•  Net convex – thicker in the middle
•  Parallel rays converge to real focus.
r1>0
r2<0
•  f > 0
f<0
Divergent lens
•  Net concave – thinner in the middle
•  Parallel rays diverge from virtual focus.
•  f < 0
r1<0
r2>0
Signs in the Lens Equation for Thin Lenses
•  p is positive for real object
•  p is negative for virtual object
•  q is positive for real image
•  q is negative for virtual image
•  m is positive if image is upright
•  m is negative if image is inverted
•  f is positive if converging lens
•  f is negative if diverging lens
Properties of Images - Summary
For converging lenses ( f > 0):
•  If the object is inside the focal point, the image is virtual (q < 0),
enlarged, has the same orientation, and farther from the lens.
•  If the object is outside the focal point, the image is real (q > 0),
reduced or enlarged (depending on the object distance), inverted,
and farther from or closer to the lens on the other side.
•  If the object is at the focal point, no image is formed.
For diverging lenses ( f < 0):
•  The image is always virtual (q < 0), reduced, has the same
orientation, and closer to the lens.
Power of a lens = 1/f (m-1) (diopters or D)
DOCCAM 2
2F---------------F------------0---------------F--------------2F
F AND 2F FOR LENSES
The Eye
≈ 2.5 cm
f depends on p
p ↓ => f↓ to
keep q at ≈
2.5 cm
NOTES
Lenses in Combination a Microscope
First lens:
Second lens:
( < f2 here)
What if p2> f2 ?
Total transverse magnification
In this example, p1> 0, q1> 0,
p2> 0, q2< 0
Corrective Lenses (Eyeball radius = 2.5 cm)
Hyperopia: A person Can’t focus near.
has near point = 75 cm
Correct to 25 cm by conv. lens
Use a lens of +2.67D
Myopia(nearsighted)
e.g., far point = 40 cm
Can’t focus far.
Diopters in meters
Corrected by diverging lens
Use a lens of -2.5D
Magnifying Lens Including Eye
An object is placed near the focal point of a magnifying lens. The
angle subtended by the image is tanθ = y/f . Without the lens, the
largest angle subtended by the object is achieved when the object
is placed at the near point, tanθ’ = y/xnp.
θ’
θ
xnp
For small angles, the
angular magnification
f
Compound Microscope
A microscope consists of two converging lenses: an objective
(the front lens) and an eyepiece.
An object is placed near the first focal point of the
objective. The separation of the lenses is adjusted so
that the image produced by the objective is formed
just inside the first focal point of the eyepiece. The
lateral magnification of the objective is
The eyepiece angular magnification (eye near point = xnp )
The overall magnifying
power is defined as
Astronomical Telescopes
Refractor Telescope
  Same combination (except image at ∞)
as compound microscope: Objective
creates a real image which allows the
eyepiece to magnify.
 The angular magnification M of the
telescope is defined as θe /θo
Aberrations
Cameras, …
Chromatic Aberration
correct
nblue > nred
Parabolic
mirrror
Mirror Telescope has No Chromatic
and Spherical aberration
Large
telescopes, …
Reflector Telescope
  No chromatic aberration
  Large mirrors can be made
 (large amount of light gathered)
  Easier to support
  View center blocked off
Whipple Telescopes
(segmented lens)
NOTES
Physics 241 Extra Quiz
An object 2 cm tall is 15 cm in front (i.e., left) of a
concave (diverging) lens of focal length −10 cm.
Which of the following is a correct description of
its image?
a) The image is enlarged.
b) The image is real.
c) The image is upright.
d) The image is in back (right) of the lens.
e) None of the above is correct.
Physics 241 Extra Quiz 3
An object 2 cm tall is 15 cm in front (i.e., left) of a
convex (converging) lens of focal length 10 cm.
Which of the following is a correct description of
its image?
a) The image is diminished.
b) The image is inverted.
c) The image is virtual.
d) The image is in front (left) of the lens.
e) None of the above is correct.