PES 2130 Fall 2014, Spendier
Lecture 20/Page 1
Lecture today: Chapter 34 Images
1) Spherical Mirror Formula
2) Thin lenses
3) Images formed by thin lenses
4) Thin lens Formula
Announcements:
- HW 8 given out
- Exam 2 graded
- Next Monday Nov 10, Prof. Tolya Pinchuk will fill in for me
Last lecture:
Images, mirror (flat, concave, convex)
Flat mirror: p = -i
p…object distance
i… image distance (i is positive for a real image and negative for virtual image)
C… center of curvature
F…focal point
r…radius of curvature
f…focal length
Ray tracing
Images: real, virtual, upright, inverted
Locating Images by Drawing Rays (Ray tracing diagrams)
We can graphically locate the image of any off-axis point of the object by drawing a ray
diagram with any two of four special rays through the point:
1. A ray that is initially parallel to the central axis reflects through the focal point F
2. A ray that reflects from the mirror after passing through the focal point emerges
parallel to the central axis.
3. A ray that reflects from the mirror after passing through the center of curvature C
returns along itself.
4. A ray that reflects from the mirror at point c is reflected symmetrically about that axis
PES 2130 Fall 2014, Spendier
Lecture 20/Page 2
1) Spherical Mirror formula
r is positive for a concave mirror and negative for a convex mirror
Equation relating the object distance p, the image distance i, and the focal length f :
This equation applies to any concave, convex, or plane mirror. For a convex or plane
mirror, only a virtual image (-i) can be formed, regardless of the object’s location on the
central axis.
Plane mirror:
Magnification:
The size of an object or image, as measured perpendicular to the mirror’s central axis, is
called the object or image height. Let h represent the height of the object, and h’ the
height of the image. Then the ratio h’/h is called the lateral magnification m produced
by the mirror. However, by convention, the lateral magnification always includes a plus
sign when the image orientation is that of the object and a minus sign when the image
orientation is opposite that of the object. For this reason, we write the formula for m as
Note: For a plane mirror, for which i = -p, we have m = +1.
The magnification of 1 means that the image is the same size as the object.
PES 2130 Fall 2014, Spendier
Example 1:
A 1.6 m Santa checks himself for soot, using hid reflection in
a shiny silvered Christmas tree ornament 0.750 m away. The
diameter of the ornament is 7.20 cm. Where and how tall is the
image of Santa? Is it upright (erect) or inverted?
Lecture 20/Page 3
PES 2130 Fall 2014, Spendier
Lecture 20/Page 4
Read section on “Spherical Refracting surfaces” on your own
2) Thin Lenses
Today, we will discuss two types of lenses:
A lens can produce an image of an object
only because the lens can bend light
rays, but it can bend light rays only if its
index of refraction differs from that of the
surrounding medium.
What happens to light when it passes through such lenses?
Convex lens:
According to Snell’s law, a light ray is bend at the surface (air into glass and then glass
into air). When we ray trace, we typically ignore what is going on inside the lens and just
show the end effect:
PES 2130 Fall 2014, Spendier
Lecture 20/Page 5
Rays initially parallel to the central axis of a converging lens are made to converge to a
real focal point F2 by the lens. The lens is thinner than drawn, with a width like that of
the vertical line through it.
Here r1 is the radius of curvature of the lens surface nearer the object and r2 is that of the
other surface.
For a lens which is curved the same way C1 and C2 are equidistant from the center of the
lens.
A convex lens is sometimes called a converging lens, r > 0 (radius of curvature)
Concave lens:
For a diverging lens the focal point is on the other side.
Parallel rays are made to diverge by a diverging lens. Extensions of the diverging rays
pass through a virtual focal point F2.
Concave lenses are also called diverging lenses, r < 0
PES 2130 Fall 2014, Spendier
Lecture 20/Page 6
Sign convention:
The sign conventions for lenses are opposite those for mirrors
- image is real ( i > 0) when on the opposite side of the lens from the object.
- image is virtual (i < 0) when on the same side of the object.
- radius of curvature, negative if C is on the opposite side of outgoing light
- radius of curvature, positive if C is on the same side of outgoing light
- convex lens: - focal length is positive ( f > 0 )
-concave lens: focal length is negative (f < 0)
Focal length of a lens (Lens maker equation):
How a lens focuses depends and its index of refraction n, and its curvatures C1 and C2.
For a thin lens in air it can be shown that
Example 2:
Suppose the absolute values of both radii of curvature of a bi-convex lens equal to 10.0
cm and the index of refraction is n = 1.52. What is the focal length f of the lens?
C1 on outgoing side of lens ==> r1 = + 0.1 m
C2 on incoming side of lens ==> r2 = - 0.1 m
 1
1
1 
 1.52  1 

  10.4 / m
f
 0.1  0.1 
f 
1
 0.0962 m  9.62cm
10.4 / m
PES 2130 Fall 2014, Spendier
Lecture 20/Page 7
3) Images formed by thin lenses
Once we have the focal length of a lens, we can figure out what kind of image it forms.
As we did with mirrors, there are 4 cases. I will show 1 case, you should proof to yourself
the other 3 cases.
i) Converging lens, p < |f|
ii) Converging lens, p = |f|
iii) converging lens, p > |f|
iv) Diverging lens (all p)
PES 2130 Fall 2014, Spendier
Lecture 20/Page 8
4) Images formed by thin lenses
Formula relating the image distance, object distance, and focal length of thin lenses:
1 1 1
 
f
p i
f ... focal length (sign depends on type of lens)
p ... object distance (always positive)
i ... image distance (+opposite side of the lens from the object, - same side than object)
Remember: The sign conventions are different for mirrors and lenses!
Summary:
Mirrors:
1 1 1 2
  
p i f r
Thin lenses:
1 1
1 1 1
    n  1   
p i f
 r1 r 
Magnification formula for mirrors and thin lenses
i h'
m 
p h
h' ... image height
h ... object height
Proofs for these equations: see last section of chapter 34 in your book.