Ch. 23
Mirrors and lenses
Conceptual questons #13, 15
Problems# 5, 7, 13, 15, 21, 25, 27, 29,
31, 37, 39, 41
Images appear in mirrors where rays of light
actually intersect or where they appear to
originate.
Real image – light actually passes through the
image point
Virtual image – light appears to come from the
image point.
Different situations will yield real or virtual
images.
See pictures on page 755.
The images are virtual.
For a flat mirror:
Magnification:
M = image height/object height = h’/h
M=1
The image appears as far behind the mirror as
the object is in front.
The image is unmagnified (M=1), virtual and
upright (not flipped upside down)
Mirrors are commonly made of a piece of glass
with a thin reflective metal coating on the back
side. (sometimes a layer of silver)
If you scratch away the metal coating you just
have piece of glass.
Some light reflects off the front surface of the
glass.
The rest of the light passes through the glass
before reflecting off the metal and going back
through the mirror.
See how a rearview mirror works with a day and
night setting.
Spherical Mirrors
Spherical mirrors have the shape of a segment
of a sphere.
Concave mirror - the reflective surface is on the
inner side of the mirror.
see picture on page 757
• The principle axis is a line drawn through the
center of the mirror.
• Use approximation that we only use light rays
that make small angles with the principle axis.
• Rays that make large angle with principle axis
have effect called spherical aberration.
• Produces blurred image because the rays are
not reflected to the exact same place.
Spherical Mirrors
M = h’/h = -q/p
q = image length
p = object length
Mirror equation
1/p + 1/q = 2/R
R = radius of curvature of mirror
When p is very large(approaches infinity)
1/p = 0.
Then the image is at q = R/2
The incoming rays are essentially parallel and are
focused at the image location. (fig. 23.10)
This point is called the focal point, f
f = R/2
Mirror equation becomes;
1/p + 1/q = 1/f
M = h’/h = -q/p
Example of flat mirror. For flat mirror, R = infinity
So 1/f = 0
Thus p = q
Convex mirror: Silvered so the light is reflected from
the outer side of the mirror. Also called diverging
mirror. Mirror equation is true for these also.
Sign conventions for mirrors
See page 760.
If p, q, and f are in front of the mirror, they are
positive. If in back, negative.
If the image is upright, positive image height.
If image is inverted, negative image height.
The big difference between concave and convex
mirrors is that concave mirrors have a positive
focal length. Convex mirrors have a negative
focal length.
Ray diagrams for mirrors
Rules for drawing ray diagrams. These are all
products of the rule of angle of incidence is
equal to angle of reflection.
1) Rays drawn parallel to principle axis are
reflected through the focal point.
2) Rays drawn through focal point are reflected
parallel to principle axis.
3) Ray drawn through center of curvature (point
C) get reflected back on itself.
4) Also can see that a ray incident at the center
of the mirror is reflected at an equal angle.
See pictures of ray diagrams and images on
page 761.
Work example 23.2
more examples:
http://www.phys.ufl.edu/~phy3054/light/mirr
or/applets/Welcome.html
http://webphysics.davidson.edu/Applets/Optics
/prb4.html
Images can be formed by refraction
Consider two mediums with different indices of
refraction. Rays originating from the object are
refracted and converge at the image point.
See figure 23.15.
Equation to find where the image is:
n1
p
n2
q
n2 n1
R
Magnification is still M = h’/h
becomes M = h’/h = -(n1q)/(n2p)
See sign conventions in table 23.2
do example 23.5
Flat refracting surface
For a flat refraction surface, all you need to do is
use: n1 n2 n2 n1
and set R =
p
q
then n1/p = -n2/q
q = -(n2/n1)p
do example 23.6
R
Thin Lenses
Lens = piece clear material (typically glass) that
is shaped so that the two refracting surfaces
are either a segment of a sphere or a plane.
Examples in fig. 23.21
Two types: Converging and diverging lenses.
Converging lenses are thicker in the center
than at the edge.
Diverging lenses are thinner in the middle.
Equation for lenses is the same as equation for mirrors.
Group of rays parallel to principle axis
converge at the focal length after passing
through the lens.
The focal length is the image distance that
corresponds to an infinite object length.
Lenses have two focal points, one on each side.
Converging lenses cause parallel rays to converge.
Diverging lenses cause parallel rays to diverge
(spread out)
see fig. 23.22
again M = h’/h = -q/p
Table 23.3 has sign conventions for lenses.
Converging lens has + focal length
Diverging lens has – focal length
Thin lenses obey the equation: 1/p + 1/q = 1/f
The focal length depends on the index of
refraction and the radii of the two sides of the
lens.
Lens maker’s formula:
1
f
1
(n 1)
R1
1
R2
Ray diagrams for lenses
1) If ray comes from parallel to axis, after being
refracted, it will pass through the focal point.
2) If ray is drawn through the center of the lens,
it keeps going straight.
3) If ray is drawn through the focal on the same
side of the object, after going through the
lens, it will be parallel to the principle axis.
See fig. 23.25
http://www.phy.ntnu.edu.tw/ntnujava/index.php?t
opic=48
see example 23.7 and 23.8
Combinations of lenses. When you have
multiple lenses, you can work each lens one at a
time. The image formed by the first lens
becomes the object for the second lens.
Some telescopes and microscopes work this way.
Lens and mirror aberrations
We used assumptions that the incoming rays
make small angles with the principle axis. This
is not always true in life. When the angles are
large, the rays are not focused to the exact
same point, and the image is fuzzy.
Two types of aberrations:
spherical and chromatic
Spherical aberration
Results from fact that the focal points of rays far
from principle axis are different from the rays
that are near the principle axis. Has to do with
the spherical shape of the lens or mirror.
See fig. 23.30
Cameras can take care of this by having an
adjustable aperture that can reduce this problem.
By narrowing the aperture, you eliminate the rays
that are far from the principle axis.
Spherical aberration in mirrors can be reduced or
eliminated by using a parabolic mirror. All rays
that are parallel to the principle axis are reflected
through the focal point.
Good for telescopes, where the object is far
enough that the rays are essentially parallel to
the axis.
Also used in flashlights. Put the bulb at the focus
and the light is parallel beam is reflected off the
parabolic surface.
High quality parabolic mirrors are expensive.
Chromatic aberration.
White light is made up of light of different colors
(wavelengths). In chapter 22 we saw that index
of refraction depends on wavelength. Therefore,
when different colors pass through a lens, they
are focused to different points. See fig. 23.31.
Chromatic aberration can be reduced by using
the right combination of converging and diverging
lenses.