Curved Mirrors

1
Curved Mirrors
2
Concave and Convex Mirrors
Concave and convex mirrors are curved mirrors
similar to portions of a sphere.
light rays
light rays
Concave mirrors reflect light from their inner
surface, like the inside of a spoon.
Convex mirrors reflect light from their outer
surface, like the outside of a spoon.
3
Concave Mirrors

• Concave mirrors are approximately spherical and
  have a principal axis that goes through the
  center, C, of the imagined sphere and ends at the
  point at the center of the mirror, A. The
  principal axis is perpendicular to the surface of
  the mirror at A.

• CA is the radius of the sphere,or the radius of
  curvature of the mirror, R .
• Halfway between C and A is the focal point of
  the mirror, F. This is the point where rays
  parallel to the principal axis will converge when
  reflected off the mirror.
• The length of FA is the focal length, f.
• The focal length is half of the radius of the
  sphere (proven on next slide).

4
r 2 f
To prove that the radius of curvature of a
concave mirror is twice its focal length, first
construct a tangent line at the point of
incidence. The normal is perpendicular to the
tangent and goes through the center, C. Here, i
r ?. By alt. int. angles the angle at C is
also ?, and a 2 ß. s is the arc length from
the principle axis to the pt. of incidence. Now
imagine a sphere centered at F with radius f.
If the incident ray is close to the principle
axis, the arc length of the new sphere is about
the same as s. From s r ?, we have s r ß
and s ? f a 2 f ß. Thus, r ß ? 2 f ß, and
r 2 f.
tangent line
?
?
s
?
?


f
C
F
r
5
Focusing Light with Concave Mirrors
Light rays parallel to the principal axis will be
reflected through the focus (disregarding
spherical aberration, explained on next slide.)


In reverse, light rays passing through the focus
will be reflected parallel to the principal axis,
as in a flood light.
Concave mirrors can form both real and virtual
images, depending on where the object is located,
as will be shown in upcoming slides.
6
Spherical Aberration
F


C
Spherical Mirror
Parabolic Mirror
Only parallel rays close to the principal axis of
a spherical mirror will converge at the focal
point. Rays farther away will converge at a
point closer to the mirror. The image formed by a
large spherical mirror will be a disk, not a
point. This is known as spherical aberration.
Parabolic mirrors dont have spherical
aberration. They are used to focus rays from
stars in a telescope. They can also be used in
flashlights and headlights since a light source
placed at their focal point will reflect light in
parallel beams. However, perfectly parabolic
mirrors are hard to make and slight errors could
lead to spherical aberration.
7
Spherical vs. Parabolic Mirrors
Parallel rays converge at the focal point of a
spherical mirror only if they are close to the
principal axis. The image formed in a large
spherical mirror is a disk, not a point
(spherical aberration).
Parabolic mirrors have no spherical aberration.
The mirror focuses all parallel rays at the focal
point. That is why they are used in telescopes
and light beams like flashlights and car
headlights.
SPHERICAL vs. PARABOLIC
8
Concave Mirrors Object beyond C
The image formed when an object is placed beyond
C is located between C and F. It is a real,
inverted image that is smaller in size than the
object.
object


C
F
9
Concave Mirrors Object between C and F
The image formed when an object is placed between
C and F is located beyond C. It is a real,
inverted image that is larger in size than the
object.
object


C
F
10
Concave Mirrors Object in front of F
The image formed when an object is placed in
front of F is located behind the mirror. It is a
virtual, upright image that is larger in size
than the object. It is virtual since it is
formed only where light rays seem to be diverging
from.
object


C
F
11
Concave Mirrors Object at C or F
What happens when an object is placed at C?


What happens when an object is placed at F?
12
Break
13
Convex Mirrors

• A convex mirror has the same basic properties as
  a concave mirror but its focus and center are
  located behind the mirror.
• This means a convex mirror has a negative focal
  length (used later in the mirror equation).
• Light rays reflected from convex mirrors always
  diverge, so only virtual images will be formed.

• Rays parallel to the principal axis will reflect
  as if coming from the focus behind the mirror.
• Rays approaching the mirror on a path toward F
  will reflect parallel to the principal axis.

14
Convex Mirror Diagram
The image formed by a convex mirror no matter
where the object is placed will be virtual,
upright, and smaller than the object. As the
object is moved closer to the mirror, the image
will approach the size of the object.
object
image


C
F
15
Mirror/Lens Equation Derivation
From ?PCO, ? ? ?, so 2? 2? 2?. From
?PCO, ? 2? ? , so -? -2? - ?.
Adding equations yields 2? - ? ?.
P
From s r ?, we have s r ß, s ? di a, and
s ? di a (for rays close to the principle
axis). Thus
?
object
s
?
?
?
?

T
O
C
image
di
do
(cont.)
16
Mirror/Lens Equation Derivation (cont.)
From the last slide, ? s / r, ? ? s / d0 , ?
? s / di , and 2 ß - ? ?.
Substituting into the
last equation yields
P
2s
s
s
-
?
object

r
s
?
di
do
?
?
?
1
2
1



r
T
do
di
O
C
image
1
1
2


2f
do
di
di
1
1
1


f
do
di
do
The last equation applies to convex and concave
mirrors, as well as to lenses, provided a sign
convention is adhered to.
17
Mirror Sign Convention
f focal length di image distance do object
distance
1
1
1


f
do
di
for real image - for virtual image
di
for concave mirrors - for convex mirrors
f
18
Magnification
By definition,
m magnification hi image height (negative
means inverted) ho object height
Magnification is simply the ratio of image height
to object height. A positive magnification means
an upright image.
19
Magnification Identity
To derive One hits the mirror on the axis. The
incident and reflected rays each make angle ?
relative to the axis. A second ray is drawn
through the center and is reflected back on top
of itself (since a radius is always perpendicular
to an tangent line of a
circle). The intersection of the reflected
rays determines the location of the tip of the
image. Our result follows from
similar triangles, with the negative sign a
consequence of our sign convention. (In this
picture hi is negative and di is positive.)
object
?
ho

C
image, height hi
di
do
20
Mirror Equation Sample Problem
Suppose AllStar, who is 3 and a half feet tall,
stands 27 feet in front of a concave mirror with
a radius of curvature of 20 feet. Where will his
image be reflected and what will its size be?
di
15.88 feet
hi
-2.06 feet
21
Mirror Equation Sample Problem 2
Casey decides to join in the fun and she finds a
convex mirror to stand in front of. She sees her
image reflected 7 feet behind the mirror which
has a focal length of 11 feet. Her image is 1
foot tall. Where is she standing and how tall is
she?
do
19.25 feet
ho
2.75 feet