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Physics 6C
Geometric Optics
Mirrors and Thin Lenses
We have already learned the basics of Reflection and Refraction.
Reflection: angle of incidence = angle of reflection
Refraction: light bends toward the normal according to Snell’s Law
Now we apply those concepts to some simple types of mirrors and lenses.
We have already learned the basics of Reflection and Refraction:
Reflection - angle of incidence = angle of reflection
Refraction - light bends toward the normal according to Snell’s Law
Now we apply those concepts to some simple types of mirrors and lenses.
Flat Mirror
This is the simplest mirror – a flat reflecting surface. The light rays bounce off and you see an image that seems
to be behind the mirror. This is called a VIRTUAL IMAGE because the light rays do not actually travel behind the
mirror. The image will appear reversed, but will be the same size and the same distance from the mirror. A typical
light ray entering the eye of the viewer is shown.
The object distance is labeled S and the image distance is labeled S’.
Real Object
Virtual Image
S
S’
Spherical Mirrors
For curved mirrors we will assume that the shape is spherical (think of a big shiny ball, and slice off any piece of
that – there’s your spherical mirror). This will make our math relatively simple, with only a couple of formulas. The
hard part will be to get the negative signs correct.
The radius of curvature describes the shape of the mirror. This is the same as the radius of the big shiny ball that
the mirror was cut from. We will have two types of mirrors, depending on which direction they curve:
CONCAVE mirrors curve toward you, and have POSITIVE R (like the inside of the sphere).
CONVEX mirrors curve away from you, and have NEGATIVE R (think of the outside of the ball).
There is a point called the FOCAL POINT which is halfway between the mirror and the center.
C
R
Concave Mirror – R is positive
R
C
Convex Mirror – R is negative
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Here are the rays for a CONCAVE mirror:
Ray 1 through
the center
Focal
Point
Optical Axis
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Here are the rays for a CONCAVE mirror:
Ray 1 through
the center
Ray 1 reflects
directly back
Focal
Point
Optical Axis
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Here are the rays for a CONCAVE mirror:
Optical Axis
Ray 2 through
the focal point
Focal
Point
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Here are the rays for a CONCAVE mirror:
Ray 2 reflects
parallel to axis
Optical Axis
Ray 2 through
the focal point
Focal
Point
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Here are the rays for a CONCAVE mirror:
Focal
Point
Ray 3 comes in
parallel to axis
Optical Axis
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Here are the rays for a CONCAVE mirror:
Ray 3 reflects
through focal point
Focal
Point
Ray 3 comes in
parallel to axis
Optical Axis
We will learn 2 techniques for dealing with mirrors (and lenses):
Graphical – draw the light rays and see where they intersect to find an image.
Formula – use a couple of formulas to locate and describe an image.
First the Graphical Method:
For a spherical mirror there are 3 basic rays that you can draw:
1)
Any ray that goes through the CENTER of the circle reflects directly back toward the light source.
2)
Any ray that goes through the FOCAL POINT is reflected back PARALLEL to the optical axis.
3)
Any ray that starts parallel to the optical axis is reflected back through the focal point. (opposite of 2)
Now here are all 3 rays with an object and its image (at the intersection)
Optical Axis
Image
Object
1
2
3
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
We have a few basic formulas for spherical mirrors.
1) The focal length is half the radius.
f=
R
2
Remember the sign convention – if the mirror is
concave R is positive. If convex, R is negative.
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
We have a few basic formulas for spherical mirrors.
1) The focal length is half the radius.
f=
R
2
2) This formula relates the object (S) and image (S’)
positions to the focal length (f) of the mirror.
1 1 1
= +
f S S'
Here S is always positive for mirrors, and
S’ is positive if the image is on the same
side as the object (a REAL image).
To remember this, just follow the light – a
real (positive) image will have light rays
passing through it.
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
We have a few basic formulas for spherical mirrors.
1) The focal length is half the radius.
f=
R
2
2) This formula relates the object (S) and image (S’)
positions to the focal length (f) of the mirror.
3) The magnification (m) of the image is related to
the relative positions of the object and image.
1 1 1
= +
f S S'
m=−
S'
S
Don’t forget the negative sign in this
formula. The sign of m tells you if the
image is upright (+) or inverted (-)
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
OK, back to the problem. We have some given information:
focal length
R = +0.5m ⇒ f = +0.25m
S = +0.2m
object distance
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
OK, back to the problem. We have some given information:
Now we can use formula 2 to locate the image (S’)
1 1 1
= +
f S S'
1
1
1
=
+ ⇒ S' = −1m
0.25 0.2 S'
focal length
R = +0.5m ⇒ f = +0.25m
S = +0.2m
object distance
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
OK, back to the problem. We have some given information:
Now we can use formula 2 to locate the image (S’)
focal length
R = +0.5m ⇒ f = +0.25m
S = +0.2m
object distance
1 1 1
= +
f S S'
1
1
1
=
+ ⇒ S' = −1m
0.25 0.2 S'
This means the image will be located 1m BEHIND the mirror. This is a VIRTUAL image.
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
OK, back to the problem. We have some given information:
Now we can use formula 2 to locate the image (S’)
focal length
R = +0.5m ⇒ f = +0.25m
S = +0.2m
object distance
1 1 1
= +
f S S'
1
1
1
=
+ ⇒ S' = −1m
0.25 0.2 S'
This means the image will be located 1m BEHIND the mirror. This is a VIRTUAL image.
For the magnification, just use formula 3.
m=−
− 1m
S'
=−
= +5
S
0.2m
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
OK, back to the problem. We have some given information:
Now we can use formula 2 to locate the image (S’)
focal length
R = +0.5m ⇒ f = +0.25m
S = +0.2m
object distance
1 1 1
= +
f S S'
1
1
1
=
+ ⇒ S' = −1m
0.25 0.2 S'
This means the image will be located 1m BEHIND the mirror. This is a VIRTUAL image.
For the magnification, just use formula 3.
m=−
− 1m
S'
=−
= +5
S
0.2m
So the image is upright (+) and 5 times as large as the object.
We could also draw the ray diagram…
Example using the Formula Method:
A concave makeup mirror with radius of curvature 0.5m is held 0.2m from a woman’s face.
Where is her image and how large is it?
Notice the 3 rays in the diagram. They all start at the object and go toward the mirror. Ray 1 through the center is
easy to draw. So is ray 2, which starts out flat, then bounces off the mirror and goes through the focal point (f).
Ray 3 is the tricky one. Since the object is inside the focal point (closer to the mirror, or S<f) we can’t draw the ray
through the focal point. Instead we pretend the ray came from the focal point and passed through the object on its
way to the mirror, then bounced off flat.
The outgoing rays do not intersect! So we have to trace them backwards to find their intersection point behind the
mirror. This is what your brain does for you every time you look in a mirror. The virtual image appears at the point
where the outgoing light rays seem to be coming from.
3
Object
1
2
f
Image
S
S’
Convex Mirrors
These will work the same way as concave, but R and f are negative. Take a look at
where the center of the sphere is – it is behind the mirror. There are no light rays there.
This is why the radius is negative. Because the light rays do not go there.
The 3 typical light rays are shown.
Ray 1 points toward the center and bounces straight back.
Ray 2 starts flat and bounces off as if it is coming from the focal point.
Ray 3 starts toward the focal point and bounces off flat.
object
R
f
C
3
Image (this is a virtual image behind
the mirror, so S’ is negative)
1
2
Convex Mirror – R is negative
SPHERICAL MIRROR EQUATIONS
AND SIGN CONVENTION
Concave Mirror Illustrated
Light In Side
S < 0 Virtual Object
S > 0 Real Object
V
F
C
Light Out Side
S’ > 0 Real Image
C This Side, R > 0
Optic Axis
S’ < 0 Virtual Image
C This Side, R < 0
Definitions: C – Center of Curvature
R – Radius of Curvature
F – Focal Point (Same Side as C)
V – Vertex
Equations: Paraxial Approximation
f=
R
2
m=
y'
S'
=−
y
S
1 1 1
+ =
S S' f
THIN LENS EQUATIONS
AND SIGN CONVENTION
Surface 1
Surface 2
Light Out Side
S < 0 Virtual Object
S’ > 0 Real Image
C1 This Side, R1 > 0
C2 This Side, R2 > 0
Light In Side
S > 0 Real Object
S’ < 0 Virtual Image
C1 This Side, R1 < 0
C2 This Side, R2 < 0
n – Index of Refraction
Definitions: C1 – Center of Curvature, Surface 1
C2 – Center of Curvature, Surface 2
Illustrated Lens is Double Convex Converging
With C1 on the Light Out Side and C2 on the Light In Side
Equations:
 1
1
1
= (n− 1)
−
f
 R1 R 2



1 1 1
+ =
S S' f
m=
y'
S'
=−
y
S
REFRACTION AT SPHERICAL INTERFACE
BETWEEN TWO OPTICAL MATERIALS
Light In Side
S > 0 Real Object
S’ < 0 Virtual Image
C This Side, R < 0
na – Index of
Refraction
Light Out Side
S < 0 Virtual Object
S’ > 0 Real Image
C This Side, R > 0
nb – Index of
Refraction
Illustrated Interface Has C, Center of Curvature,
On The Light Out Side, Thus R > 0
A Flat Interface Has R =
n a n b nb − n a
+
=
S S'
R
∞
m=
n S'
y'
=− a
y
nb S