Transcript p>|f

mirrors and lenses
PHY232
Remco Zegers
[email protected]
Room W109 – cyclotron building
http://www.nscl.msu.edu/~zegers/phy232.html
an important point
 objects do not emit rays of light that get ‘seen’ by your
eye. Light (from a bulb or the sun) gets reflected off the
object towards your eye.
PHY232 - Remco Zegers - Mirrors and lenses
2
we saw…
 that light can be reflected or refracted at boundaries
between material with a different index of refraction.
 by shaping the surfaces of the boundaries we can make
devices that can focus or otherwise alter an image.
 Here we focus on mirrors and lenses for which the
properties can be described well by a few equations.
PHY232 - Remco Zegers - Mirrors and lenses
3
the flat mirror
p
 in the previous chapter we
already saw flat mirrors.
 The distance from the object to
the mirror the object distance p
 The distance from the image to
the mirror is the image distance q
 in case of a flat mirror, an
observer sees a virtual image,
meaning that the rays do not
actually come from it.
 the image size (h’ ) is the same as
the object size (h), meaning that
the magnification h’/h=1
 the image is not inverted
q
NOTE: a virtual image
cannot be projected on
a screen but is ‘visible’ by
the eye or another optical
instrument.
PHY232 - Remco Zegers - Mirrors and lenses
4
question
 You are standing in front (say 1 m) of a mirror that is less high than
your height. Is there a chance that you can still see your complete
image?
 a) yes b) no
PHY232 - Remco Zegers - Mirrors and lenses
5
ray diagrams
 to understand the properties of optical elements we use
ray diagrams, in which we draw the most important
elements and parameters to understand the elements
h
h’
p
q
PHY232 - Remco Zegers - Mirrors and lenses
6
concave mirrors
M
F
C
C: center of mirror curvature
F: focal point
a light ray passing through the center of curvature will be
reflected back upon itself because it strikes the mirror
normally to the surface.
a light ray traveling parallel to the central axis of the mirror
will be reflected to the focal point F, with FM=CM/2
The distance FM is called the focal length f.
PHY232 - Remco Zegers - Mirrors and lenses
7
concave mirrors: an object outside F
O
I
F
step 1: draw the ray from the top of the object parallel to the central
axis and its reflection (through F).
step 2: draw the ray from the top of the object through F and its
reflection (parallel to the central axis)
the image of the top of the object is located where the reflected
rays meet
step 3: note that a ray from the bottom of the object just reflects back.
construct the image I
PHY232 - Remco Zegers - Mirrors and lenses
8
concave mirrors: an object outside F
O
I
F
The image is:
a) inverted (upside down)
b) real (light rays pass through it)
c) smaller than the object
PHY232 - Remco Zegers - Mirrors and lenses
9
concave mirrors: an object outside F
O
I
F
distance object-mirror: p
distance image-mirror: q
distance focal point-mirror: f
mirror equation: 1/p + 1/q = 1/f
given p,f this equation can be used to calculate q
magnification: M=-q/p
can be used to calculate magnification.
• if negative: the image is inverted
• if smaller than 1, object is demagnified
PHY232 - Remco Zegers - Mirrors and lenses
10
example
 An object is placed 12 cm in front of a a concave mirror
with focal length 5 cm. What are:
 a) the location of the image
 b) the magnification
PHY232 - Remco Zegers - Mirrors and lenses
11
concave mirrors: an object inside F
the image is:
a) not inverted
b) virtual
c) magnified
F
O
I
step 1: draw the ray from the top of the object parallel to the central
axis and its reflection (through F).
step 2: draw the ray from the top of the object through F and its
reflection (parallel to the central axis)
the image of the top of the object is located where the reflected rays
meet: in this you must draw virtual rays on the other side of the lens
step 3: note that a ray from the bottom of the object just reflects back.
create the image
PHY232 - Remco Zegers - Mirrors and lenses
12
concave mirrors: an object inside F
the image is:
a) not inverted
b) virtual
c) magnified
F
O
I
The lens equation and equation for magnification are still
valid. However, since the image is now on the other
side of the mirror, its sign should be negative
PHY232 - Remco Zegers - Mirrors and lenses
13
example
 an object is placed 2 cm in front of a lens with a focal
length of 5 cm. What are the a) image distance and b) the
magnification?
PHY232 - Remco Zegers - Mirrors and lenses
14
demo: the virtual pig
PHY232 - Remco Zegers - Mirrors and lenses
15
convex mirrors: an object outside F (p>|f|)
O
I
F
F is now located on the other
side of the mirror
step 1: draw the ray from the top of the object parallel to the central
axis and its reflection (through F).
step 2: draw the ray from the top of the object through F and its
reflection (parallel to the central axis)
the image of the top of the object is located where the reflected
rays meet
step 3: note that a ray from the bottom of the object just reflects back.
construct the image I
PHY232 - Remco Zegers - Mirrors and lenses
16
convex mirrors: an object outside F (p>|f|)
O
I
F
F is now located on the other
side of the mirror
the image is:
a) not inverted
b) virtual
c) demagnified
The lens/mirror equation and equation for magnification are
still valid. However, since the image and focal point are now
on the other side of the mirror, their signs should be negative
PHY232 - Remco Zegers - Mirrors and lenses
17
example
 an object with a height of 3 cm is placed 6 cm in front of a
convex mirror, with f=-3 cm. What are a) the image
distance and b) the magnification?
PHY232 - Remco Zegers - Mirrors and lenses
18
convex mirrors with p < |f|
 the situation is exactly the same as for the situation with
p > |f|. The demagnification will be different though…
F
O
PHY232 - Remco Zegers - Mirrors and lenses
I
F
19
Mirrors: an overview
type
p?
image
image
direction
M
q
f
concave p>f
real
inverted
|M|>0 M -
+
+
concave p<f
virtual
not
inverted
|M|>1 M +
-
+
convex
p>|f|
virtual
not
inverted
|M|<1 M +
-
-
convex
p<|f|
virtual
not
inverted
|M|<1 M +
-
-
 mirror equation 1/p + 1/q = 1/f
 f=R/2 where R is the radius of the mirror
 magnification: M=-q/p
PHY232 - Remco Zegers - Mirrors and lenses
20
lon-capa
 now do problems 7,8,11 of lon-capa 8
PHY232 - Remco Zegers - Mirrors and lenses
21
Lenses
 Lenses function by refracting light at their surfaces
 Their action depends on
 radii of the curvatures of both surfaces
 the refractive index of the lens
 converging (positive lenses) have positive focal length
and are always thickest in the center
+
 diverging (negative lenses) have negative focal length
used in
and are thickest at the edges
-
drawings
PHY232 - Remco Zegers - Mirrors and lenses
22
lensmakers equation
object
R2
1
2
R1
f: focal length of lens
n: refractive index of lens
R1 radius of front surface
R2 radius of back surface
R2 is negative if the center of the circle is on the left of
curvature 2 of the lens
R1 is positive if the center of the circle is on the right of
curvature 1 of the lens
if the lens is not in air then (nlens-nmedium)
PHY232 - Remco Zegers - Mirrors and lenses
23
example
object
R2
1
2
R1
 Given R1=10 cm and R2=5
cm, what is the focal length?
The lens is made of glass
(n=1.5)
PHY232 - Remco Zegers - Mirrors and lenses
24
example 2
object
R1
1
2
R2
 Given R1=5 cm and R2=10
cm, what is the focal length?
The lens is made of glass
(n=1.5)
PHY232 - Remco Zegers - Mirrors and lenses
25
example 3
object
R1
1
2
R2
 Given R1=5 cm and R2=,
what is the focal length? The
lens is made of glass (n=1.5)
PHY232 - Remco Zegers - Mirrors and lenses
26
question
 A person is trying to make a lens but decides to make
both surfaces flat, resulting in essentially a flat piece of
glass on both sides. What is the focal length of this ‘lens’?
 a) infinity
 b) 0
 c) cannot say, depends on the index of refraction n
PHY232 - Remco Zegers - Mirrors and lenses
27
converging lens p>f
I
O
F
F
+
1) A ray parallel to the central axis will be bend through the focal point
2) A ray through the center of the lens will continue unperturbed
3) A ray through the focal point of the lens will be bend parallel to the
central axis
4) the image is located at the crossing of the above 3 rays (you need
just 2 of them).
A real inverted image is created. The magnification
depends on p: |M| can be <1, 1 or >1
PHY232 - Remco Zegers - Mirrors and lenses
28
lens equation
I
O
F
F
+
The equation that connects object distance p, image
distance q and focal length f is (just like for mirrors):
1/p + 1/q = 1/f
Similarly for the magnification:
M=-q/p
q is positive if the image is on the opposite side of the lens as the object
NOTE THAT THIS IS DIFFERENT THAN THE CASE FOR MIRRORS
PHY232 - Remco Zegers - Mirrors and lenses
29
example
 an object is put 20 cm in front of a positive lens, with focal
length of 12 cm. a) What is the image distance q? b) What
is the magnification?
PHY232 - Remco Zegers - Mirrors and lenses
30
converging lens p<f
I
F
O
F
+
1) A ray parallel to the central axis will be bend through the focal point
2) A ray through the center of the lens will continue unperturbed
3) A ray through the focal point of the lens will be bend parallel to the
central axis
4) the image is located at the crossing of the above 3 rays (you need
just 2 of them).
A virtual non-inverted image is created.
Magnification >1
PHY232 - Remco Zegers - Mirrors and lenses
31
example
 an object is put 2 cm in front of a positive lens, with focal
length of 3 cm. a) What is the image distance q? b) What
is the magnification?
PHY232 - Remco Zegers - Mirrors and lenses
32
question
 An object is placed in front of a converging (positive) lens with the
object distance larger than the focal distance. An image is created
on a screen on the other side of the lens. Then, the lower half of the
lens is covered with a piece of wood. Which of the following is true:
 a) the image on the screen will become less bright only
 b) half of the image on the screen will disappear only
 c) half of the image will disappear and the remainder of the image
will become less bright.
PHY232 - Remco Zegers - Mirrors and lenses
33
NOT CORRECT
PHY232 - Remco Zegers - Mirrors and lenses
34
diverging lens p>|f|
O
F
F
I
-
1) A ray parallel to the central axis will be bend so that the ray passes
through the focal point IN FRONT of the lens
2) A ray through the center of the lens will continue unperturbed
3) A ray aimed at the focal point on the other side of the lens will be
bent parallel to the central axis
4) the image is located at the crossing of the above 3 rays (you need
just 2 of them).
A virtual non-inverted image is created.
The magnification |M|<1
PHY232 - Remco Zegers - Mirrors and lenses
35
example
 an object is put 5 cm in front of a negative lens, with focal
length of -3 cm. a) What is the image distance q? b) What
is the magnification?
PHY232 - Remco Zegers - Mirrors and lenses
36
diverging lens p<|f|
F
O
F
I
-
1) A ray parallel to the central axis will be bend so that the ray passes
through the focal point IN FRONT of the lens
2) A ray through the center of the lens will continue unperturbed
3) A ray aimed at the focal point on the other side of the lens will be
bent parallel to the central axis
4) the image is located at the crossing of the above 3 rays (you need
just 2 of them).
A virtual non-inverted image is created.
The magnification |M|<1 similar to case with p>|f|
PHY232 - Remco Zegers - Mirrors and lenses
37
example
 an object is put 2 cm in front of a negative lens, with focal
length of -3 cm. a) What is the image distance q? b) What
is the magnification?
PHY232 - Remco Zegers - Mirrors and lenses
38
lenses, an overview
type
p?
image
image
direction
M
q
f
converging
p>f
real
inverted
|M|>0 M -
+
+
converging
p<f
virtual
not
inverted
|M|>1 M +
-
+
diverging
p>|f|
virtual
not
inverted
|M|<1 M +
-
-
diverging
p<|f|
virtual
not
inverted
|M|<1 M +
-
-
 mirror equation 1/p + 1/q = 1/f
 magnification: M=-q/p
 lens makers equation: 1/f=(n-1)(1/R1-1/R2)
PHY232 - Remco Zegers - Mirrors and lenses
39
spherical aberrations: Hubble space telescope
spherical aberrations are due to the rays hitting the lens
at different locations have a different focal point
perfect
distorted
example: Hubble
before
PHY232 - Remco Zegers - Mirrors and lenses
after
correction
40
chromatic aberrations
Chromatic aberrations are due to light of different
wavelengths having a different index of refraction
Can be corrected by combining lenses/mirrors
If n varies with wavelength, the focal length f
changes with wavelength
PHY232 - Remco Zegers - Mirrors and lenses
41
two lenses
 an object, 1 cm high, is placed 5 cm in front of a
converging mirror with a focal length of 3 cm. This setup is
placed in front of a diverging mirror with a focal length of
–5 cm. The distance between the two lenses is 10 cm.
Where is the image located, and what are its properties?
+
3cm
5 cm
5cm
15 cm
PHY232 - Remco Zegers - Mirrors and lenses
42
lon-capa
 now do problems 9,10,12 of lon-capa 8
PHY232 - Remco Zegers - Mirrors and lenses
43