Transcript Phys132 Lecture 5 - University of Connecticut

Physics 1402: Lecture 30
Today’s Agenda
• Announcements:
– Midterm 2: Monday Nov. 16 …
– Homework 08: due Friday
• Optics
– Mirrors
– Lenses
– Eye
h
q
q
R-i
h’
o-R
i
o
&
h
i
o
f
h’
The Mirror Equation
• We will now transform the geometric drawings into algebraic
equations:
from triangles,
eliminating a,
a
a
q
g
b
object
h
image
i
o
Now we employ the small angle approximations:
Plugging these back into the above equation relating the angles, we get:
Defining the focal
length f = R/2,
This eqn is known as the mirror eqn. Note that there is no mention of q in
this equation. Therefore, this eqn works for all q, ie we have an image!
Magnification
• We have derived the mirror eqn which determines the image
distance in terms of the object distance and the focal length:
• What about the size of the image?
h
• How is h’ related to h??
• From similar triangles:
q
q
h’
o
Now, we can introduce a sign convention. We can
indicate that this image is inverted if we define its
magnification M as the negative number given by:
i
More Sign Conventions
• Consider an object distance s which is less than the focal
length:
Ray Trace:
• Ray through the center of
the sphere (light blue) is
q
reflected straight back.
h q o
• Ray parallel to axis (red)
i
passes through focal point f.
f
h’
• These rays diverge! ie
these rays look they are
coming from a point behind
the mirror.
• We call this a virtual image, meaning that no light from the object passes
through the image point.
• Proof left to student: This situation is described by the same mirror
equations as long as we take the convention that images behind the mirror
have negative image distances s’. ie:
In this case, i < 0, which leads to M > 0,
indicating that the image is virtual (i<0)
and not inverted (M>0).
Concave-Planar-Convex
• What happens as we change the curvature of the mirror?
IMAGE:
– Plane mirror:
virtual
» R=
upright (non-inverted)
– Convex mirror:
» R<0
q
q
h
h’
o
i
f
IMAGE:
virtual
upright (non-inverted)
Lecture 30, ACT 1
• In order for a real object to create a real, inverted enlarged image,
a) we must use a concave mirror.
b) we must use a convex mirror.
c) neither a concave nor a convex mirror can produce this image.
Mirror – Lens Definitions
•
Some important terminology we introduced last class,
– o = distance from object to mirror (or lens)
– i = distance from mirror to image
o positive, i positive if on same side of mirror as o.
– R = radius of curvature of spherical mirror
– f = focal length, = R/2 for spherical mirrors.
– Concave, Convex, and Spherical mirrors.
– M = magnification, (size of image) / (size of object)
negative means inverted image
a
a
q
object
h
g
b
image
i
o
Lenses
• A lens is a piece of transparent material shaped such that
parallel light rays are refracted towards a point, a focus:
– Convergent Lens
» light moving from air into glass
will move toward the normal
» light moving from glass back into
air will move away from the normal
» real focus
– Divergent Lens
» light moving from air into glass
will move toward the normal
» light moving from glass back into
air will move away from the normal
» virtual focus
Converging Lens Principal Rays
F
Object
P.A.
Image
F
1) Rays parallel to principal axis pass through focal point.
2) Rays through center of lens are not refracted.
3) Rays through F emerge parallel to principal axis.
Image is: real, inverted and enlarged (in this case).
Assumptions:
• monochromatic light incident on a thin lens.
• rays are all “near” the principal axis.
ACT 2: Converging Lens
Which way should you move object so image
is real and diminished?
Demo
Object
(1) Closer to lens
F
(2) Further from lens
(3) Converging lens can’t create real
diminished image.
F
P.A.
The Lens Equation
•
We now derive the lens equation which determines the image distance in terms of the
object distance and the focal length.
– Convergent Lens:
h
i
o
f
h’
Ray Trace:
• Ray through the center of the lens (light blue) passes through undeflected.
• Ray parallel to axis (white) passes through focal point f.
two sets of similar triangles:
eliminating h’/h:
magnification: also same as mirror eqn!!
M < 0 for inverted image.
same as mirror eqn
if we define
i>0
f>0
Summary
• We have derived, in the paraxial (and thin lens) approximation, the
same equations for mirrors and lenses:
when the following sign conventions are used:
Variable
Mirror
Lens
f>0
f<0
concave
convex
converging
diverging
o>0
o<0
real (front)
virtual (back)
real (front)
virtual (back)
i>0
i<0
real (front)
virtual (back)
real (back)
virtual (front)
3 Cases for Converging Lenses
Past 2F
Image
Object
Between
F & 2F
Image
Object
Inside F
Image Object
Inverted
Reduced
Real
This could be used in a
camera. Big object on
small film
Inverted
Enlarged
Real
This could be used as a
projector. Small slide
on big screen
Upright
Enlarged
Virtual
This is a magnifying
glass
Diverging Lens Principal Rays
F
Object
F
P.A.
Image
1) Rays parallel to principal axis pass through focal point.
2) Rays through center of lens are not refracted.
3) Rays toward F emerge parallel to principal axis.
Image is virtual, upright and reduced.
ACT 3: Diverging Lenses
Which way should you move object so image
is real?
F
Demo
Object
1)
2)
3)
F
Closer to lens
Further from lens
Diverging lens can’t create real image.
P.A.
Lecture 30, ACT 4
• A lens is used to image an object on a
screen. The right half of the lens is
covered.
– What is the nature of the image on the
screen?
(a) left half of image disappears
(b) right half of image disappears
(c) entire image reduced in intensity
object
lens
screen
Multiple Lenses
• We determine the effect of a system of lenses by considering the
image of one lens to be the object for the next lens.
-1
0
+1
+2
o1 = +1.5, f1 = +1
\
For the second lens: o2 = +1, f2 = -4
\
+4
f = -4
f = +1
For the first lens:
+3
+5
+6
Multiple Lenses
• Objects of the second lens can be virtual. Let’s move the second lens
closer to the first lens (in fact, to its focus):
-1
+1
0
f = +1
For the first lens:
+2
+3
f = -4
o1 = +1.5, f1 = +1
\
For the second lens:
o2 = -2, f2 = -4
\
Note the negative object distance for the 2nd lens.
+4
+5
+6
Multiple Lenses
• If the two lenses are thin, they can be touching – i.e.
in the same position. We can treat as one lens.
ftotal = ??
?
For the first lens:
o=o1, i1 and f1
For the second lens:
Adding,
As long as,
o2 = -i1, i2=i, f2
The Lens Equation
– Convergent Lens:
h
i
o
f
h’
The Lensmaker’s Formula
• So far, we have treated lenses in terms of their focal lengths.
• How do you make a lens with focal length f ?
• Start with Snell’s Law. Consider a plano-convex lens:
Snell’s Law at the curved surface:
q
b
light ray
q
Assuming small angles,
a
h
air
N
air
The bend-angle b is just given by:
The bend-angle b also defines the focal length f:
The angle q can be written in terms of R, the radius of curvature of the lens :
Putting these last equations together,
More generally…Lensmaker’s Formula
Two curved surfaces…
Two arbitrary indices
of refraction
The complete generalized case…
Note: for one surface Planar,
R > 0 if convex when light hits it
R < 0 if concave when light hits it
Compound Microscope
Objective
(fob< 1cm)
o1
L
i1
feye
I1
h
O
h2
fob
Eyepiece
(feye~5cm)
I2
Magnification:
h1
Refracting Telescope
Objective
(fob~ 250cm)
Eyepiece
(feye~5cm)
fob
Star
i1
q
q
h2
feye
I2
Angular
Magnification:
I1
h1q
q
I2
I1
~fe
~fo
objective
L
eyepiece
The Eye
• What does the eye consist of?
– Sphere (balloon) of water.
- An aperture that controls how much light gets through –
the Iris/pupil
- Bulge at the front – the cornea
- A variable focus lens behind the retina – the lens
- A screen that is hooked up to your brain – the retina
Retina
Cornea
Iris
To brain
Lens