Lecture 4
Physics 1502: Lecture 31
Today’s Agenda
• Announcements:
– Midterm 2: Monday Nov. 16 …
– Homework 08: due Wednesday (after midterm 2)
• Optics
– Lenses
– Eye
h
θ
θ
R
R-i
h’
o-R
i
o
&
h
i
o
f
h’
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Lecture 4
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
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Lecture 4
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.
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
+4
+5
+6
f = -4
f = +1
For the first lens:
+3
o1 = +1.5, f1 = +1
∴
For the second lens: o2 = +1, f2 = -4
∴
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Lecture 4
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
+4
+5
+6
f = -4
o1 = +1.5, f1 = +1
∴
For the second lens:
o2 = -2, f2 = -4
∴
Note the negative object distance for the 2nd lens.
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:
o2 = -i1, i2=i, f2
Adding,
As long as,
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Lecture 4
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Lecture 4
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:
Assuming small angles,
θ
light ray
h
θ
R
air
β
N
α
air
The bend-angle β is just given by:
The bend-angle β also defines the focal length f:
The angle θ can be written in terms of R, the radius of curvature of the lens :
Putting these last equations together,
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Lecture 4
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
e
h
T E
EY
I2
I1
~fe
~fo
objective
L
eyepiece
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Lecture 4
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
The Eye
• The “Normal Eye”
– Far Point ≡ distance that relaxed eye can focus onto retina
=∞
– Near Point ≡ closest distance that can be focused on to the retina
= 25 cm
Therefore the normal eye acts as a lens with a focal length which
can vary from 2.5 cm (the eye diameter) to 2.3 cm which allows
objects from 25 cm → ∞ to be focused on the retina!
2.5cm
25cm
this is called “accommodation”
Diopter: 1/f Eye = 40 diopters, accommodates by about 10%, or 4 diopters
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Lecture 4
Lecture 31, ACT 1
When your eye adjusts to read versus see far objects,
its muscles adjust so that the lens bulges and
elongates. To read a book do we want a bulged lens or
an elongated lens ?
D
Far Away Case
Cornea Lens
D = FF
Near Case
Cornea Lens
F<D
FN < FF
Now since,
We have f1 = fcornea, f2 = flens
For F to get smaller, so must flens
Smaller f means more curvature (see lensmakers formula)
Bonus: Calculate how much the radius of curvature of the lens changes as
the eye adjusts from the far to the near point.
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Lecture 4
Getting Old
• As you age, the lens loses its ability to change its shape.
• It gets stuck in its relaxed position, the far point.
• Thus the eye is now just an unadjustable lens. Objects at different
distances will focus at different places.
• Only objects at infinity will focus on the retina.
2.5cm
25cm
This is called presbyopia, it is not necessarily
“farsightedness”.
An intuitive way to view eye corrections
Near-sighted eye is elongated, image forms in front of retina
Add diverging lens, image forms on retina
Far-sighted eye is short, image forms behind retina
Add converging lens, image forms on retina
Note: for old age (presbyopia), this sort of correction can only make one
point in focus. If your relaxed eye naturally focuses either at infinity (for
driving) or the near point (reading) then you only need one lens. Otherwise
bifocals are needed. Could you design multifocals ??
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Lecture 4
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Lecture 4
Magnification
• Our sense of the size of an object is determined by the size of
image on the retina.
– Consequently, the relevant magnification factor of a lens is
just the ratio of the angular size with the lens to the angular
size without the lens.
Lnp
h
α
h
β
~f
Object at Near Point
Object just inside Focal Point
of simple magnifier
Define Angular Magnification:
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Lecture 4
Compound Microscope
Objective
(fob< 1cm)
fob
o1
Eyepiece
(feye~5cm)
L
feye
i1
I1
h
h1
O
h2
I2
Magnification:
Refracting Telescope
Objective
(fob~ 250cm)
Eyepiece
(feye~5cm)
fob
Star
i1
I1
θο
θο
h2
feye
h1θ
θ
I2
Angular
Magnification:
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