physics 1402: lecture 31 today’s agenda announcements: –midterm 2: monday nov. 16 …...
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Physics 1402: Lecture 31Today’s Agenda
• Announcements:
– Midterm 2: Monday Nov. 16 …
– Homework 08: due Wednesday (after midterm 2)Homework 08: due Wednesday (after midterm 2)
• Optics – Lenses
– Eye
oi
fh’
h
Rh
h’o-R
R-i
o
i
&
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
f > 0f < 0
o > 0o < 0
i > 0i < 0
Mirror
concaveconvex
real (front)virtual (back)
real (front) virtual (back)
Lens
convergingdiverging
real (front)virtual (back)
real (back) virtual (front)
This could be used as a projector. Small slide on big screen
This is a magnifying glass
This could be used in a camera. Big object on small film
Upright
Enlarged
Virtual
Inverted
Enlarged
Real
Inverted
Reduced
Real
Image Object
Inside F
Object
Image
Past 2F
Image
Object
BetweenF & 2F
3 Cases for Converging Lenses
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.
F
F
Object
P.A.
Image is virtual, upright and reduced.
Image
Diverging Lens Principal Rays
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.
For the first lens: o1 = +1.5, f1 = +1
For the second lens: o2 = +1, f2 = -4
f = +1 f = -4
-1 +3+10 +2 +6+5+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):
For the first lens: o1 = +1.5, f1 = +1
For the second lens: o2 = -2, f2 = -4
Note the negative object distance for the 2nd lens.
f = +1 f = -4
-1 +3+10 +2 +6+5+4
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 = ???
Adding,
For the first lens: o=o1, i1 and f1
For the second lens: o2 = -i1, i2=i, f2
As long as,
The Lens Equation
– Convergent Lens:
i
fh’
o
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:
The bend-angle is just given by:
The bend-angle also defines the focal length f:
The angle can be written in terms of R, the radius of curvature of the lens :
Putting these last equations together,
RNair air
h
light ray
Assuming small angles,
More generally…Lensmaker’s Formula
Two curved surfaces…
Two arbitrary indices of refraction
R > 0 if convex when light hits it
R < 0 if concave when light hits it
The complete generalized case…
Note: for one surface Planar,
~fe
I1
eyepiece
I2
~fo
objectiveL
The
EYE
Retina
To brain
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
Cornea
IrisLens
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 cmTherefore 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
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 ?
Cornea LensF < DFN < FF
Near Case
We have f1 = fcornea, f2 = flens
For F to get smaller, so must flens
Smaller f means more curvature (see lensmakers formula)
Cornea LensD = FF
Far Away CaseD
Bonus: Calculate how much the radius of curvature of the lens changes as the eye adjusts from the far to the near point.
Now since,
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 correctionsNear-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 retinaNote: 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 ??
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
Object at Near Point
~f
h
Object just inside Focal Pointof simple magnifier
Define Angular Magnification:
Compound Microscope
o1
h
O
I2h2
feye
h1
I1
i1
Objective(fob< 1cm)
fob
L
Eyepiece(feye~5cm)
Magnification:
Refracting Telescope
Star
feye
I2h2
fob
Objective(fob~ 250cm)
Eyepiece(feye~5cm)
i1
I1h1
AngularMagnification: