Download 20. Electric Charge, Force, & Field

31. Images & Optical Instruments
1.
2.
3.
4.
Images with Mirrors
Images with Lens
Refraction in Lenses: The Details
Optical Instruments
How does laser surgery provide permanent vision correction?
Ans: Laser light reshapes cornea to adjust the focal point.
Light ray = line (or curve)  wave front.
Geometrical (or ray) optics:
Light ray in homegeneous medium = straight line.
Valid when L >> .
Real image: image location is the point of convergence of actual light rays.
Can be shown on screen.
Virtual image: some or all of the light rays that converge to form the image are
virtual (straight line extension of the actual rays).
Can’t be shown on screen.
31.1. Images with Mirrors
Virtual
Short-cut:
OP = PO
Front-to-back reversal.
Right-to-left-handed coord.
GOT IT? 31.1.
You stand in front of a plane mirror whose top
is at the same height as the top of your head.
Approximately how far down must the mirror
extend for you to see your full image?
Ans. Half your height.
Curved Mirrors
Spherical aberration
equal angles
conic sections:
f=ed
Normal
f
d
Parabolic mirror
Parabola: e = 1
Spherical aberration is small for paraxial rays.
Hubble telescope:
Spherical mirror (upper)
Parabolic mirror (lower)
Tactics 31.1: Paraxial Ray Tracing with Spherical Mirrors
C
1.
2.
3.
4.
F
Ray // axis reflected through focus.
Ray through focus reflected // axis.
Ray striking through mirror mid-point reflects symmetrically.
Ray through mirror center reflects upon itself.
Concave Spherical Mirror
Bear and image are both in front of the mirror.
Convex Mirrors
Convex mirror.
Image always virtual, upright,
and reduced in size.
Wide-angle view.
The Mirror Equation
Shaded triangles are similar:
h
s

 M  Magnification 
h
s
h < 0
s
 h s  f


s
h
f
1

s
positive if
s
always
s, f
same side of outgoing ray
h, h
above axis
f
s  1  1
f s
f
1
1 1 1
 
s s f
f 
1
R
2
Mirror Equation
R = radius of mirror
Table 31.1. Image Formation with Mirrors:
Sign Conventions
Example 31.1. Hubble Space Telescope
During assembly, a technician stood 3.85 m in front of the concave mirror of the HST.
Let the focal length of the telescope be 5.52 m. Find
(a) the location, and
(b)the magnification
of the technician ‘s image.
A technician standing in front of the
Hubble Space Telescope mirror.

1 
 1
s  


 5.52 3.85 
M 
1
1
1
 
3.85 s 5.52
1
 12.7 m
s
12.7

 3.30
s
3.85
( Virtual image; behind mirror )
( Upright; enlarged )
Example 31.2. Jurassic Park
Convex side-view mirror:
Objects in mirror are closer than they seem.
If the curvature radius of the mirror is 12 m and the T. Rex is 9.0 m from the mirror,
by what factor does the dinosaur appear reduced in size?
1 1
1
 
9 s
12 / 2
1 1

s    
9 6
18
2
s
M     5   0.4
5
s
9

1

18
5
(image is upright & smaller)
31.2. Images with Lens
Convex lens
Concave lens
Thin lens: Light rays bend just once going through the lens.
Tactics 31.2. Ray Tracing with Thin Lens
passes thru undeflected
entering // axis
passes thru focus
entering thru center
Lens Images by Ray Tracing
Getting Quantitative: The Lens Equation
positive if
s
always
s, f
same side of outgoing ray
h, h
above axis
Shaded triangles similar:
1 1 1
 
f s s
M
h
s

h
s
 h s  f
s


h
f
s
lens equation
 s  f  s  f  
f2
Table 31.2. Image Formation with Lens:
Sign Convention
GOT IT?. 31.3.
You look through a lens at this page
and see the words enlarged and right-side up.
Is the image you observe real or virtual?
Is the lens concave or convex?
Example 31.3. Fine Print
You ‘re using a magnifying glass (converging lens)
with a 30-cm focal length to read a telephone book.
How far from the page should you hold the lens in
order to see the print enlarged three times?
Image is upright enlarged so it must be virtual.
Hence, h > 0, s < 0.
M
1 1 1
 
f s s
h
 s
3
h
s
1
1
1


30 3 s s
s  30 
2
3
 20 cm
31.3. Refraction in Lenses: The Details
Refraction at a Curved Surface in the Paraxial Approximation
Snell’s law:
n1 sin 1  n2 sin  2

n11  n2 2
paraxial approx.
Green line is tangent to lens surface at point A.
It merges with segment AB in the paraxial approx.
AB
s
AB
T(IBA):   tan  
T(BCI):  2    
s
 AB AB 
 AB AB 
n1 


n

Snell’s law:
2 


R 
s 
 s
 R
T(BOC): 1    
T(BOA):   tan  
T(BAC):   tan  
n1 n2 n2  n1
 
s s
R
AB
R
Example 31.4. Cylindrical Aquarium
An aquarium consists of a thin-walled plastic tube 70 cm in diameter.
For a cat looking directly into the aquarium,
what is the apparent distance to a fish 15 cm from the aquarium wall?
positive if
n1 n2 n2  n1
 
s s
R
s
always
s, f, R
same side of outgoing ray
h, h
above axis
Plastic wall thin  negligible.
nwater = 1.333
1.333 1. 1.  1.333
 
15
s
70 / 2
 0.333 1.333 
s  


15 
 35
Top view
1
 12.6 cm
Lenses, Thick & Thin
n1 n2 n2  n1
 
s s
R
O 1  I 1 = O2  I 2
positive if
LHS (O1  I1 ) :
1 n n 1
 
s1 s1
R1
RHS (O2  I2 ) :
n
1 1 n
 
 s1  t s2
R2
Thin lens (t  0) :
 1
1 1
1 
   n  1  
 R1 R 2 
s s


s
always
s, f, R
same side of outgoing ray
h, h
above axis
s1  0 , R1  0
R2  0
 1
1
1 
  n  1  
 R1 R 2 
f


Lensmaker’s formula
Common Lens Types
Example 31.5. Plano-Convex Lens
Find an expression for the focal length of the plano-convex lens,
given refractive index n and radius R for the curved surface.
 1
1
1 
  n  1  
 R1 R 2 
f


positive if
s
always
s, f, R
same side of outgoing ray
h, h
above axis
Object on left hand side:
1
1 1
  n  1   
f
R 
Object on right hand side:
1
1 
1
  n  1  

f
  R 
f 
R
n 1
same result
Lens Aberrations
Spherical aberration
Stopped down  better focus
Chromatic aberration: cause: n = n( ).
Minimized by using composite lens.
Astigmatism: cause: different R in different direction.
31.4. Optical Instruments
The Eye
Corrective power P = 1 / f.
[P] = diopter = m1
Myopic (nearsighted)
Corrective: Divergent lens.
Hyperopic (farsighted)
Corrective: Convergent lens.
Application: Laser Vision Correction
LASIK procedure
Conceptual Example 31.6. Contact Lens Mix-Up
You and your roommate have gotten your boxes of disposable
contact lenses mixed up.
One box is marked “1.75 diopter”, the other “+2.5 diopter”.
You are farsighted and your roommate is nearsighted.
Which lenses are yours?
Converging lens correct far-sightedness.

f >0

P=1/f >0
Making the Connection
What’s the focal length of your contact?
P
f 
1
 2.5
f
1
m
2.5
 40 cm
Example 31.6. Lost Your Glasses!
You’ve lost your reading glasses;
without them, your eyes can’t focus closer than 70 cm.
Nonprescription reading glasses come in 0.25-diopter increments.
Which glasses should you buy so you can focus at the standard 25-cm near point?
Aim: make object at 25 cm appears to be at 70 cm.
i.e.,
s = 25 cm , s = 70 cm.
P
1
1
1
1 1
 2.57 diopters


 
f
0.25 m 0.70 m
s s
Ans: buy glasses with P = 2.5 diopters
Cameras
Works like the eye.
Magnifiers & Microscopes
Closest distance eye can focus is 25 cm (near point).
Angular magnification :
m


Angle subtended by image

Angle subtended by object at near point 
h/ f
25 cm

h / 25 cm
f
Simple magnifier
mmax  4
s1  fo
s2   f e
from eyepiece
Compound microscope
For
fo , f e
s1  L
L
from objective
For the eyepiece:
me 
Overall magnification:

Mo  
L
f0
25 cm
fe
M  M ome  
L 25 cm
fo f e
Compound
telescope
Telescopes
Refracting telescope
fo
h1 / f e


m

fe

h1 / f o
Object inverted
Extra diverging eyepiece or set of reflecting prisms needed to get upright image.
Reflecting Telescope
Advantages:
1.No chromatic aberration.
2.Mirror (adjustable) can be much larger (~10m)
than lens (~1m).
Only light gathering power is important for
astronomical telescopes.
Cassegrain design
(large telescopes)
Newtonian design
(small telescopes)
Giant Magellan Telescope (2016)