Transcript Chapter 12 - GEOCITIES.ws

Physics Beyond 2000
Chapter 12
Optical Instruments
Geometric Optics
• In this chapter, the lenses and mirrors have
dimension much longer than the wavelength
of light.
• Effect of diffraction can be ignored.
• Light is regarded as ray.
http://webphysics.davidson.edu/physletprob/ch18_v4_physlets/optics4/default.html
http://www.phy.ntnu.edu.tw/demolab/index.html
Reflection
• Laws of reflection.
http://www.netzmedien.de/software/download/java/brechung/
Plane mirror
• What are the properties of the image?
http://www.continental.clara.net/physics/lt31.htm
Note that we cannot capture a virtual image on a
screen.
Locate a virtual image
• Method of no parallax
Use a long search pin to locate the image behind the
mirror.
In front of the mirror, view the image in the mirror and
the search pin.
search pin
image
plane mirror
object
Locate a virtual image
• Method of no parallax
If the search pin is at the exact position of the image, the
image in the mirror and the search pin always coincide
even if we change the angle of view.
search pin
eye
image in
the mirror
eye
Locate a real image
• The method of no parallax can be applied to
locate the position of real images.
search pin
O
I
real image
Rotation of a plane mirror
Normal 1
Normal 2
Reflected ray 1
Fixed incident ray
θ
Reflected ray 2
2θ
Mirror 1
Mirror 2
Rotate the plane mirror from position 1 to position 2
by an angle θ. The reflected ray will turn through 2θ.
Rotating a plane mirror
• Light-beam galvanometer.
A moving mirror
position 2
position 1
image 2
fixed object
2v
v
If the plane mirror moves at a speed v,
the image moves at speed 2v.
image 1
Concave mirrors
• http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/
lens_e.html
Spherical aberration
• If the aperture of the mirror is large, the
reflected rays do not all passes through the
focus.
• This is called spherical aberration.
Spherical aberration
• It can be corrected by using a parabolic
mirror.
Focal length f and
radius of curvature R
Show that r = 2f for small angles.
paraxial ray
θ
θ
h
2θ
θ
C
F
r
f
Images for concave mirror
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
Example 1
• The rays from the sun are parallel and the
image of the sun is on the focal plane.
C
F
h
f θ
θ
Mirror formula
1 1 1
 
u v f
for small angles
θ
α
O
θ
γ
β
C
I
F
r
u
v
h
Mirror formula
1 1 1
 
u v f
for small angles
Real-is-positive convention:
Nature of
object/image
Real
Object distance
u
positive
Image distance
v
positive
Virtual
negative
negative
Mirror formula
1 1 1
 
u v f
for small angles
Real-is-positive convention:
Nature of mirror
focal length f
concave
positive
convex
negative
Linear magnification m
height
m
height
of
of
image v

object u
I
O
C
u
F
v
Example 2
• Justify the nature of the image from the sign
of image distance v.
Variation of image with
object distance
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
Variation of magnification with
object distance
f
m
u f
Convex mirror
1 1 1
 
u v f
Nature of
mirror
convex
for small angles
focal
length f
negative
object
distance u
positive
image
distance v
negative
(virtual)
Convex mirror
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
http://www.iln.net/html_p/c/453262/453270/453373/454123/56652_2079292.asp
Example 3
O
I
u
v
I
F
Measure the focal length of
concave mirror
Method A
•Object at infinity.
•Image is at the focal plane.
•Measure the distance between the mirror and the screen.
C
θ
θ
F
h
f
Measure the focal length of
concave mirror
Method B
•Object at the radius of curvature.
•Image is at the radius of curvature.
•Use the method of no parallax to locate the image.
•Adjust the position of the object so that its image is
coincide with the object.
•Measure the distance between the object and the mirror.
http://www.usafa.af.mil/dfp/physics/webphysics/Physlet_examples/concave_mirror_f.html
Measure the focal length of
concave mirror
Method C
•Object at different positions to produce real images.
•Images are captured by a screen.
1
v
1 1 1
 
u v
f
1
f
0
1
f
1
u
Measure the focal length of
concave mirror
Method D
•Object at different positions to produce real images.
•Images are captured by a screen.
•Calculate the linear magnification m.
m
1
m  .v  1
f
slope =
0
-1
v
1
f
Measure the focal length fm
of a convex mirror
• It is not possible to capture a virtual image on a screen.
• Put a converging lens of focal length flens in front of the
convex mirror.
• Adjust the position of the object so that a real image is at
the same position as the object.
O
P
Q
s
I
lens
flens
C
2.fm
mirror
Measure the focal length fm
of a convex mirror
• Measure s, the separation between the lens and the
convex lens.
• 2  focal length of the convex mirror is flens – s.
O
P
Q
s
I
lens
flens
C
2.fm
mirror
Refraction
http://www.fed.cuhk.edu.hk/sci_lab/download/project/Lightrefraction/LightRefract.html
http://www.netzmedien.de/software/download/java/brechung/
sin 1 1 c1 n2

 
sin  2 2 c2 n1
Medium 1:
1, 1, c1 and n1
1
2
Medium 2:
2, 2, c2 and n2
Refraction
Snell’s law:
n1.sin1 = n2.sin2
Medium 1:
1, 1, c1 and n1
1
2
Medium 2:
2, 2, c2 and n2
Refraction
If n2 > n1,
then medium 2 is an optically denser medium
and medium 1 is an optically less dense medium
Medium 1:
1, 1, c1 and n1
1
2
Medium 2:
2, 2, c2 and n2
Total internal reflection
This occurs when light travels from an optically denser
medium to an optically less dense medium and the angle of
incidence > critical angle c.
refraction with 1 < c.
critical case with 2 = c.
medium 1
medium 2
1
2
3
total internal
reflection
with 3 > c.
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/light/flashLight.html
Total internal reflection
n2
The critical angle c is given by sin c 
n1
refraction with 1 < c.
critical case with 2 = c.
medium 1 of n1
medium 2 of n2 
1
2
3
total internal
reflection
with 3 > c.
http://www.continental.clara.net/physics/lt23.htm
Fish-eye view
• http://www.fed.cuhk.edu.hk/sci_lab/ntnujav
a/fishEye/fishEye.html
Example 4
• The critical angle of glass with n = 1.5 is
about 42o in air.
• It depends also on the medium in which the
glass is immersed.
Reflecting prism
• Angle of incidence = 45o > Critical angle = 42o.
• Total internal reflection occurs inside the glass
prism.
• The glass prism can be used as a reflecting mirror.
45o
45o o
45
45o
Optical fibre
• There is total internal reflection inside the
optical fibre.
• Light is guided to travel in the optical fibre.
Real depth and apparent depth
• The image I is displaced upwards relative to
the object O.
air
medium with
refractive index
n
B
C
apparent
depth
I
real depth
O
Real depth and apparent depth
real depth
n
apparent depth
air
medium with
refractive index
n
for small angles.
B
C
apparent
depth
I
real depth
O
Real depth and apparent depth
Where would be the image I
if we are inside the medium?
Suppose that the angles
are small.
O
air
medium with
refractive index
n
B
C
Measure the refractive index
of glass
Find the real depth and apparent depth from h1 and h2.
eye
travelling
microscope
h2
h1
h
h
h
glass block
I
O1
O1
O2
real depth = h2
apparent depth = h2 – h2
Rectangular glass block
The incident ray and the emergent ray are parallel.
w. sin( i  r )
The lateral displacement is
d
cos r
incident ray
w
i
r
r
i
d
emergent ray
Prism
Find the angle of deviation D in terms of angles of
incidence (1and 2) and angles of refraction (1 and 2).
D = (1 - 1)+(2 - 2)
A
1
1
2
D
2
Prism
Find the refracting angle A of the prism in terms of the
angles of incidence and angles of refraction.
A = 1+ 2
A
1
1
2
D
2
Prism
The angle of deviation is a minimum Dmin when the
light ray is symmetrical.
Find the refractive index n of the glass prism.
n
A
1
1
2
D
2
sin
1
( A  Dmin )
2
A
sin
2
Small-angled prism
• For a prism with small refracting angle A.
• The angle of deviation
D = (n - 1).A
• The angle of deviation is independent of the
angle of incidence.
A
D
Convex lenses
The focal lengths on both sides are equal.
F
f
F
f
F’
F’
Aberration of lens
• Spherical aberration.
• Parallel incident rays far from the centre do
not meet at the same focus as paraxial rays.
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/thickLens/thickLens.html
Aberration of lens
• Chromatic aberration.
• Violent light bends more than red light in glass.
 fviolet < fred
white parallel light
Location of image for
convex lens
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
Ray diagrams.
Lens formula
1 1 1
 
u v f
for thin lens.
Real-is-positive convention:
Nature of
object/image
Real
Object distance
u
positive
Image distance
v
positive
Virtual
negative
negative
Lens formula
1 1 1
 
u v f
for thin lens.
Real-is-positive convention:
Nature of lens
focal length f
convex
positive
concave
negative
Lens formula: proof
BP
The angle of deviation D = (n – 1).A 
f
where A is the refracting angle
Note that for small angle prism (thin lens), D is
independent of the angle of incidence.
B
D
P
f
F
Lens formula: proof
Suppose that there is a real image.
Prove that
1 1 1
 
u v
f
B

O
u
F’
for thin lens.
D
P
f

F
I
v
Interchange of locations of
object and real image
I
O
u
v
v
u
1 1 1
 
u v f
I
O
Object-image distance
for real image
• To produce a real image, the object-image
distance d must be longer than or equal to
2.f.
• Prove it.
I
O
d
Thin lenses in contact
• Let f1 be the focal length of the first thin lens and
f2 be the focal length of the second thin lens .
• When they are in contact, the combined focal length
f is given by
1 1 1
 
f
f1 f 2
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/thinLens/thinLens.html
Example 5
• The image formed by the first lens is the
object of the second lens.
The converging power of a lens
• Definition of converging power
1
P=
f
Unit: dipotre (D)
• For two thin lenses in contact, the combined power is
D1 + D2.
Concave lens
• For a real object, virtual image is always
formed.  v is negative.
• Focal length is negative.
Example 6
I1 is the virtual object of the concave lens.
u is negative 
The concave lens produces a real image I2.
v is positive.
u
I1
v
convex
lens
concave
lens
I2
Measure the focal length
of convex lens
• Method A.
Object at infinity  Image is at the focal
plane.
focal plane
parallel rays from
distant object
f
I
Measure the focal length
of convex lens
Method B.
Place the convex lens on a plane mirror.
Adjust the position of the object so that it coincides
with the image. This is method no parallax.
eye
object
O
image
convex lens
plane mirror
Measure the focal length
of convex lens
Method B.
The distance between the object/image and the lens = f.
I
O
f
Measure the focal length
of convex lens
• Method C.
• Produce different real images.
• Measure the object distance u and the real image
distance v.
1
v
1
f
1
f
1
u
Measure the focal length
of convex lens
• Method D.
Without changing the positions of the object and the image,
find the two possible positions of the lens.
Measure a and d.and find f from 1  4d
f
object
1st position
of lens
d 2  a2
2nd position
of lens
O
d
a
Screen
position
of image
Measure the focal length
of concave lens
• Use another convex lens to help producing a
virtual object for the concave lens.
• Use the lens formula to calculate the focal length
of the concave lens.
u
v
O
I1
I2
Optical instruments
• Human eye: a convex lens with variable
focal length.
• Far point of a normal eye = infinity.
• Near point of a normal eye = 25 cm.
• Least distance of distinct vision = 25 cm.
Short-sightedness
• A short-sighted eye can focus objects in the
range from 25 cm to 200 cm.
• The far point of the short-sighted eye is 200
cm
• Find the focal length of the spectacle to
correct the defect.
• Wearing a pair of spectacles, what is the
new near point?
Long-sightedness
• A long-sighted eye can focus objects in the
range from 200 cm to infinity.
• The near point of the long-sighted eye is
200 cm
• Find the focal length of the spectacle to
correct the defect.
• Wearing a pair of spectacles, what is the
new far point?
Visual angle
• Visual angle  of an object is the angle subtended
by the object at the eye.
• The bigger the visual angle, the bigger the
apparent size of the object.
• Most optical instruments are designed to magnify
the visual angle.
object

eye
Angular magnification M
It is used to measure the magnification of an optical
instrument,

M

•  is the visual angle of the final image
•  is the visual angle of the object
Note that the visual angles are usually small so   tan 
 sin  and   tan   sin .
Normal adjustment
• An optical instrument is in normal adjustment
when it forms the final image at a position
which the user expects to see.
• Telescope: final image at infinity (far point).
• Magnifying glass: final image at 25 cm (near
point).
• Microscope: final image at 25 cm (near point).
Magnifying glass
• A magnifying glass is a convex lens.
• It produces an enlarged virtual image.
Magnifying glass
• Without the magnifying glass, the largest visual
angle of the object is  with the object at the least
distance of distinct vision D = 25 cm.
h
object
h
  tan  
D

eye
D
visual angle without optical instrument
Magnifying glass
• With the magnifying glass in normal adjustment,
the final image is also at D.
image
h

object u
v=D
h
  tan  
u
eye
visual angle with the optical instrument
Magnifying glass
Apply lens formula,
With the visual angles,
1
1
1


u D f
h

D
h

u
 D
 M   1
 f
(1)
(2)
(3)
the angular magnification
of a magnifying glass
Compound microscope
• It is used to view small objects.
• It consists of two convex lenses.
• The objective lens and the eye-piece. Both are of
short focal lengths.
eye
object
objective lens
eye-piece
History of compound microscope:
http://www.utmem.edu/~thjones/hist/hist_mic.htm
Compound microscope
• The objective lens produces a magnified real
image. The object is placed near the focus of the
objective lens.
• This image is the object of the eye-piece.
1st image
eye
object
objective lens
eye-piece
Compound microscope
• The eye-piece is a magnifying glass.
• It produces a magnified virtual image at D = 25
cm from the eye-piece.
1st image
eye
object
final
image
objective lens
eye-piece
D
Note that the final image is an inverted image.
Compound microscope
h

D
object

D
h
h
object
eye
1st image

h
final
h2 objective lens
image
D
eye
1
eye-piece
h2

D
Compound microscope
The angular magnification M of a compound microscope
is
 h2 h2 h1
M    .  me .mo
 h h1 h
where me is the linear magnification of the eye-piece
and mo is the linear magnification of the objective
M can be increased by using lenses of short focal lengths.
Example 7
• The angular magnification = 5.5
Refracting telescope
• It is used to view distant objects e.g. stars.
• Two convex lenses.
• The objective lens: Pointing to the object, with
very long focal length.
• The eyepiece: A magnifying glass. Its focal length
is short
eye
Objective lens
Eyepiece
Refracting telescope
• The object is at infinity. The incident rays
are parallel
• The objective lens produces a real image I1
on its focal plane.
fo
I1
Objective lens
real image
eye
Eyepiece
Refracting telescope
• The first image I1 is the object of the
eyepiece.
• The eyepiece produces a virtual image at
infinity. This is the normal adjustment.
fe
fo
I1
Objective lens
Image at infinity
real image
eye
Eyepiece
Refracting telescope
In normal adjustment, the angular magnification is
 f0
M 
 fe
Refracting telescope
• The length of the refractive telescope is fo +
fe
• The image is inverted.
fe
fo
I1
Objective lens
Image at infinity
real image
eye
Eyepiece
Refracting telescope
• The aperture of the objective lens is large
– to collect more light.
– to reduce the diffraction effect.
Example 8
• Note the focal lengths of the lenses.
Eye ring
• Eye ring is the position of the eye, at which
most light enters the eye when using an
optical instrument.
• The image is brightest when the eye is at the
eye ring.
Eye ring
All the light collected by the objective lens passes through
the position of the eye ring.
All the light enters your eye if you place your eye at the
eye ring.
Locating the eye ring
• As the light comes from the objective lens, you may take
the objective lens as the object.
• The position of the eye ring is the position of the image.
• Find the position of the eye ring from lens formula.
1 1 1
 
f u v
where u is the distance from the
objective lens to the eyepiece,
f is the focal length of the eyepiece
and
v is the distance from the eye ring
to the eyepiece.
Locating the eye ring
The position of the eye ring is
L. f e
d
L  fe
Reflecting telescope
• It is similar to a refracting telescope.
• Light is collected by a concave mirror.
plane mirror
F
objective
(concave mirror)
F’
eye piece
eye
Reflecting telescope
fo
• The angular magnification is M 
fe
plane mirror
F
objective
(concave mirror)
F’
eye piece
eye
Reflecting telescope
• Advantages of using a reflecting telescope:
The mirror reduces less light intensity than a
lens.
There is not any problem of chromatic
aberration and spherical aberration.
It is easier to produce a large mirror than a
large lens.
Example 9
• Locating the eye ring.
Spectrometer
• It is used for spectral analysis with the aids of a
diffraction grating or a glass prism.
• It consists of three parts: collimator, diffraction
grating/glass prism and a telescope.
Spectrometer
• Collimator: Place the source S near the slit.
The collimator produces a parallel beam of
light.
collimator
S
Spectrometer
• Diffraction grating: Use a diffraction grating to
produce a spectrum of fringes.
• The diffraction grating is on a turntable to measure
the angles of bright fringes.
m=2
diffraction grating
m=1
collimator
S
m=0
turntable
m=1
m=2
Spectrometer
• Diffraction grating: Use a diffraction grating to
produce a spectrum of fringes.
• The diffraction grating is on a turntable to measure
the angles of diffraction.
m=2
diffraction grating
m=1
collimator
S
m=0
turntable
m=1
m=2
Spectrometer
• Telescope: It can be rotated to any angle to
view the diffraction spectrum.
eye
telescope
diffraction grating
m=1
collimator
S
m=0
turntable
m=1
m=2
Spectrometer
Telescope: the image is formed at the position of
cross-wire.
objective lens
eye piece
light from
diffraction grating
eye
cross-wire
Spectrometer
• It is necessary to make a horizontal turntable.
Adjust the levelling screws and use a spirit
level to check the turntable.
spirit level
turntable
levelling screw