chapter 12 optical instruments physics beyond 2000
TRANSCRIPT
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Chapter 12
Optical Instruments
Physics Beyond 2000
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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://www.phy.ntnu.edu.tw/demolab/index.html
http://webphysics.davidson.edu/physletprob/ch18_v4_physlets/optics4/default.html
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Reflection
• Laws of reflection.
http://www.netzmedien.de/software/download/java/brechung/
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Plane mirror
• What are the properties of the image?
Note that we cannot capture a virtual image on a screen.
http://www.continental.clara.net/physics/lt31.htm
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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
object
plane mirror
image
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Locate a virtual image• Method of no parallaxIf 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
image in the mirroreye eye
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Locate a real image
• The method of no parallax can be applied to locate the position of real images.
O I
search pin
real image
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Rotation of a plane mirror
Rotate the plane mirror from position 1 to position 2 by an angle θ. The reflected ray will turn through 2θ.
Normal 2
Reflected ray 2
2θ
θ
Fixed incident ray Reflected ray 1
Normal 1
Mirror 1
Mirror 2
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Rotating a plane mirror• Light-beam galvanometer.
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A moving mirror
If the plane mirror moves at a speed v,the image moves at speed 2v.
fixed object image 1
v
2v
image 2
position 1position 2
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Concave mirrors
• http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
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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.
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Spherical aberration
• It can be corrected by using a parabolic mirror.
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Focal length f and radius of curvature R
Show that r = 2f for small angles.
θθ
C Fr
f
2θθ
h
paraxial ray
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Images for concave mirror
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
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Example 1• The rays from the sun are parallel and the
image of the sun is on the focal plane.
C Fθθ
h
f
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Mirror formula
fvu
111 for small angles
θ θ
C Fr
v
O
u
α β γI
h
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Mirror formula
fvu
111 for small angles
Nature of object/image
Object distance
u
Image distance
v
Real positive positive
Virtual negative negative
Real-is-positive convention:
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Mirror formula
fvu
111 for small angles
Nature of mirror focal length f
concave positive
convex negative
Real-is-positive convention:
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Linear magnification m
u
v
objectofheight
imageofheightm
C F
v
O
u
I
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Example 2
• Justify the nature of the image from the sign of image distance v.
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Variation of image with object distance
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
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Variation of magnification with object distance
fu
fm
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Convex mirror
fvu
111 for small angles
Nature of mirror
focal length f
object distance u
image distance v
convex negative positive negative
(virtual)
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Convex mirror
http://www.iln.net/html_p/c/453262/453270/453373/454123/56652_2079292.asp
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
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Example 3
F
v
O
u
II
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Measure the focal length of concave mirror
•Object at infinity.•Image is at the focal plane.•Measure the distance between the mirror and the screen.
Method A
C Fθθ
h
f
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Measure the focal length of concave mirror
•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.
Method B
http://www.usafa.af.mil/dfp/physics/webphysics/Physlet_examples/concave_mirror_f.html
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Measure the focal length of concave mirror
•Object at different positions to produce real images.•Images are captured by a screen.
Method C
v
1
u
1
f
1
f
10
fvu
111
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Measure the focal length of concave mirror
•Object at different positions to produce real images.•Images are captured by a screen.•Calculate the linear magnification m.
Method D
0
m
v
-1
1.1
vf
m
slope = f
1
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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 conv
ex mirror. • Adjust the position of the object so that a real image is at the
same position as the object.
O
I
lens mirror
C
P Q
flens
2.fm
s
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Measure the focal length fm of a convex mirror
O
I
lens mirror
C
P Q
flens
2.fm
s
• Measure s, the separation between the lens and the convex lens.
• 2 focal length of the convex mirror is flens – s.
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Refraction
1
2
2
1
2
1
2
1
sin
sin
n
n
c
c
Medium 1:1, 1, c1 and n1
Medium 2:2, 2, c2 and n2
1
2
http://www.netzmedien.de/software/download/java/brechung/
http://www.fed.cuhk.edu.hk/sci_lab/download/project/Lightrefraction/LightRefract.html
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Refraction
Medium 1:1, 1, c1 and n1
Medium 2:2, 2, c2 and n2
1
2
Snell’s law:n1.sin1 = n2.sin2
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Refraction
Medium 1:1, 1, c1 and n1
Medium 2:2, 2, c2 and n2
1
2
If n2 > n1, then medium 2 is an optically denser medium and medium 1 is an optically less dense medium
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Total internal reflectionThis occurs when light travels from an optically densermedium to an optically less dense medium and the angle of incidence > critical angle c.
medium 1
medium 2
refraction with 1 < c.
total internalreflection with 3 > c.
critical case with 2 = c.
12 3
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/light/flashLight.html
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Total internal reflection
medium 1 of n1
medium 2 of n2
refraction with 1 < c.
total internalreflection with 3 > c.
critical case with 2 = c.
12 3
The critical angle c is given by 1
2sinn
nc
http://www.continental.clara.net/physics/lt23.htm
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Fish-eye view
• http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/fishEye/fishEye.html
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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.
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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
45o
45o
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Optical fibre
• There is total internal reflection inside the optical fibre.
• Light is guided to travel in the optical fibre.
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Real depth and apparent depth• The image I is displaced upwards relative to
the object O.
O
Iapparentdepth
real depth
air
medium withrefractive indexn
B C
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Real depth and apparent depth
depthapparent
depthrealn for small angles.
O
Iapparentdepth
real depth
air
medium withrefractive indexn
B C
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Real depth and apparent depth
air
medium withrefractive indexn
Where would be the imageif we are inside the medium?Suppose that the anglesare small.
O
I
B C
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Measure the refractive index of glass
O1 O1
IO2
h1
h2
Find the real depth and apparent depth from h1 and h2.
real depth = h2
apparent depth = h2 – h2
glass blockhh h
travelling microscope
eye
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Rectangular glass blockThe incident ray and the emergent ray are parallel.The lateral displacement is
r
riwd
cos
)sin(.
ir
ri
d
wincident ray
emergent ray
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PrismFind the angle of deviation D in terms of angles ofincidence (1and 2) and angles of refraction (1 and 2).
D = (1 - 1)+(2 - 2)
A
D1
212
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PrismFind the refracting angle A of the prism in terms of theangles of incidence and angles of refraction.
A = 1+ 2
A
D1
212
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PrismThe angle of deviation is a minimum Dmin when thelight ray is symmetrical.Find the refractive index n of the glass prism.
2sin
)(21
sin min
A
DAn
A
D1
212
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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.
D
A
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Convex lenses
F’F
F F’
The focal lengths on both sides are equal.
f
f
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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
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Aberration of lens• Chromatic aberration.• Violent light bends more than red light in glass. fviolet < fred
white parallel light
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Location of image for convex lens
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/Lens/lens_e.html
Ray diagrams.
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Lens formula
fvu
111 for thin lens.
Nature of object/image
Object distance
u
Image distance
v
Real positive positive
Virtual negative negative
Real-is-positive convention:
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Lens formula
fvu
111 for thin lens.
Nature of lens focal length f
convex positive
concave negative
Real-is-positive convention:
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Lens formula: proof
DF
f
B
P
The angle of deviation D = (n – 1).Af
BP
where A is the refracting angle
Note that for small angle prism (thin lens), D is independent of the angle of incidence.
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Lens formula: proof
fvu
111 for thin lens.
Suppose that there is a real image.
Prove that
O I
D
u v
B
PFF’
f
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Interchange of locations of object and real image
O
O
I
I
u
u
v
v fvu
111
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Object-image distancefor real image
• To produce a real image, the object-image distance d must be longer than or equal to 2.f.
• Prove it.
O
I
d
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Thin lenses in contact
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/thinLens/thinLens.html
• Let f1 be the focal length of the first thin lens andf2 be the focal length of the second thin lens .• When they are in contact, the combined focal lengthf is given by
21
111
fff
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Example 5
• The image formed by the first lens is the object of the second lens.
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The converging power of a lens
• Definition of converging power
P = f
1Unit: dipotre (D)
• For two thin lenses in contact, the combined power isD1 + D2.
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Concave lens
• For a real object, virtual image is always formed. v is negative.
• Focal length is negative.
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Example 6I1 is the virtual object of the concave lens. u is negative
The concave lens produces a real image I2. v is positive.
I2I1
convex lens
concave lens
v
u
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Measure the focal length of convex lens
• Method A.
Object at infinity Image is at the focal plane.
f
I
focal plane
parallel rays fromdistant object
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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 coincideswith the image. This is method no parallax.
eye
object image
convex lensplane mirror
O
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Measure the focal length of convex lens
Method B.The distance between the object/image and the lens = f.
O
f
I
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Measure the focal length of convex lens
• Method C. • Produce different real images.• Measure the object distance u and the real image distance v.
v
1
u
1
f
1
f
1
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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
22
41
ad
d
f
O
Screenpositionof image
ad
object
1st positionof lens
2nd positionof lens
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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.
I2I1O
u
v
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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.
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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?
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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?
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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.
objecteye
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Angular magnification M
is the visual angle of the final image is the visual angle of the object
M
Note that the visual angles are usually small so tan sin and tan sin .
It is used to measure the magnification of an opticalinstrument,
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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).
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Magnifying glass
• A magnifying glass is a convex lens.
• It produces an enlarged virtual image.
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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.
objecteye
D
h
D
h tan visual angle without optical instrument
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Magnifying glass• With the magnifying glass in normal adjustment,
the final image is also at D.
object eye
v = D
h
image
u
h tan visual angle with the optical instrument
u
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Magnifying glass
Apply lens formula,fDu
111
(1)
D
h (2)
u
h (3)
1f
DM
the angular magnification
of a magnifying glass
With the visual angles,
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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.
objective lens eye-piece
objecteye
History of compound microscope:http://www.utmem.edu/~thjones/hist/hist_mic.htm
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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.
objective lens eye-piece
objecteye
1st image
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Compound microscope• The eye-piece is a magnifying glass.• It produces a magnified virtual image at D = 25 cm
from the eye-piece.
objective lens eye-piece
objecteye
1st image
finalimage
D
Note that the final image is an inverted image.
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Compound microscope
eyeD
h
object
objective lens eye-piece
objecteye
1st image
finalimage
D
h
h1
h2
D
h
D
h2
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Compound microscope
oe mmh
h
h
h
h
hM .. 1
1
22
where me is the linear magnification of the eye-pieceand mo is the linear magnification of the objective
The angular magnification M of a compound microscope is
M can be increased by using lenses of short focal lengths.
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Example 7
• The angular magnification = 5.5
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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
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Refracting telescope• The object is at infinity. The incident rays
are parallel
• The objective lens produces a real image I1 on its focal plane.
eye
Objective lens Eyepiecereal image
I1
fo
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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.
eye
Objective lens Eyepiecereal image
I1
fo
fe
Image at infinity
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Refracting telescopeIn normal adjustment, the angular magnification is
ef
fM 0
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Refracting telescope• The length of the refractive telescope is fo +
fe
• The image is inverted.
eye
Objective lens Eyepiecereal image
I1
fo
fe
Image at infinity
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Refracting telescope• The aperture of the objective lens is large
– to collect more light.– to reduce the diffraction effect.
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Example 8• Note the focal lengths of the lenses.
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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.
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Eye ringAll the light collected by the objective lens passes throughthe position of the eye ring.All the light enters your eye if you place your eye at theeye ring.
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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.
vuf
111
where u is the distance from theobjective lens to the eyepiece,f is the focal length of the eyepieceand v is the distance from the eye ringto the eyepiece.
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Locating the eye ring
The position of the eye ring is
e
e
fL
fLd
.
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Reflecting telescope• It is similar to a refracting telescope.
• Light is collected by a concave mirror.
Fobjective(concave mirror)
plane mirror
eye piece
F’
eye
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Reflecting telescope
• The angular magnification is
Fobjective(concave mirror)
plane mirror
eye piece
F’
eye
e
o
f
fM
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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.
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Example 9
• Locating the eye ring.
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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.
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Spectrometer• Collimator: Place the source S near the slit.
The collimator produces a parallel beam of light.
S
collimator
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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.
S
collimator
m = 0
m = 1
m = 1
m = 2
m = 2
diffraction grating
turntable
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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.
S
collimator
m = 0
m = 1
m = 1
m = 2
m = 2
diffraction grating
turntable
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Spectrometer• Telescope: It can be rotated to any angle to
view the diffraction spectrum.
S
collimator
m = 0
m = 1
m = 1
m = 2
diffraction grating
turntable
telescope
eye
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SpectrometerTelescope: the image is formed at the position of cross-wire.
light fromdiffraction grating cross-wire
objective lenseye piece
eye
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Spectrometer
• It is necessary to make a horizontal turntable. Adjust the levelling screws and use a spirit level to check the turntable.
levelling screw
turntablespirit level