lecture 25 - phys.lsu.eduphys.lsu.edu/~jdowling/phys21024sp07/lectures/lecture25.pdf · lecture 25...
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Lecture 25Lecture 25
Physics 2102Jonathan Dowling
Optics: ImagesOptics: Images
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Thin LensesThin Lenses
fip
111=+
For small angles and thin lenses,
Convergent: f positive
Divergent: f negative
!"
#$%
&''=21
11)1(
1
rrn
fLens maker’s equation
Convergent lens
Divergent lens
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Images due to lenses:
• An object placed beyond a convergent lenses’ focal point, will produce a real, inverted image on the other side of the lens. This is the principle used in projectors.
• An object placed between a convergent lens and its focal point will produce a virtual image on the same side as the object.
•Divergent lenses always produce a virtual image on the same side asthe object.
• Real images have i positive in formulas, virtual images have i negative.
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Locating images by drawing rays:
• A ray of direction initially parallel to the axis will pass through the focal point.
• A ray that initially has a direction thatpasses through the focal point will emerge parallel to the central axis.
• A ray going through the center of thelens will be undeflected.
• The image of a point appears where all rays emanating from a point intersect.
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ExampleExample• An object 1.2cm high is placed 4cm from a bi-convex lens withr1=10cm and r2=15cm. Find the position and size of the image.
• A second lens of focal length +6cm is placed 12cm to the right ofthe first lens. Find the position and size of the new image.
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Images fromImages fromspherical mirrorsspherical mirrors
fip
111=+
Consider an object placed between the focalpoint and the mirror. It will produce a virtualimage behind the mirror.
When the object is at the focal point theimage is produced at infinity.
If the object is beyond the focal point, a realimage forms at a distance i from the mirror.
Check the signs!!
p
im !=
lateralmagnification
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Thin LensesThin Lenses
fip
111=+
For small angles and thin lenses,
Convergent: f positive
Divergent: f negative
!"
#$%
&''=21
11)1(
1
rrn
fLens maker’s equation
Convergent lens
Divergent lens
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Locating images by drawing rays:
• A ray of direction initially parallel to the axis will pass through the focal point.
• A ray that initially has a direction thatpasses through the focal point will emerge parallel to the central axis.
• A ray going through the center of thelens will be undeflected.
• The image of a point appears where all rays emanating from a point intersect.
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Images due to lenses:
• An object placed beyond a convergent lenses’ focal point, will producea real, inverted image on the other side of the lens. This is the principleused in slide projectors.
• An object placed between a convergent lens and its focal point willproduce a virtual image on the same side as the object.
•Divergent lenses always produce a virtual image on the same side asthe object.
• Real images have i positive in formulas, virtual images have i negative.
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ExampleExample• An object 2cm high is placed 4cm from a bi-convex lens withr1=10cm and r2=15cm, and index of refraction n=1.5. Find theposition and size of the image.
• A second lens of focal length +6cm is placed 12cm to the right ofthe first lens. Find the position and size of the new image.
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The human eyeThe human eye consists of avariable-geometry lens (crystalline)which produces a real image on a“screen” (retina) which is transmittedto the brain via the optical nerve.
The cristalline automatically adjusts itself so we see well any objectplaced between infinity and a distance called “near point” (about 25cmfor a typical 20 year old). The “image distance” is the eye diameter~2cm.
Optical Instruments: the human eyeOptical Instruments: the human eye
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Optical instruments: combinationOptical instruments: combinationof several (thin) lensesof several (thin) lenses
F1 F1
F2 F2
If lenses are very close, the compound lens has 1/f~1/f1+1/f2
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Corrective GlassesCorrective Glasses
A farsighted person needs a convergent lens.
A nearsighted person needs a divergent lens.
The “power” of a lens is measured in dioptres: P=1/f with f is in m.Glasses with -6D are divergent glasses with f=−1/6D =−0.17m=−17cmThe dioptres add! Two lenses have 1/f=1/f1+1/f2 → D=D1+D2
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The magnification of an object is m=i/p~iθ/h,but i=eye diameter.Maximum magnification: m~2cm/25cm (!?)
Angular magnification (different from lateral): mθ=θ’/θ.
f
cm
f
h
cm
h 25m '
25=!= """
Magnifying lensMagnifying lens
We’d like to make p smaller (move the object closer). We use amagnifying lens to produce a (larger) image than our eye can see:
Very near the focus!
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Microscope:
To increase the magnification of a lens, one wants to have a shortfocal length. That means small radii of curvature (very curved lens).This, in turn implies a lot of aberration (one is immediately out ofthe thin lens approximation). A solution to this is obtained bycombining two lenses. The resulting device is called microscope.
p
im !=
Object O is magnifiedby the objective:
And its image is magnifiedby the eyepiece:
f
cmm
25=!
Total magnification:
eyob f
cm
f
smmM
25!== "
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Telescope:
Telescopes are arrangement of lenses that improve vision of objectsvery far away. They are configured like a microscope. However, theobjective forms an image essentially at its focus, and therefore theeyepiece’s focus has to be placed at that same point.
The magnification is given bythe ratio θey/θob, and since
obob fh /'=! eyey fh /'=!
ey
ob
f
fm !=
Refracting telescopes are of limiteduse (chromatic aberration). Reflectingtelescopes built with mirrors are preferred in astronomy.
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ExampleExampleThe world’s largest refracting telescope is at the YerkesObservatory of the University of Chicago at Williams Bay,Wisconsin. The objective has a diameter of 102cm and a focallength of 19.5m. The focal length of the eyepiece is 10cm. What isits magnifying power?
1951.0
5.19!=!=!=
m
m
f
fm
ey
ob
Why so large (102cm)? Because thelarger the objective, the more light itgathers.
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Reflective telescopesReflective telescopes
Keck observatory (MaunaKea, Hawaii) and theHale-Bopp comet.
Largest optical telescope,composed of 36 (!)hexagonal mirrorsegments performing as asingle mirror 10m wide.