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Lenses and image formation
Object
Image
Lens
Object
Image
Thurs. Sep. 17, 2009 Physics 208, Lecture 5
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 2
1) Rays parallel to optical axis pass through focal point.2) Rays through center of lens are not refracted.
3) Rays through F emerge parallel to optical axis.
Thin-lens approximation: Ray tracing
F
F
Object
ImageOptical Axis
Here image is real, inverted, enlarged
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 3
Making an image
s’s
Object Image
Image distanceObject distance
f f
focal length
How are all these related?
€
1
s+
1′ s
=1
f
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 4
QuestionA magnifying glass of diameter 2 cm and focal
length 5 cm is used to form an image of the sun. Approximately what is the image distance?
A. 0.2 cm
B. 1 cm
C. 2 cm
D. 2.5 cm
E. 5 cm
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 5
Image size vs object size: Magnification
= M = Magnification
€
image height
object height=
′ s
s=
image distance
object distance
Image height
Image distance
Objectdist.
Objectheight
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 6
Far away objectsThe moon is 3.8x108 m away, and 3.5x106 m diameter. I use a
1 m focal length lens to make an image of the moon. About what diameter is this image of the moon?
A. 0.5 cm
B. 1 cm
C. 2 cm
D. 10 cm
E. 1 m
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 7
Different object positions
Image (real, inverted)
Image (real, inverted)Object
Image (virtual, upright)
These rays seem to originatefrom tip of a ‘virtual’ arrow.
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 8
Virtual images
Virtual image can’t be recorded on film, Can’t be seen on a screen.
But rays can be focused by another lens e.g. lens in your eye (focus on retina) e.g. lens in a camera (focus on film plane)
Image (virtual, upright)
These rays seem to originatefrom tip of a ‘virtual’ arrow.
objects closer to a converging lens than the focal length form a virtual image
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 9
Do these rays come from real image, a virtual image, or an object?
Can’t tell. Rays are exactly equivalent, and can be imaged by a lens in exactly the same way.
Virtual Image : thin-lens equation
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 10
Image (virtual)
Object
Object distance sFocal length f ( >0 for converging lens )Image distance s’
€
1
s+
1′ s
=1
f
€
⇒1′ s
=1
f−
1
sObject distance < focal length
€
s < f ⇒ ′ s is negative
Negative image distance: image on same side as object
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 11
Object at near point –
biggest it can appear when in focus
Magnifying glass
o
• Object closer than focal point
—Lens produces virtual image
• Light rays appear to originate from virtual image
• Virtual image is used as object for eye lens.
• Have moved object ‘closer’, while permitting eye to focus
Image (virtual, upright)
ss’
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 12
Magnifying glassA magnifying glass has a focal length of 8 cm.
It is 1 cm in front of your eye.
What is the closest that you can hold it to a bug so that the bug is in focus (your eye has a near point of 25cm).
A. 4 cm
B. 6 cm
C. 8 cm
D. 12 cm
E. 25 cm
€
1
s=
1
f−
1′ s
1
s=
1
8cm−
1
−24cm=
3
24cm+
1
24cm=
1
6cm
1cm
s=?
Magnifying glass: angular magnification
Without magnifying glass, object is biggest at near point, ~ 25 cm. Subtends angle
With magnifier Object can be closer, at object distance s Subtends angle
Angular magnification is Increases with decreasing obj. dist. Smallest obj. dist. is for image at near point:
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 13
€
NP ~ h /25cm
€
1
s=
1
f−
1
−25cm€
mag ~ h /s
€
M = θmag /θNP = 25cm /s
€
Mmax = θmag /θNP = 25cm1
f+
1
25cm
⎛
⎝ ⎜
⎞
⎠ ⎟=
25cm
f+1
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 14
Diverging lens
OpticalAxis
Then thin-lens equation can be used:
€
1
s+
1′ s
=1
f
Object Image
Focal length defined to be negative
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 15
Virtual image and diverging lens
Example: object at infinity Rays appear to originate from
focal point. Result
Object has been (virtually) transported to a new location
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 16
Object
Nearsightedness
I can’t focus on this
This, I can see
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 17
Object
Fixing nearsightedness
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 18
Reading glasses
Without my glasses, my far point is about 25 cm. What is the weakest (longest focal length) corrective lens (located at my eye) would let me read a newspaper holding it 50 cm away?
A. -25 cm
B. +25 cm
C. -50 cm
D. +50 cm
E. -100 cm
F. +100 cm
Lens should form a virtual image closer to my eye.
I can focus on image only if it is less than 25 cm away.Weakest lens moves it least, so image distance = -25 cm.
€
1
f=
1
50cm+
1
−25cm=
1
50cm−
2
50cm=
1
−50cm
Diopters Two lenses close together ~
Single lens, “effective” focal length feff
Lens power P Defined as with f in meters Units of P are diopters Two lenses close together
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 19
€
1
feff
=1
f1
+1
f2
€
P =1/ f
€
Peff = P1 + P2
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 20
Far away objectsThe moon is 3.8x108 m away, and 3.5x106 m diameter. I use a
1 m focal length lens to make an image of the moon. About what diameter is this image of the moon?
A. 0.5 cm
B. 1 cm
C. 2 cm
D. 10 cm
E. 1 m
Not a very big image. How can It be made ‘bigger’?
Look at the image with a magnifying glass!
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 21
Telescope: two lenses, object far away
Eyepiece forms virtual image Real image formed on retina by your eye lens.
Objective: Forms real image of far-away objectEyepiece: Used as magnifying glass to examine image
EyepieceObjective
Telescope angular magnification Without telescope,
distant planet subtends angle
Objective lens forms real image Height Used as object by eyepiece lens
Eyepiece Forms virtual image at infinity (for relaxed eye) Object must be at focal point Subtends angle
Angular mag: Thurs. Sep. 17, 2009 Physics 208, Lecture 5 22
€
planet
€
himageobj = fobjθ planet
€
image = himageobj / feyepiece
€
image /θ planet = fobj / feyepiece
Thurs. Sep. 17, 2009 Physics 208, Lecture 5 23
Virtual image ‘Object’
Eyepiece
Compound Microscope
qp
Object
Real, inverted, image
Objective
Object outside focal pointForms a real image
Real image used as object for eyepiece.Eyepiece forms virtual image for eye.