Find image with a thin lens
ho
F
do
Object
ho
F
do
Object
f
hi
hi
F
di
F
di
f
1
Analytical calculations
Thin lens equation.
ho
hi
1 1 1
+
=
di do f
hi
di
magnification : m ≡ = −
ho
do
2
Analytical calculations
Lens maker’s equation:
The formula for a lens in vacuum (air):
1
1
1 
= ( n − 1 ) − 
f
 R1 R2 
F
near
surface
far
surface
n : index of refraction of
the lens material.
R1 : radius of near surface.
R2 : radius of far surface.
The near or far surface is with respect to the focal point F. Near side is
surface 1, far side is surface 2.
The sign of the radius is then defined as
“+” if the center is on the far side; “-” if the center is on the near side.
In this convention, positive f means converging lens, negative f means
diverging lens.
3
Analytical calculations
Lens maker’s equation:
The formula for a lens (nlens) in medium nmedium :
nmedium
F
near
surface
R1 : radius of near surface.
R2 : radius of far surface.
far
surface
1
nlens
1
1 
=(
− 1 ) − 
f
nmedium
 R1 R2 
4
Sign convention table
5
Angular size
The height of an object is measured by a meter stick. The height of the same
object we see through our eyes depends on the how far away the object is to
our eyes.
6
Angular size
Angular size is defined to be:
h
θ≡
d
7
Angular magnifying power
Angular magnifying power: the ratio of the image angular size over the object
angular size.
θ
M≡
θ0
8
Human eyes
The human eye is modeled in physics as a simple thin lens system with a
fixed image distance, but the focal length can change in a range.
f ∈ ( f min , f max )
di
9
Human eyes
The focal length range correspond to a person’s near point and far point:
Near point: when the object is pushed as close as one can have
clear image. This is the point when the eye’s focal lens is at its
minimum value.
10
Human eyes
The focal length range correspond to a person’s near point and far point:
Far point: when the object is pushed as far as one can have
clear image. Optically this is the point when the eye’s focal lens
is at its maximum value. For healthy eyes, this far point is
usually almost at infinity.
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Vision corrections
Nearsightedness (myopia):
12
Vision corrections
Farsightedness (hyperopia):
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Vision correction examples
Refractive power: The reciprocal of the focal length. Often used by opticians
and optometrists, who specify it in diopters (unit: 1/m).
1
P≡
f
Nellie is nearsighted. She cannot focus on objects farther than 40.0 cm from
her unaided eye. What focal length must her corrective contact lens have to
bring into focus the most distant objects?
Far point = 40.0 cm, a correcting lens needs to generate the image of an
object at infinity at this far point for her to see clearly. Contact lens means
the correcting lens and the lens in the eye has zero distance between them.
f c = −40.0cm
So the contact lens is a diverging lens with a focal length of -40.0 cm.
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Vision correction examples
Elizabeth is nearsighted. Without glasses, she can see objects clearly when they are
between 15.0 cm and 90.0 cm away from her eyes. Her glasses are designed to be
worn 2.00 cm from her eyes, and have a focal length so that objects at infinity produce
images at her far point. When she is wearing these glasses, how close to her eye can
an object be before it appears out of focus?
Far point = 90.0 cm, near point = 15.0 cm. A correcting lens needs to
generate the image of an object at infinity at this far point minus the 2.00
cm for her to see clearly. Contact lens means the correcting lens and the
lens in the eye has zero distance between them.
f c = −88.0cm
Far point
2.00 cm
Near point = 15.0 cm. The image distance has to be -(15.0 – 2.0) cm = 13.0 cm of an object placed at the new near point with the correcting lens.
Use the lens equation to find this new near point to be 15.25 cm.
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Multiple lens system
Microscope:
θ0
θ
Two lenses − objective and eyepiece
Objective focal length very short
First image real, near eyepiece focal point
Final image inverted, magnified, virtual
θ
Angular magnifying power is
M≡
θ0
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The magnifying power of a microscope
M = mob M eye
LN
M ≅−
f ob f eye
M = overall magnification
mob = objective lateral magnification
Mey = eyepiece angular magnification
L = distance between the lenses
N = near point distance of your eye
fob = focal length of objective
fey = focal length of eyepiece
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Multiple lens system
Telescope
θ0
θ
Two converging lenses
Focal points at same location
Final image inverted, at infinity, virtual
θ
Angular magnifying power is M ≡
θ0
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The magnifying power of a refracting telescope
θ
f ob
M≡
=−
θ ob
f eye
Image inverted
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