PHY 2054C – College Physics B
Electricity, Magnetism, Light, Optics and Modern Physics
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L10—Ch24 cont.
Spring 2010
Geometrical Optics
Chapter 24
Today’s Lecture: purpose & goals
1) Snell’s law, and Total Internal
Reflection
2) Focussing, Lenses, the “Lens
Equation” and ray tracing
3) Problems solving
Why do we care?:
We want to be able to see this!!
Xnjoy the Ueautiful fpringtime!!
Index of Refraction


When light enters a solid transparent material, or
medium, its speed changes,  it slows down!
How much it slows down is defined by the material’s
index of refraction, n c , n 1
E
v
material
+ - + - + - +
v

examples:
v
c
n
e.g. the speed of light in water
8
3 10 m s
v
2.25 108 m s
1.33
n
Vacuum
1.0000
Air
1.0003
Water
1.3300
Glass
1.5000
Diamond
2.4200
m
e
ov
d
Snell’s Law


The change in the speed of light from the vacuum to the
material changes the direction of the wave
From the geometry of the waves in the material,
sin 1 



c
sin 2
v
The ratio c/v is called the index of refraction and is denoted by
n where n = c/v,
whre n is unitless
for any two materials with indices of refraction n1 and n2
n1 sinθ1 = n2 sinθ2
This relationship is called Snell’s Law
Section 24.3
Snell's Law

If light passes from a “thinner”
to a “denser” medium, (n2>n1)
it is refracted toward the normal:
n 1 sin 1
sin
sin


1
2
n 2 sin
n2
n1
2
Refraction is due to the wavefronts “bunching together”
as they enter the higher
index material.
Notice: the shorter
wavelength is in the
higher index material.
Total Internal Reflection
basis of “fiber-optical” communication
When light passes from “denser”
to “thinner” material, it still
follows Snell's law:

n 1 sin 1 n 2 sin 2
but whensin
sending
light
n 2 from higher
1
index material to lower index material…
n 1θ ,
there is asin
maximum
2


1
for which refraction is
possible (θC):
90
2
n 2 sin 2 n 1 sin 1
sin 90
1
n2
sin C
n1
Beyond θC, there is total internal reflection. (No light can
be refracted ‘out’ through the surface, only relfected!!)

m
e
ov
d
Spherical Lenses



The simplest lenses have
spherical surfaces
The radii of curvature of the
lenses are called R1 and R2
The radii are not necessarily equal
Types of Lenses

Converging lenses


All the incoming rays
parallel to the principal
axis intersect at the focal point on the opposite side
Diverging lenses

All the incoming rays parallel to the principal axis intersect at the
focal point on the same side as the incident rays
Section 24.5
Lenses
Lenses use
refraction
according to Snell's
law to create images.


Converging (convex) lenses focus
parallel light into one focal point F.
focal length f > 0
(primary focus is ‘behind’ lens)


Diverging (concave) lenses disperse
parallel light as if it was “coming from”
one focal point F.
focal length f < 0
(primary focus is ‘in front of’ lens)
Real or Virtual Images


1) Real image: rays converge toward image.
Sign convention: image distance sI>0, positive.
2) Virtual image: rays diverge coming from image
sign convention: image distance sI<0, negative
Images from Converging Lenses:
Ray Tracing
Object standing on Principal
Axis, Image Construction:
1) “Parallel Ray” -- ray from,

object parallel to principal axis,
then leaving through the (primary)
focal point F.
2) “Focal Ray” -- ray from
object, through the (secondary)
focal point F’, then leaving
parallel to axis.
3) “Center Ray” -- ray from object,
directly towards lens center, then
undeflected out at same angle
the 3 rays meet at Image point I



“real image”
A beautiful website for showing and visualizing ray tracing is at
http://www.mtholyoke.edu/~mpeterso/classes/phys301/geomopti/twolenses.html
Example:
Let us see how to use what we know:
Magnifying Glass
-- object is closer to a converging lens
than the lens’ focal length





Draw three principal rays
1) “parallel”
2) “focal”
3) “central”
f
so
1
f
1
so
1 1
f so
1
si
si
2
1
F’
1
f
1
so
si
so
3
11
< 0!!!
Negative image distance: Image is on the left of the lens
(same side as the object)!

Image is virtual, the rays diverge from the image.

The image will be close to the eye's “near point” =25 cm
F
Images from Diverging Lenses




Draw three principal rays
 1) “parallel”
 2) “focal”
 3) “central”
A diverging lens always creates a “virtual” image on the
same side as the object.
“Virtual Image” = image would not appear on a projection
screen, but does appear to an observer who views it
directly.
The image is also always upright and de-magnified
“the Lens Equation”
1
f
m
1
so
1
si
hi
ho
si
so
Notice: (all distances measured from center of the lens)


e.g. Camera Lens: f = 50 mm, so = 80 cm,
What is the image distance and magnification? si = ?
1
f
1
so
1
0.05 m
m
hi
ho
1
si
1
0.8 m
si
so
1
si
si
1
0.05 m
0.053 m m
0.8m
1
0.8 m
0.06625
1
si
0.0533 m
…and a 20cm high
object would produce
a 1.325 mm image
Sign Conventions
1
f

1
so




1
si
hi
ho
m
e.g. Converging lenses: f > 0
so
for the Lens Equation
si
so
e.g. Diverging lenses: f < 0

si
so
Object left: sO>0
Image right: si>0
Image real: si>0
Image inverted: m<0
si
Object left: sO>0
Image left: si<0
Image virtual: si<0
Image upright: m>0




Can also do multiple lens systems by making the image position of the first lens
be the object position for the
second lens -- and start over,
measuring all distances now from
the position of the second lens,
and so on…(remember: distances are
all measured from center of each lens)
Sign Conventions
for the Lens Equation
1
f
1
so
1
si
m
hi
hO
si
so
(all distances measured from center of the lens)
 the focal length f is positive (+) for converging (convex) lenses
 and negative (-) for diverging (concave) lenses
 This puts the primary focus F “behind the lens” for
converging* and “in front of the lens” for diverging*
 The object distance so is positive (+) if it is on the same
side of the
lens from which the light is coming (which is the usual case;)
 otherwise it is negative (-)
 The image distance si is positive (+) if it is on the opposite side of the
lens from which the light is coming (called a real image);*
 otherwise it is negative (-); (called a virtual image)*
 The height of the image hi and the magnification m are positive (+)
if the image is upright; (called an erect image);
 and negative (-) if the image is upside down (called inverted image).
* reversed for mirrors
A beautiful website for showing and visualizing ray tracing is at
http://www.mtholyoke.edu/~mpeterso/classes/phys301/geomopti/twolenses.html
Example:
Converging Lens
Note: We suggest you use ray diagrams to qualitatively understand these questions. A
candle 6.70 cm high is placed in front of a thin converging lens of focal length 30.5 cm.
What is the image distance i when the object is placed 104.5 cm in front of the same
lens? NOTICE: The object is ‘far’ outside focal length
What is the size of the image? (Note: an inverted image will have a `negative' size.)
Is the image real(R) or virtual(V); upright(U) or inverted(I); larger(L) or smaller(S) or
unchanged(UC); in front of the lens(F) or behind the lens(B)? Answer these questions in
the order that they are posed. (for example, if the image is real, inverted, larger and
behind the lens then enter `RILB'.)
Ray tracing:
3 – center ray
Calculation:
1
f
1
so
1 – parallel ray
2 – focal ray
object
F
F
1
si
1
1
1
0.305 m 1.045 m s i
image
/
secondary
si
1
0.305 m
1
1.045 m
.
primary
1
0.4307 m
si
si positive real,
0.4307 m
hi
si
behind lens
0.4122
m
m h
1.045m
so
o
m negative inverted
.
m
less
than one smaller
h
=
-0.0276m
hi = m . ho = -0.4122 0.067m i
“RISB”
Example:
Converging Lens(2)
Note: We suggest you use ray diagrams to qualitatively understand these questions. A
candle 6.70 cm high is placed in front of a thin converging lens of focal length 30.5 cm.
What is the image distance i when the object is placed 45.0 cm in front of the same
lens? NOTICE: The object is ‘near’ but outside focal length
What is the size of the image? (Note: an inverted image will have a `negative' size.)
Is the image real(R) or virtual(V); upright(U) or inverted(I); larger(L) or smaller(S) or
unchanged(UC); in front of the lens(F) or behind the lens(B)? Answer these questions in
the order that they are posed. (for example, if the image is real, inverted, larger and
behind the lens then enter `RILB'.)
Ray tracing:
1 – parallel ray
Calculation:
1
f
1
so
si
1
si
object
1
1
1
d
0.305 m 0.450 m I
1
0.305 m
image
F
F
/
2 – focal ray
3 – center ray
1
1
si
0.9466 m
0.450 m
0.9466 m
2.1034
m
0.450m
hi
si
si positive real, behind
ho
so
m negative inverted
m
>
than one larger
.
hi = m . ho = -2.1034 0.067m hi = -0.1409m
m
“RILB”
Example:
2 – focal ray
Converging Lens(3)
Note: We suggest you use ray diagrams to qualitatively understand these questions. A
candle 6.70 cm high is placed in front of a thin converging lens of focal length 30.5 cm.
What is the image distance i when the object is placed 24.5 cm in front of the same
lens? NOTICE: The object is inside focal length
What is the size of the image? (Note: an inverted image will have a `negative' size.)
Is the image real(R) or virtual(V); upright(U) or inverted(I); larger(L) or smaller(S) or
unchanged(UC); in front of the lens(F) or behind the lens(B)? Answer these questions in
the order that they are posed. (for example, if the image is real, inverted, larger and
behind the lens then enter `RILB'.)
Ray tracing:
Calculation:
1
f
1
so
1
si
1 – parallel ray
image
1
0.305m
1
0.305m
si
object
1
1
0.245m s i
F/
1
1
-1.2454m
si
0.245 m
1.2454 m
m = +5.0833
0.245m
h
si
m hi
so
o
hi = m . ho = +5.0833 . 0.067m hi = +.3406m
Example:
F
3 – center ray
si negative
Virtual,
in front of lens
m positive
m > than one
upright
larger
“VULF”
Diverging Lens
Note: We suggest you use ray diagrams to qualitatively understand these questions. A
candle 6.70 cm high is placed in front of a thin diverging lens of focal length 30.5 cm.
What is the image distance i when the object is placed 104.5 cm in front of the same
lens? NOTICE: The object is ‘far’ outside focal length
What is the size of the image? (Note: an inverted image will have a `negative' size.)
Is the image real(R) or virtual(V); upright(U) or inverted(I); larger(L) or smaller(S) or
unchanged(UC); in front of the lens(F) or behind the lens(B)? Answer these questions in
the order that they are posed. (for example, if the image is real, inverted, larger and
behind the lens then enter `RILB'.)
Ray tracing:
Calculation:
1
f
1
so
1 – parallel ray
2 – focal ray
object
F
1
si
1
1
1
-0.305m 1.045 m s i
image
F
/
secondary
principal
si
1
-0.305m
1
1.045 m
1
si
3 – center ray
-0.2361m
si negative virtual,
(-.236) m
hi
si
in front of lens
0.2259
m
m h
1.045m
so
o
m positive upright
.
m
less
than one smaller
h
=
+0.0276m
hi = m . ho = +0.2259 0.067m i
“BUSF”
Maxwell’s equations:
Stay tuned. . . .And in the beginning…God said:


Wednesday:
ch25 Wave Nature of Light
Next Wedn.: Mini-exam 5
(Chs. 24,25): Light
& Geometrical Optics
div E
1
4
0
B
t
curl E
div B 0
curl B
0
j
0 0
E
t
…and there was light!
Sorry; dumb inside joke.
L13—Ch23
PHY 2054C – College Physics B
Summer 2007
Electricity, Magnetism, Light, Optics
and Modern Physics
Dr. David M. Lind
Sorry, Dr. Lind can’t be here for today’s class.
Please watch the Video today called
“Mauna Kea” about applied optics used
in astronomical telescopes. Look in the video
for how optics and light are applied in
astronomy. (HINT: there will be questions on the
next test on these ideas.)