Free Tutoring offered by Phi Eta Sigma

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Free Tutoring offered by Phi Eta Sigma

• What Tutoring for MATH 221, CHEM 102, and PHYS
  212 there will also be tutors available for
  help on writing term papers
• When Wednesday, Nov. 28
• Where Basement of ISR
• Time 7-9 p.m
• Come by yourself or with a group of friends!

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Mirrors
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Today . . .

• Overview Nothing new here!
• objects and images
• Concave Spherical Mirrors
• The Mirror Eqn, Magnification, Sign Conventions
• Planar Convex Spherical Mirrors

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A few more reflections

• We already know two ways to get a reflection
• Off a conductor (e.g., aluminum or silver mirror)
• Total internal reflection (e.g., binoculars)
• There is at least one other common mechanism
• Fresnel reflection at an interface between two
  dielectrics with different indices of refraction,
  n1 and n2
• For light at normal incidence (q 0)
• e.g., nair 1, nglass 1.5 ? R 4
• For light at glancing angle (q 90), R ? 1.

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Selective reflection color in metals

• In some systems, e.g., metals and the atmosphere,
  transparency and color influenced by the
  oscillation of free (mobile) charges

Al
Cu
plasma edge
Au
Ag
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How to get really good mirrors

• Although we are most familiar with metal mirrors,
  e.g., aluminum or silver, these still have some
  absorption R lt 95.
• One can achieve much higher reflectivities by
  using many layers of thin dielectric films
• Constructive interference between Fresnel
  reflections at interfaces separated by 1/2 the
  wavelength of light
• ?R gt 99.9995 (used to couple photons to atoms)
• One can similarly achieve very low reflectivites
  
• Destructive interference between Fresnel
  reflections at interfaces separated by 1/4 the
  wavelength of light
• ?R lt 0.1 (anti-reflection coatings on
  glasses, camera lenses,...) recall that
  uncoated glass had 4 reflection per surface
• These topics will be covered more in Physics
  214...

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Cloaking

• By properly designing materials
  (metamaterials), it may be possible to create a
  substance that directs all incident light around
  a region of space.
• Thus far demonstrated for microwaves, in 2-D.
• Only works (thus far) at a single wavelength.
• (See James Scholar Assignment 5 for more
  info.)

Theory
Cloak
Experiment
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Reflections
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Nothing New!

• For the next two lectures we will be studying
  geometric optics. You already know the
  fundamentals of what is going on!!!
• Light propagates as rays in situations in which
  the length scales are gtgt than the lights
  wavelength

• Reflection

• Refraction

• We will use these laws to understand the
  properties of mirrors (perfect reflection) and
  lenses (perfect refraction).
• We will also discover properties of combinations
  of lenses which will allow us to understand such
  applications as microscopes, telescopes, and
  eyeglasses.

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Objects and Images
image
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Flat mirrors, common misbeliefs

• One-way mirrors or one-way glass
• These are often shown in interrogation rooms
  or police line-ups. A more familiar example is
  mirrored sunglasses.
• In all cases the transmission through the system
  has to be the same no matter which way the light
  is going. The sunglasses only appear (to the
  person looking at them) not to transmit any light
  because there is no light source behind them.
• Mirrors reverse right-left, but not up-down
• Actually, mirrors dont invert up-down or
  left-right! They invert the other axis --
  distance from the mirror

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Focal Length

• One very common way to characterize an optical
  element is in terms of its focal length.

• Imagine we have parallel rays incident on the
  element. The focal length is the distance from
  the element where the rays converge or diverge
  (in the focal plane).
• Rays converge ? positive focal length
• Rays diverge ? negative focal length

focusing element
defocusing element
focal plane
parallel rays
focal length
focal length
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Concave Spherical Mirrors

• We start by considering the reflections from a
  concave mirror in the paraxial approximation
  (i.e., small angles of incidence close to a
  single axis)

• First draw a ray (white) from the tip of the
  arrow parallel to the axis. This ray is reflected
  with angle q as shown it passes through the
  focal point (f) of the mirror.

f

• Next draw a ray (green) from the tip of the
  arrow through the focal point. This ray is
  reflected back parallel to the axis.

• Note that the green and white rays intersect in
  a point, suggesting an inverted image. We can
  check this by drawing a third principle ray

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Concave Spherical Mirrors continued

• A third principle ray draw a ray (light blue)
  from the tip of the arrow through the center of
  the sphere. This ray is reflected straight back
  since the angle of incidence 0. Note This
  third ray only works for a spherical mirror. A
  more reliable ray bounces off the center of the
  mirror that works for parabolic mirrors too!

f

• Note that this ray intersects the other two at
  the same point, as it must if an image of the
  arrow is to be formed there.

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Preflight 25
2) The diagram below shows three light rays
reflected off of a concave mirror. Which ray is
NOT correct?
A) B) C)
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X

• Ray A goes through focal pt and is reflected
  parallel to the axis.

• Ray B has angle incidence angle reflection

• Ray C goes through center of sphere.
• Therefore it has normal incidence
• Should be reflected straight back

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Lecture 25, ACT 1

• Where do the rays which are reflected from the
  convex mirror shown physically intersect?

(a) Inverted and in front of the mirror
(b) Inverted and in back of the mirror
(c) Upright and in back of the mirror
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Lecture 25, ACT 1

• Where do the rays which are reflected from the
  convex mirror shown physically intersect?

(b) to right of
(a) to left of
(c) they dont intersect

• The angle of incidence angle of reflection.
• Blue ray has normal incidence and is reflected
  straight back.

• White ray is reflected at larger angle than
  blue ray.

• Therefore the reflected rays are diverging!!
  They do not intersect!

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Lecture 25, ACT 1

• What is the nature of the image of the arrow?

(a) Inverted and in front of the mirror
(b) Inverted and in back of the mirror
(c) Upright and in back of the mirror

• Have you ever been to a 7-11, and looked at
  those mirrors in the corner?
• What about the right-side mirror on a car or
  Jeep (see Jurassic Park / T-Rex scene)

• Well, you are looking at a convex mirror like
  this. So, what is the answer?

• To find the image of the arrow, we need to find
  the point where the reflected rays APPEAR to come
  from.
• The intersection of the extrapolations of the
  reflected rays gives the image position as shown.

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The Mirror Equation

• We will now transform the geometric drawings into
  algebraic equations !! we want to relate s,
  s, and R !!

from triangles,
eliminating a,
Plugging these back into the above equation
relating the angles, we get
This eqn is known as the mirror eqn. Note that
there is no mention of q in this equation.
Therefore, eqn works for all small q, i.e., we
have an image!
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Magnification

• We have derived the mirror eqn which determines
  the image distance in terms of the object
  distance and the focal length

• What about the size of the image?
• How is h related to h??
• Use similar triangles .

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Magnification continued

• What about the size of the image?
• How is h related to h??
• From similar triangles

h
h
s
s
Now, we can introduce a sign convention. We can
indicate that this image is inverted if we define
its magnification M as the negative number given
by
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More Sign Conventions

• Consider an object distance s which is less than
  the focal length

• Ray Trace
• Ray through the center of the sphere (light
  blue) is reflected straight back.

• Ray parallel to axis (white) passes through
  focal point f.

• These rays diverge! I.e., these rays look they
  are coming from a point behind the mirror.

• We call this a virtual image, meaning that no
  light from the object passes through the image
  point.
• This situation is described by the same mirror
  equations as long as we take the convention that
  images behind the mirror have negative image
  distances s?. I.e.

s? lt 0 ? virtual image M gt 0 ? not inverted
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Concave-Planar-Convex

• What happens as we change the curvature of the
  mirror?
• Plane mirror
• R

IMAGE virtual upright (non-inverted)

• Convex mirror
• R lt 0

IMAGE virtual upright (non-inverted)
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4). The image produced by a concave mirror of a
real object is a) always real.
b) always virtual. c)
sometimes real and sometimes
virtual. 6). The image produced by a concave
mirror of a real object is a)
always upright. b) always inverted.
c) sometimes upright
and sometimes inverted.
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Is image of a real object from a concave mirror
real or virtual?

• It depends on the position of the object relative
  to the focal point!
• Draw Rays or..
• 1/s 1/f 1/s (concave implies f gt 0)

Is image of a real object from a concave mirror
upright or inverted?

• Once again, it depends on position relative to
  focal point!
• If s gt 0, real and inverted
• If s lt 0, virtual and upright

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7). The image produced by a convex mirror of a
real object is a) always real.
b) always virtual. c)
sometimes real and sometimes
virtual. 9). The image produced by a convex
mirror of a real object is a)
always upright. b) always inverted.
c) sometimes upright
and sometimes inverted.
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Is image of a real object from a convex mirror
real or virtual?

• Same PROCEDURE as for concave mirrors we did
  earlier!
• Draw Rays or..
• 1/s 1/f 1/s (convex implies f lt 0)

Is image of a real object from a convex mirror
upright or inverted?

• Once again, depends on sign of s (real/inverted
  or virtual/upright)
• Here s lt 0 ALWAYS therefore virtual and upright

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Lecture 25, ACT 2

• In order for a real object to create a real,
  inverted enlarged image,

a) we must use a concave mirror.
b) we must use a convex mirror.
c) neither a concave nor a convex mirror can
produce this image.
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Lecture 25, ACT 2

• In order for a real object to create a real,
  inverted enlarged image,

a) we must use a concave mirror.
b) we must use a convex mirror.
c) neither a concave nor a convex mirror can
produce this image.

• A convex mirror can only produce a virtual image
  since all reflected rays will diverge.
  Therefore, b) is false.
• To create a real image with a concave mirror,
  the object must be outside the focal point.

• The example we just did gave a real, inverted
  reduced image.
• Is it possible to choose the parameters such
  that the image is enlarged??

• The easy (but clever) answer
• h is a real image.
• Therefore consider the OBJECT to be h. The
  IMAGE will be h !!!
• Therefore it certainly IS POSSIBLE!!

equations follow
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Lecture 25, ACT 2

• In order for a real object to create a real,
  inverted enlarged image,

a) we must use a concave mirror.
b) we must use a convex mirror.
c) neither a concave nor a convex mirror can
produce this image.
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Summary

• We have derived the equations for mirrors

when the following sign conventions are used
Principal rays connect object and image one
goes through the center of the mirror the other
goes parallel to the optic axis and then is
reflected through a focal point the third one is
like the second one.
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Concave Spherical Mirrors

• We start by considering the reflections from a
  concave spherical mirror in the paraxial
  approximation (i.e., small angles of incidence
  close to a single axis)

• First draw a ray (light blue) from the tip of
  the arrow through the center of the sphere. This
  ray is reflected straight back since the angle of
  incidence 0.

f

• Now draw a ray (white) from the tip of the
  arrow parallel to the axis. This ray is reflected
  with angle q as shown.

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Concave Spherical Mirrors continued

• Note that the blue and white rays intersect in a
  point, suggesting an inverted image.
• To check this, draw another ray (green) which is
  incident at angle a, as shown.

f

• Note that this ray intersects the other two at
  the same point, as it must if an image of the
  arrow is to be formed there.