Download Physics 116 Mirrors and ray tracing

A
equal angles
B
Physics 116
Lecture 15
Mirrors and ray tracing
Oct 24, 2011
R. J. Wilkes
Email: [email protected]
Announcements
•! Guest lecturer today: Prof. Victor Polinger
Lecture Schedule
(up to exam 2)
Today
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Rays vs waves
•! We know light travels in the form of electromagnetic waves
•! We can picture light rays as lines perpendicular to the wavefronts,
indicating the direction the light wave is traveling
wavefronts
rays
•! If light were a stream of particles (as Newton thought), rays would
describe particle paths
•! Ray optics (or geometric optics) is very convenient for analyzing optical
systems that are very large compared to the wavelength of light
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Reflection of light rays
•! Light rays reflected off a smooth surface (specular reflection)
will obey the simple law
The angle of reflection = angle of incidence
–! Both are measured relative to normal to surface at point of reflection
A
equal angles
Line normal to surface
B
If the surface is not
smooth – light rays
arriving at nearby
points get reflected at
very different angles –
reflection will be
diffuse
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Reflection (Deep Thought)
•! Reflection law is also consistent with the “principle of least time”
–! In going from point A to point B, reflecting off a mirror, the ray
path actually traveled is also the fastest (shortest) route
Distance traveled from A to B is longer for any other path
•! Nature automatically finds the most economical path !
A
longer
fastest path:
equal angles
B
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Looking in the mirror
•! It’s helpful to think in terms of images
Looking into a mirror, light rays appear to come from behind the mirror
Actual incident ray
Apparent path of ray
line of sight to image’s head
h/2
real you
R
h
R
Mirror need only
be half as high as you are
tall to see your whole body.
Your image will appear same
distance behind mirror as you are
in front of it.
“image” you
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Plane Reflection: examples
•! Angle of incidence is 55 deg, find angle of 2nd reflection
•!
What is the minimum value of h to see the tabletop in the mirror below?
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Curved mirrors
•! What if mirror isn’t flat?
–! Same rules, but must use local surface normal for ray
•! Spherical mirrors focus parallel incoming parallel rays to a point
–! Just as (we’ll soon see) a lens does for rays passing through it
–! Parallel rays = rays coming from a very distant source (eg, a star)
C = center of curvature of mirror
C
o
Ray through C is just reversed
f
o
Optic axis
(symmetry axis of system)
f = focal point = halfway between C and mirror
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Ray tracing diagrams
•! For mirrors (and lenses) we can use simple rules to trace the paths
of certain rays
–! C rays – rays that pass through the center of curvature of the mirror
–! P rays – rays that are parallel to the optic axis of the mirror
–! F rays – rays that pass through the focal point after (or before*)
reflection
* Deep thought: ray paths have “time-reversal” symmetry – work
just as well backwards
C = center of curvature of mirror
P rays pass through F
Any ray through C is just reversed
C
o
f = focal point = halfway to C
So f = R/2
Optic axis
o
f
(symmetry axis of system)
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Two kinds of images
•! Real image
–! Rays of light from point on object actually converge on image plane
•! You can “catch” a real image on a screen: object is “reconstructed”
when rays from points on the object re-converge on image plane
•! Examples: movie on theater screen, this slide on lecture-hall screen,
images projected by a slide projector, or darkroom enlarger
–! Real images can be captured directly on photo film or camera chip
•! Virtual image
–! Rays of light appear to come from image plane when viewed, but never
actually converge in space
•! You can view a virtual image, but cannot capture it on a screen (or
film or video camera)
–! Of course, a camera (lens system) can view it, and record it that way
•! Examples: your image in bathroom mirror, image in a magnifying
glass, image in a “Galilean” telescope
–! You can only view virtual images by looking through the mirror or lens
–! The image is an “optical illusion”: your brain reconstructs the rays as
coming from the virtual image location
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Real vs virtual
•! Concave mirror can form a real image if object is farther away
than its focal length
object
Object size =h
C
o
Ray through C
is just reversed
f’
o
image
Image size =h’
h’ = Mh
Here, M is negative
(inverted image)
–! Ray tracing diagram shows two rays, from tip of object (arrow)
•! Ray passing through center of curvature is reflected on itself
•! Ray arriving parallel to optic axis is reflected to pass through f
–! Rays from any other point on object would converge on
corresponding point on image
–! Image formed is real and inverted
–! Image is magnified: h’ = Mh (in this case, smallified !)
•! Searchlight Mirror demo: object is almost at C
•! Convex mirror images are always virtual
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Virtual image
•! Spherical mirror forms virtual image if object is closer than f
object
o
C
image
o
f’
–! Now, reflected rays never actually converge, but appear to
converge if extended behind the mirror (appear to come from
virtual image)
•! Person viewing mirror would “see” magnified image behind it
–! This is a “shaving mirror” or magnifying mirror
–! You have to put your face close to it to see your magnified image
•! Home experiment: Put shaving mirror on floor, facing up, and
hold your hand above it while gazing into mirror: if f is right,
you will see a real image of your hand apparently floating in
space above the mirror!
–! Looks spooky because you expect images in mirrors to be behind
the glass, not in front.
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Mirror equation
•! Euclid tells us how to find the relations between image, object and
mirror locations for reflection, using the law of reflection, and ray
tracing rules: it’s all about similar triangles
Works for convex mirrors also:
Just remember, R is negative for them:
f = – R/2
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Examples
•! Object is placed 1.5R in front of
concave mirror: what is image type,
location, and magnification?
Minus means inverted;
rays converge at image, so it is real
•! Object is placed 1.5R in front of
convex mirror: what is image type,
location, and magnification?
Plus sign means erect image;
rays appear to emerge from image, so it is virtual
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