Transcript Lecture 12 - New Mexico State University

Final Exam
Lectures
EM Waves and Optics
Electromagnetic Spectrum
Traveling EM Wave
• Maxwell’s equations predict the existence of em
waves propagating through space at the speed of
light
• The waves consist of oscillating E and B fields that
are perpendicular to each other and the direction of
wave propagation
EM Waves cont
• EM waves generated with transformers and
LC circuits
• EM waves is composed of changing E and B
fields and will therefore travel in a vacuum
• Maxwell’s equations can be used to develop
a wave equation from which the form of the
waves can be deduced
 E
 E
  o o 2
2
x
t
2
2
 B
 B
  o o 2
2
x
t
2
2
E  Emax sin kx  t 
B  Bmax sin kx  t 
1
E
c

 0 0 B
Properties of EM Waves
• The solutions of Maxwell’s equations are
wavelike, with both B and E satisfying a
wave equation.
• EM waves travel through a vacuum at the
speed of light.
• The components of the E and B fields of
plane em waves are perpendicular to each
other and to the direction of propagation
(transverse waves)
• The magnitudes of E and B in empty space
E
are related by the expression
c
B
• EM waves obey the principle of
superposition
Energy Transport
• Poynting vector—the rate of energy
transport per unit area in an em wave
 1  
S  EB
o
Watts
• Its units are
m2
• The direction of the Poynting vector is
the direction of wave propagation
• Intensity—the time averaged value of S
over one or more cycles
I
1
co
2
Erms

c
o
2
Brms
Ps
I
A
Radiation Pressure
• Radiation pressure is the linear momentum
transported by an em wave
• If the surface absorbs all the incident energy
S
P
c
• An example of this type of surface is a black
body
• If the surface is perfectly reflecting for a
normally incident wave
2S
P
c
• An example of this type of surface is a mirror
Optics Definitions
• Geometrical optics—the study of the
properties of light waves under the
approximation that it travels as a
straight line (plane wave)
• Reflection—when light hits a surface
and bounces back
• Refraction—travel of light through a
surface (or interface) that separates 2
media. Light is bent at the surface, but
inside the medium it travels in a straight
line
• Index of refraction n—associated with a
medium of travel. It also depends on the
wavelength of light for all media except
vacuum.
• Angle of incidence I—the angle the light
makes to the normal to the surface when it hits
the surface
• Angle of reflection r —the angle the light
makes to the normal to the surface when it
bounces back
• Angle of transmission t —the angle the light
makes to the normal to the surface inside the
surface
Polarization
• Polarization – em waves
which vibrate randomly in
all directions are made to
vibrate in one direction
• An E field component
parallel to the polarizing
direction is passed
(transmitted) by a
polarizing sheet; a
component perpendicular
to it is absorbed
Reflection
i   r
nt
sin  c 
ni
• Law of reflection – the angle of incidence
equals the angle of reflection
• Total internal reflection – when all light
incident on a surface is reflected
Refraction
vvacuum c
c
n
 v
vmedium v
n
c
sin  t vt
nt ni
 

sin  i vi c
nt
ni
 nt sin  t  ni sin  i
• Refraction – the travel of light through an
interface (bending of light by an interface)
• Law of refraction (Snell's Law)
Definitions
• Image—the reproduction derived from light
of an object. Images are located either at a
point from which light rays actually diverge
or at the point from which they appear to
diverge.
• Virtual image—image perceived to be on the
opposite side of the mirror from the object
and observer (no actual light)
• Real image—image perceived to be on the
same side of the mirror as the object and
observer (light)
More Definitions
• Mirror—a surface which reflects a beam of
light in one direction, not scattering or
absorbing it
• Plane mirror—a flat reflecting surface
(mirror). Light diverges after reflection from
this type of mirror.
• Spherical mirror—a mirror with a reflecting
surface like a section of a sphere. This
mirror focuses incoming parallel waves to a
point
More Definitions
• Image length (i ) – the perpendicular
distance of an image from the center of the
mirror
• Object length (p)—the perpendicular
distance of the object from the center of the
mirror
• Magnification (M)—a measure of the size of
the image compared to the size of the object
height image 
i
M

height object 
p
Facts About all Mirrors
• the angle of incidence equals the angle
of reflection
• p is positive for all images. Using the
convention an object or image in front of
the mirror (or the side light or an
observer is) is positive and an object or
image behind the mirror would be
negative.
• i is negative for virtual images, and
positive for real images
Plane Mirrors
• The magnification is
always 1.
• The image is as far
behind the mirror as the
object is in front of it
(p = -i).
• The image is virtual and
upright (same orientation
as the object).
• The image has front-back
reversal
•
Point Source:
Finding Images
1. Draw 2 rays extending from the object to the
mirror
2. Using law of reflection, reflect the 2 rays off the
mirror
3. Extend the reflections back till the point where
they join
4. This is the image of the point
•
Extended Source:
1. Do the above steps for a point at the top of the
object and for a point at the bottom of the
object
2. Draw in the rest
Spherical Mirror Definitions
• Concave—caved in spheres,
looking from the interior of
the sphere. Light rays
converge to a real point after
reflection; therefore there is
a real focus
• Convex—flexed out spheres,
looking from the exterior of
the sphere. Light rays
diverge after reflection;
therefore there is a virtual
focus
More Spherical Mirror Definitions
• Central (principal) axis—extends through the
center of curvature of the sphere and through the
center of the mirror
• Paraxial rays—rays which diverge from the object
to make a small angle with the principal axis
• Focus (focal point)—point through which all
paraxial rays parallel to central axis reflect through
(a point on the central axis), or their extensions for
a convex mirror
• Focal length (f)—the distance of the focus from the
center of the mirror
1
f  r
2
Concave Mirror Facts
•
•
•
•
There is a smaller field of view than with plane mirrors.
The image is greater in size than the object.
The focus is real.
As the object is moved closer to the focal point, the
real, inverted image moves to the left. When the
object is on the focal point the image is infinitely far to
the left. When the object moves past the focal point
toward the mirror, the image is virtual, upright, and
enlarged.
• For a concave mirror the image goes out to infinity for
p<f (m increases) and image comes in from infinity for
p>f (m increases from -infinity to 0)
Convex Mirror Facts
• There is a greater field of view than with
plane mirrors.
• ip
• The image is smaller in size than the object.
• The focus is virtual.
• As the object distance increases, the virtual
image decreases in size and approaches the
focal point as the object distance
approaches infinity
Locating Images By Drawing Rays
• A ray parallel to the central axis reflects
through the focal point.
• A ray passing through the focal point
reflects parallel to the central axis.
• A ray passing through the center of
curvature reflects along itself.
• A ray reflecting at the center of the
mirror is reflected symmetrically about
the central axis
Mirror Type
Plane
Concave Concave Convex
i = -p
p<f
p>f
i<p
Magnification
M=1
M>1
M<0
0<M<1
Image
Virtual
Virtual
Real
Virtual
Orientation
Same
Same
Inverted Same
Sign of f
No f
+
+
1 1 1
 
f p i
-
1
f  r
2
Lens Definitions
• Lens—a transparent object with two refracting
surfaces whose central axes coincide (image formed
by first serves as the object for the second)
• Converging lens—causes a light ray that is initially
parallel to the central axis to converge to a point
• Diverging lens—causes a light ray that is initially
parallel to diverge
• Thin lens—thickness of lens is much less than p, i, r1,
r2 (r1 is the radius of curvature of the first lens surface
and r2 is the radius of curvature of the other lens
surface)
Refraction
•
•
•
•
•
•
•
•
If nt  ni
Then  t   i
If nt  ni
Then  t   i
Bend toward normal
If nt  ni
Then  t   i
Bend away from
normal
Refraction from • If rays are bent toward
the central axis, they
Spherical
form a real image on that
Surfaces
axis on the opposite side
of the surface from the
object (+ i)
• If rays are bent away
from the central axis,
they form a virtual image
on that axis on the same
side of the surface from
the object (- i)
na
nb
nb  na


Spherical Surface
s
s
R
na
nb


0
Planar Surface
s
s
Refraction cont
1 1 1
 
f p i
1 1
 n  1  
 r1 r2 
• convex surface is a converging lens
• concave surface is a diverging lens
Images from Thin Lenses
• A ray initially parallel to the central axis
will pass through the focal point f.
• A ray initially passing through the focal
point f (or its backward extension)
emerges parallel to the central axis.
• A ray initially directed toward the center
of the lens will emerge with no direction
change
Lens Type
Converging
(Convex)
p > f1
Diverging
(Concave)
p < f1
Magnification M < 0
M>1
0<M<1
Image
Real
Virtual
Virtual
Orientation
Inverted
Same
Same
Sign of f
+
+
-
Object produces image in 1st lens which is the
object for the 2nd.
Two Lens Systems
• Find the image formed by the first lens as if
the second lens is not present
• Draw a ray diagram for the second lens with
the image of lens 1 as the object of lens 2
• The second image formed is the final image
for the system
• One configuration of this is if the image
formed by the first lens is behind the second
lens and is used as a virtual object for the
second lens
• The total magnification of the system will be
M total  M1  M 2
Human Eye
Human Eye
• Light enters the eye through the cornea, a
transparent structure.
• Behind the cornea is a clear liquid called
the aqueous humor.
• Next is a variable aperture called the pupil,
which is an opening in the iris.
Human Eye cont
• Next is a crystalline lens. The purpose of the
crystalline lens is to allow the eye to focus on
an object through a process called
accommodation. The ciliary muscle is situated
in a circle around the lens. Thin filaments
called zonules run from the muscle to the lens
1. To focus the eye on a far object, the ciliary muscle
is relaxed which tightens the zonules on the lens
forcing it to flatten and increase its focal length
2. To focus the eye on a near object, the ciliary
muscle is tightened which relaxes the zonules on
the lens allowing it to bulge and decrease its focal
length
Human Eye cont
• Most refraction occurs at the outer surface of the eye,
where the cornea is covered with a film of tears. Very
little occurs in the lens because the aqueous humor
and the lens have very similar index of refractions
• The iris is a muscular diaphragm that controls the
pupil size and therefore the intensity of light that gets
into the eye
• The cornea lens system of the eye focuses light onto
the back surface of the eye called the retina,
consisting of millions of little receptors called rods and
cones. When these receptors are stimulated by light
they send a signal to the brain by way of the optic
nerve
• In the brain the image is perceived and analyzed
Nearsightedness
• In nearsightedness the rays converge before
they meet the retina. A nearsighted person
sees close objects but not far. This means
the far point is much closer than infinity. A
diverging lens before the eye corrects this
condition
Farsightedness
• In farsightedness the light rays reach the
retina before they converge. A farsighted
person can see far away objects but not
near objects. That means their near point is
much farther away than 25 cm. The
condition is corrected by putting a
converging lens before the eye
Two Lens Systems
• Find the image formed by the first lens as if
the second lens is not present
• Draw a ray diagram for the second lens with
the image of lens 1 as the object of lens 2
• The second image formed is the final image
for the system
• One configuration of this is if the image
formed by the first lens is behind the second
lens and is used as a virtual object for the
second lens
• The total magnification of the system will be
M total  M1  M 2
Microscope
• Microscope – used to view small objects
with a combination of two lenses to get
greater magnification
• One lens is called the objective and has
a very short focal length (< 1 cm)
• The second lens is called the eyepiece
and has a longer focal length of a few
centimeters
Telescope
• Two types of telescopes are used to view
distant objects, such as the planets in our Solar
System
– The refracting telescope uses a combination of
lenses to form an image (uses two lenses, the
objective and the eyepiece)
– The reflecting telescope uses a curved mirror and a
lens
fo
m
fe
Aberrations
•
Two types:
1. Spherical aberrations occur because the focal
points of rays far from the principal axis of a
spherical lens are different from the focal points
of rays of the same wavelength passing near the
axis (paraxial rays) Minimized by adjustable
apertures or parabolic reflecting surfaces
2. Chromatic aberrations occur because different
wavelengths of light refracted from a lens focus
at different points Minimized by use of a
combination of a converging lens made of one
type of glass and a diverging lens made of
another type of glass
Interference
• Interference phenomena occur when 2
waves combine.
• The effects occur where light reflected from
the front and back surfaces of a film
interfere with each other.
• Examples are colors seen in oil films or
soap bubbles.
r2  r1  m  m  0, 1, 2,...

r2  r1  m  1
2   m  0, 1, 2,...
Diffraction
• Diffraction occurs when many sources are
present.
• These effect occur whenever a wave passes
through an aperture or around an obstacle.
Relativity Lecture
•
•
•
•
•
Relativity
Time Dilation
Length Contraction
Transformation Equations
Review
Postulates
• Relativity postulate – the laws of physics are the
same for observers in all inertial reference frames
• Einstein extended this from Galileo (laws of
mechanics) to include electromagnetism and optics
• Speed of light postulate – the speed of light in
vacuum has the same value c in all directions and in
all inertial reference frames
• Ultimate speed-no entity which carries energy or
information can exceed this limit c=299792458
• Inertial reference frame – frames in which Newton’s
laws are valid (nonaccelerating)
Events
• Event – something that happens to which an
observer can assign a set of coordinates:
– Space, time, or spacetime
Construction to help
picture spacetime
X coordinate from
measuring rods and
time coordinate from
clocks
Relativity
• Relativity deals with the measurement of
events and how they are related
• If two observers are in relative motion, they
will not, in general, agree as to whether two
events are simultaneous
Relativity - Simultaneity
• Consider Sam and Sally to the
left
• Blue and Red events occur
• Sam sees them as simultaneous
• Sally sees the red event first
(before Sam does), and the blue
event later
• Note both measure themselves
halfway in between (Sam
conclude simultaneous and
Sally concludes red event
happens first)
Time Dilation
• The time interval between two events
depends on how far apart they occur, in
both space and time
• Proper time interval – the time interval
between two events, which occur at the
same location in an inertial reference frame,
measured in that frame
• Measurements of the same time interval in
any other inertial reference frame are
always greater
Time Dilation cont
Length Contraction
• The length of an object depends on which
reference frame it is measured in
• Proper length (rest length) – the length of an
object measured in the rest frame of the object
• Measurements of the same length in any other
inertial reference frame are always less
• Length contraction occurs only along the
direction of relative motion
Transformation Equations
Lorentz
Transformation
Equations
Galilean
x  x  vt
Transformation
t  t
Equations
u  u  v
Velocities
• Using the
Lorentz eqs.
we can
compare the
velocities
observed by 2
observers in
frames
moving
relative to
each other
Momentum
• Momentum is also effected by speed
• Classically: p=mv
• Relativistically:
Mass Energy
• Mass and energy are conserved together not
separately as assumed classically
• Nuclear reactions show us this
• Rest energy or mass energy
E0  mc2
• Use units
Energy cont
It is impossible
to increase speed to
c because it would
require an infinite
amount of energy
• The total energy (without potential energy)
Review
• Ch 22 – 26 deals with electrostatics
(charges that are not moving)
• Ch 26 – 28 deals with electrodynamics
(moving charges)
• Ch 29 – 31 are dealing with magnetism an
effect of moving charges
• Ch 32 – 33 deals with combining electricity
and magnetism plus some of the uses of
these concepts
• Ch 34 – 37 deals with geometric optics
• Ch 38 deals with relativity
Radiation Lecture
•
•
•
•
•
Nuclear Physics
Nuclear Properties
Radioactive Decay
Radioactive Dating
Radiation Dosage
Nuclear Physics History
• Nuclear Physics – the study of the nucleus of the
atom
• Plum pudding model – the original theory of atom
structure, postulated by JJ Thompson. The
positive charge of the atom is spread throughout
the entire atom volume. The electrons vibrated at
fixed points within the sphere of positive charge.
• Nuclear model – positive charge of atom is
densely concentrated at the center of the atom
(nucleus), postulated by Ernest Rutherford.
Experiment for Nuclear model
• An alpha particle source (radon gas) shot alpha
particles at a gold foil.
• The angle of deflection of these particles was
studied.
– Most particles were deflected through small angles
– A few were deflected through large angles
approaching 180 degrees.
• Analysis of the data implied the radius of the
nucleus was ≈104 times smaller than the radius
of the atom
Nuclear Properties
• Nucleus made up of protons and neutrons
– Atomic number Z - # of protons
– Neutron number N - # of neutrons
– Mass number A - # of both protons & neutrons
AZ N
197
A
Z Element
Gold for example
Au
Z  79
N  A Z
 197  70  118
or
197
79 Au
Isotopes
• Isotopes – nuclide with same Z but different
A (different # of neutrons)
• For a given element, they have the same #
electrons and therefore the same chemical
properties
• The nuclear properties vary from 1 isotope
to another.
• Usually an element has one stable isotope
and the rest are radioactive and decay by
emitting a particle.
Nuclidic Chart
• There is a well
defined band of
stable nuclides
(green) with
unstable above,
below, and the
upper end of the
chart.
• Light stable N~Z
• Heavy stable N>Z
Binding Energy
• Binding energy – difference between mass M
of a nucleus and the sum of the masses of its
individual protons and neutrons
• Binding energy is a convenient measure of
how well a nucleus is held together
Eben
Ebe

A
• The nuclei high on the plot are very tightly
bound. (Ni)
• Those low on the plot are less tightly bound. (H
& U)
• Consequence:
– Right side nuclei would be more tightly bound if split
into 2 nuclei farther up the plot in the process fission.
– Left side nuclei would be more tightly bound if
combined to form nucleus closer to top in the process
fusion
Radioactive Decay
• Radioactive decay follows statistical laws.
– A 1 mg sample of U has 1018 atoms. During
any second only 12 of them will decay and it is
impossible to predict which 12 will do it. All
have the same chance.
• Decay rate
 is decay constant (value is
characteristic of every radio nuclide
N is # in the sample at a given time
R is the decay rate at a given time
N
R
t
Activity of a Sample
• R is called the activity of a sample
– 1 bacquerel = 1 Bq = 1 decay/s
– 1 curie = 1 Ci = 3.7x1010 Bq
• Half life ( T12 ) – the amount of time in which
  1to
both N & R are reduced
 half their original
value
• Mean life () – the amount of time in which
both N & R are reduced to e of their original
value
1
 
Decay
• Alpha Decay – nucleus emits an alpha
particle
• Beta Decay – nucleus emits an electron or
positron
• Gamma Decay – nucleus emits a photon or
gamma ray
Alpha Decay
• The nucleus emits an alpha particle and
transforms to a different nuclide.
U 23490Th 24He
238
92
• Spontaneous because total mass of the
decay products is less than the mass of the
original
• Disintegration energy (Q) – the difference
between the initial mass energy and the total
final mass energy
Q  Mc 2   mc2
Beta Decay
• The nucleus emits an electron or positron
•  is a neutrino
Radiation Dosage
• Absorbed Dose – a measure of the radiation
dose actually absorbed by a specific object
• SI unit: 1 gray = 1Gy = 1 J/kg
• Dose Equivalent – the biological effect of a
radiation source (found by multiplying
absorbed dose by RBE)
• SI unit: 1 sievert = 1 Sv = 100 rem
• RBE (Relative Biological Factor)
•
•
•
•
Radiation
Electron & x rays
Slow neutrons
Alpha particles
RBE
1
5
10
• A whole body short term gamma ray dose
of 3 Gy will cause death in 50% of the
population exposed to it.
• Recommended radiation exposure is
< 5mSv in a year.