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

2

2

2

2

2

2

2

2

t

B

x

B

t

E

x

E

oo

oo

tkxEE sinmax

tkxBB sinmax

BE

c 00

1

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 are related by the expression

• EM waves obey the principle of superposition

BE

c

Energy Transport• Poynting vector—the rate of energy

transport per unit area in an em wave

• Its units are

• The direction of the Poynting vector is

the direction of wave propagation

• Intensity—the time averaged value of S over one or more cycles

BESo

1

2m

Watts

221rms

orms

o

Bc

Ec

I

AP

I s

Radiation Pressure• Radiation pressure is the linear momentum

transported by an em wave

• If the surface absorbs all the incident energy

• An example of this type of surface is a black body

• If the surface is perfectly reflecting for a normally incident wave

• An example of this type of surface is a mirror

cS

P

cS

P2

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

• Law of reflection – the angle of incidence equals the angle of reflection

• Total internal reflection – when all light incident on a surface is reflected

ri i

tc n

nsin

Refraction

• Refraction – the travel of light through an interface (bending of light by an interface)

• Law of refraction (Snell's Law)

nc

vvc

vv

nmedium

vacuum

iitt

t

i

i

t

i

t

i

t

nn

nn

ncn

c

vv

sinsin

sinsin

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

p

iobjectheightimageheight

M

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

Finding Images• Point Source:

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

rf21

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.

•

• 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

pi

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 + + -

ipf111

pi

objectheightimageheight

M rf21

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)

• If

• Then

• If

• Then

• Bend toward normal

• If

• Then

• Bend away from normal

it nn

it

it nn

it

it nn

it

Refraction

Refraction from Spherical Surfaces

• If rays are bent toward the central axis, they form a real image on that 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)

0

a b b a

a b

n n n n

s s Rn n

s s

Spherical Surface

Planar Surface

Refraction cont

• convex surface is a converging lens

• concave surface is a diverging lens

21

111

111

rrn

ipf

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)

Diverging (Concave)

p > f1 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 21 MMM total

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 21 MMM total

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

eyob fcm

fs

mmM25

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

e

o

ff

m

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.

2 1

2 1

0, 1, 2,...

1 0, 1, 2,...2

r r m m

r r m m

sin 0, 1, 2,...

1sin 0, 1, 2,...2

d m m

d m m

2 2 2

2 cos2

4 cos2

P

P

E E

E E

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 spacetimeX 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

020

1t

cv

tt

22 1

1

1

1

cv

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

02

0 1L

LL

Transformation Equations

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tt

vtxx

Galilean TransformationEquations

Lorentz TransformationEquations

2

2

1c

vuvu

u

c

vxtt

zz

yy

vtxx

Velocities• Using the

Lorentz eqs. we can compare the velocities observed by 2 observers in frames moving relative to each other

x x v t

t t v xc

x

t

x v t

t v xc

xt v

v xtc

u vvu

c

FH IK

b g2

22

211

Momentum

• Momentum is also effected by speed

• Classically: p=mv

• Relativistically: p mx

tm

x

t

t

tmv

0 0

Mass Energy

• Mass and energy are conserved together not separately as assumed classically

• Nuclear reactions show us this

• Rest energy or mass energy

• Use units

E mc02

1 166 10

1 16 10

9 315 10

27

19

2 8

u kg

eV J

c eVu

.

.

.

Energy cont

• The total energy (without potential energy)

E E K mc K mctotal 02 2

K mv E E mc mc mctotal 12 12

02 2 2 b g

It is impossibleto increase speed toc because it wouldrequire an infinite amount of 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

Element

NZAAZ

Au197 Au197

79or

11870197

79

ZAN

Z

Gold for example

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

22 McmcEbe

AE

E beben

• 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

t

t

eRR

eNN

tN

N

0

0

is decay constant (value is characteristic of every radio nuclideN is # in the sample at a given timeR is the decay rate at a given time

tN

R

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 ( ) – the amount of time in which both N & R are reduced to half their original value

• Mean life () – the amount of time in which both N & R are reduced to e of their original value

21T

2ln2ln2

1 T 1

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.

• 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

HeThU 42

23490

23892

22 mcMcQ

Beta Decay

• The nucleus emits an electron or positron

is a neutrino

eNiCu

eSP

16428

6429

13216

3215

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 RBE• Electron & x rays 1• Slow neutrons 5• Alpha particles 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.