Waves can be represented by simple harmonic motion

Standing wave

y = Asin(kx − ωt) + Asin(kx + ωt)

The amplitude of a wave is a measure of the maximum disturbance in the medium during one wave cycle. (the maximum distance from the highest point of the crest to the equilibrium).

The wavelength (denoted as λ) is the distance between two sequential crests (or troughs). This generally has the unit of meters.

A wavenumber 2k

Period 2T

Phase velocity: fTk

v

)sin()( tkxAty

Electromagnetic waves

c

f

Light as a Wave (1)

• Light waves are characterized by a wavelength and a frequency f.

f = c/

c = 300,000 km/s = 3*108 m/s

• f and are related through

The Electromagnetic Spectrum

Need satellites to observe

Wavelength

Frequency

High flying air planes or satellites

Dual, wave-particle nature of light

c

hhfE sec Joule106.6 34 h

1 eV = 1.6x10-19 J

c = 3x108 m/s

1 Angstrom = 10-10 m

Speed of light in matter:

cm = c/n, wheren is refractive index

Note: n is a function of

Light as a Wave (2)

• Wavelengths of light are measured in units of nanometers (nm) or Ångström (Å):

1 nm = 10-9 m

1 Å = 10-10 m = 0.1 nm

Visible light has wavelengths between 4000 Å and 7000 Å (= 400 – 700 nm).

Light as Particles

• Light can also appear as particles, called photons (explains, e.g., photoelectric effect).

• A photon has a specific energy E, proportional to the frequency f:

E = h*f

h = 6.626x10-34 J*s is the Planck constant.

The energy of a photon does not depend on the intensity of the light!!!

Maxwell’s Equations

0Q

SdE

0 SdB

SdEdt

dirdB

000

SdBdt

drdE

0

1 E

0 B

Bt

E

Et

jB

000

Information Age

The cost of the transmission, storage and processing of data has been decreasing extremely fast

Information is available anytime, any place, and for everyone

Information and knowledge became a capital asset

All of this became possible because of several revolutionary ideas

Telecommunications

Samuel Morse's telegraph key, 1844. Today's information age began with the telegraph. It was the first instrument to transform information into electrical signal and transmit it reliably over long distances.

Alexander Graham Bell’s commercial telephone from 1877.

How it all started …

Speaking into the handset's transmitter or microphone makes its diaphragm vibrate. This varies the electric current, causing the receiver's diaphragm to vibrate. This duplicates the original sound.

• Telephone connection requires a dedicated wire line;• Only one communication is possible at a time

How many channels are possible? How many signals can be transmitted at the same time??

Radio: communication through radio waves

1895

Alexander PopovGuglielmo Marconi

www.nrao.edu

Frequency measured in Hertz1 Hz = 1 cycle/second1 kHz = 1000 cycles/second

Radio stations have to broadcast at different carrier frequencies to avoid cross-talk

Range of frequencies (Bandwidth) needs to be at least 20 kHz for each station

Human ear: 10 Hz-20 kHz

Frequencies of different stations should be at least 20 kHz apart

Higher carrier frequencies

Wider bandwidth

Higher data rate, more channels

Need more channels? Need higher speed?Use higher frequencies for transmission!

Using light? Optical frequencies ~ 1014 Hz !

How can we send light over long distances?

Air? Only within line of sight; High absorption and scattering, especially when it rains

Are there any “light wires” (optical waveguides)?

Copper wire? High absorption, narrow bandwidth 300 MHz

Glass? Window glass absorbs 90% of light after 1 m.Only 1% transmission after 2 meters.

Sand?!

Transmisson 95.5% of power after 1 km 1% of power after 100 km: need amplifiers and repeaters

Total bandwidth ~ 100,000 GHz!!

Ultra-low absorption in silica glasses

Silica (Silicon dioxide) is sand – the most abundant mineral on Earth

Predicted 1965, in first low-loss fiber in 1970

Total internal reflection!

n1 > n2

How to trap light with transparent material??

Light coming from more refractive to less refractive medium experiences total reflection – get trapped there!

No charges, no real currents

0 SdE

0 SdB

SdBdt

drdE

SdEdt

drdB

00

zyyx iitxEiE

0),(0

zzyx itxBiiB

),(00

Wave equation

2

2

002

2

t

E

x

E yy

2

2

002

2

t

B

x

B zz

2

2

002

2

t

E

x

E yy

)sin( tkxAEy

2

kT

2

k is a wave number, is a wave length, T is the period

Velocity of propagation

ck

v 00

1

Coulomb’s Law

Charge

Charge 1q

2q

2

21

04

1

x

qqFE

229229

0

/109/1094

1CmNcoulombmeterNewton

Conservation of electric charge

Charge is conserved: in any isolated system, the total charge cannot change. If it does change, then the system is not isolated: charge either went somewhere or came in from somewhere

0 is the permittivity of free space

Charge 1q

2qCharge

21r̂12r̂

2112 ˆˆ rr

Let’s denote the force that exerts on as and force exerted by on as ; r is the distance between charges.

1q 2q 2F

2q 1q 1F

221221

0122

21

01 ˆ

4

1ˆ

4

1Fr

r

qqr

r

qqF

(Newton’s third law works!)

Like charges repel; opposites attract

Exercise: If two electrons are placed meters apart, what is the magnitude of the Coulomb force between them? Compare this to the gravitational force between them.

810

Solution: The magnitude of electric force

NNx

qFE

1228

2199

2

2

0

103.2)10(

)106.1(109

4

1

The magnitude of gravitational force

NNx

mmGFG

5528

23111

221 104.5

)10(

)109(1067.6

43

210

2

104.5

3.2

4

mGm

q

F

F

G

E

(no matter what the separation is)

r

Gauss’s Law

0Q

SdE

A conducting sphere, conducting shell, insulating sphere, shell …..

d

+

+

+

+

+

+

a

l

-

-

-

cap

EaEdSSdE

0a

Ea 0

E (the total field at any point

between the plates)

Two parallel conducting plates

CapacitorsConsider two large metal plates which are parallel to each other and separated by a distance small compared with their width.

Area A

The field between plates is 0

E

VL

y

LdydyEbottomVtopVLL

y00 00

)]()([

A

QLL

A

ALbottomVtopV

000

)]()([

0AQL

V

The capacitance is:

L

A

AQL

Q

V

QC 0

0

L

A

AQL

Q

V

QC 0

0

Capacitors in series: ...1111

321

CCCCtot

Capacitors in parallel: ...321 CCCCtot

22

2

1

2

1Q

CCVW

faradC ][

Current, Ohm’s Law, Etc.

dt

dQi

)(;:' VoftindependenConstRi

VRLawsOhm

A

lR

Ej

jE

Current Density

S

Sdji

Consider current flowing in a homogeneous wire with cross sectional area A.

jAdSjjdSSdjiA A A

A

ij

For steady state situation

0 Sdj

0 rdE

Circuits will be included!

Joule’s Law

R

VRiViP

22

The force on a charge q moving with a velocity F

v

)( BvEqF

sec)//(][ meterCoulombNewtonsB

gausssmCNewtonmwteslaT 42 10//1/1)(1

TGaussBEarth4101

If the magnitude of the force

sinqvBF

0E

qvBFBv

amF

0 FmaF rr

r

vmmrqvB

22 ri

qB

mvr

The angular velocity

m

qB

qBmvv

r

v

0|| FBv

Using Crossed and Fields E

B

Velocity selector

0 qEqvB

vBE

B

Ev independent of the mass of the particle!

Ampere’s Law

irdB 0

The field produced by an infinite wire

a

iB

2

0

Biot-Savart Law

3

)(

r

rsdiBd

Infinitesimally small element of a current carrying wire produces an infinitesimally small magnetic field

Sd

i

(Also called Ampere’s principle)

30 )(

4 r

rsdiBd

r

0 is called permeability of free space

2770 )/(104)/(104 ampNmeterampwebers

Force exerted on a current carrying wire

BsidFd

For a straight, finite wire of length and uniform magnetic field l

BliF

Faraday’s Law of Induction

The induced EMF in a closed loop equals the negative of the time rate of change of magnetic flux through the loop

dt

dEMF B

SdBdt

d

dt

drdE B

There can be EMF produced in a number of ways:

• A time varying magnetic field• An area whose size is varying• A time varying angle between and • Any combination of the above

B

Sd

R

From Faraday’s law: a time varying flux through a circuit will induce an EMF in the circuit. If the circuit consists only of a loop of wire with one resistor, with resistance R, a current

R

EMFi

Which way?

Lenz’s Law: if a current is induced by some change, the direction of the current is such that it opposes the change.

dt

drdE B

A Simple Generator

Faraday’s Law is used to obtain differential equations for some simple circuits.

SdBrdE

Self-inductance L

LiSdBB

Displacement current

SdEdt

diD

0

SdEdt

dirdB

000

Thank you for a great semester!