Lecture 22: TUE 13 APR Lecture 22: TUE 13 APR 20102010

Ch.32.1–5: Maxwell’s Ch.32.1–5: Maxwell’s equationsequations

Ch.33.1–3: Electromagnetic Ch.33.1–3: Electromagnetic WavesWaves

James Clerk Maxwell (1831-1879)

Physics 2102

Jonathan Dowling

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EXAM 03: 6PM THU 15 APR LOCKETT 6

The exam will cover: Ch.28 (second half) through Ch.32.1-3 (displacement current, and Maxwell's equations). The exam will be based on: HW07 – HW10.

The formula sheet for the exam can be found here:http://www.phys.lsu.edu/classes/spring2010/phys2102/formulasheet3.pdf

You can see examples of old exam IIIs here:http://www.phys.lsu.edu/classes/spring2009/phys2102/Test3.oldtests.pdf

Maxwell I: Gauss’ Law for E-Maxwell I: Gauss’ Law for E-Fields:Fields:

charges produce electric fields,field lines start and end in charges

∫ =•S

qdAE 0/ ε

S

S S S

Maxwell II: Gauss’ law for B-Maxwell II: Gauss’ law for B-Fields:Fields:

field lines are closedor, there are no magnetic monopoles

∫ =•S

dAB 0

S

S

S

S

S

S

Maxwell III: Ampere’s law:Maxwell III: Ampere’s law:electric currents produce magnetic fields

∫ =•C

idsB 0μ

C

Maxwell IV: Faraday’s law:Maxwell IV: Faraday’s law:changing magnetic fields produce (“induce”)

electric fields

∫∫ •−=•SC

dABdtddsE

Maxwell Equations I – Maxwell Equations I – IV:IV:

∫ =•S

qdAE 0/ ε

∫ =•S

dAB 0

∫ =•C

idsB 0μ

∫∫ •−=•SC

dABdtddsE

∫ =•S

dAB 0

∫ =•C

dsB 0

∫∫ •−=•SC

dABdtddsE

In Empty Space with No Charge or In Empty Space with No Charge or CurrentCurrent

…very suspicious…NO SYMMETRY!

?qq=0=0

ii=0=0

E • dA=0S—∫

Maxwell’s Displacement CurrentMaxwell’s Displacement CurrentIf we are charging a capacitor, there is a current left and right of the capacitor.

Thus, there is the same magnetic field right and left of the capacitor, with circular lines around the wires.

But no magnetic field inside the capacitor?

With a compass, we can verify there is indeed a magnetic field, equal to the field elsewhere.

But there is no current producing it! ?

EEBB BB

The missingMaxwellEquation!

Maxwell’s Fix

dtd

dtEAd

dtEdd

dA

dtdVC

dtCVd

dtdqi EΦ

====== 000 )()()( εεε

We can write the current as:

We calculate the magnetic field produced by the currents at left and at right using Ampere’s law :

∫ =•C

idsB 0μ

q=CV V=EdC=ε0A/d ΦE=∫E•dA=EA

EE

id=ε0dΦ/dt

∫∫ •=•SC

dAEdtddsB 00εμ

B !

E

i

B

i

B

Displacement CurrentDisplacement CurrentMaxwell proposed it based on symmetry and math — no experiment!

∫ ≠•C

dsB 0

Maxwell’s Equations I – Maxwell’s Equations I – V:V:

∫ =•S

qdAE 0/ ε

∫ =•S

dAB 0

idAEdtddsB

SC000 μεμ +•=• ∫∫

∫∫ •−=•SC

dABdtddsE

II

IIII

IIIIII

IVIV

V

∫ =•S

dAE 0

∫ =•S

dAB 0

∫∫ •=•SC

dAEdtddsB 00εμ

∫∫ •−=•SC

dABdtddsE

Maxwell Equations in Empty Maxwell Equations in Empty Space:Space:

Fields withoutsources?

Changing E gives B.Changing B gives E.

A solution to the Maxwell equations in empty space is a “traveling wave”…

c =1μ0ε0

=3×108μ /s

The electric-magnetic waves travel at the speed of light?Light itself is a wave of

electricity and magnetism!

Maxwell, Waves, and LightMaxwell, Waves, and Light

d 2Edx2 =−μ0ε0

d 2Edt2

⇒ E =E0 sink(x−ct)

electric and magnetic fields can travel in EMPTY SPACE!

∫∫ •=•SC

dAEdtddsB 00εμ ∫∫ •−=•

SC

dABdtddsE

First person to use electromagnetic waves for communications:Guglielmo Marconi (1874-1937), 1909 Nobel Prize

(first transatlantic commercial wirelessservice, Nova Scotia, 1909)

Electromagnetic wavesElectromagnetic wavesFirst person to prove that electromagnetic waves existed:

Heinrich Hertz (1875-1894)

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Electromagnetic Waves: Electromagnetic Waves: One Velocity, Many Wavelengths!One Velocity, Many Wavelengths!

with frequencies measured in “Hertz” (cycles per second)and wavelength in meters.

http://imagers.gsfc.nasa.gov/ems/http://www.astro.uiuc.edu/~kaler/sow/spectra.html

How do E&M Waves How do E&M Waves Travel?Travel?

Is there an “ether” they ride on? Michelson and Morley looked and looked, and decided it wasn’t there. How do waves travel???Electricity and magnetism are

“relative”: Whether charges move or not depends on which frame we use…This was how Einstein began thinking about his “theory of special relativity”… We’ll leave that theory for later.

A solution to Maxwell’s equations in free space:

)sin( txkEE m ω−=

)sin( txkBB m ω−=n.propagatio of speed ,c

k=ω

c =Eμ

Bμ

=1μ0ε0

=299,462,954 μs = 187,163 μ ilεs/sεc

Visible light, infrared, ultraviolet,radio waves, X rays, Gammarays are all electromagnetic waves.

Electromagnetic WavesElectromagnetic Waves

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Radio waves are reflected by the layer of the Earth’s atmosphere called the ionosphere.

This allows for transmission between two points which are far from each other on the globe, despite the curvature of the earth.

Marconi’s experiment discovered the ionosphere! Experts thought he was crazy and this would never work.

Fig. 33-1

The wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum.

Maxwell’s Rainbow

Fig. 33-2

(33-2)

An LC oscillator causes currents to flow sinusoidally, which in turn produces oscillating electric and magnetic fields, which then propagate through space as EM waves.

Fig. 33-3Oscillation Frequency:

1LC

ω =

Next slide

The Traveling Electromagnetic (EM) Wave, Qualitatively

(33-3)

c =Eμ

Bμ

=1μ0ε0

c =Eμ

Bμ

=1μ0ε0

Fig. 33-5

Mathematical Description of Traveling EM Waves

Electric Field: ( )sinmE E kx tω= −

Magnetic Field: ( )sinmB B kx tω= −

Wave Speed:0 0

1cμ ε

=

Wavenumber: 2k πl

=

Angular frequency:2πωt

=

Vacuum Permittivity:0ε

Vacuum Permeability: 0μ

All EM waves travel a c in vacuum

Amplitude Ratio: m

m

E cB

= Magnitude Ratio: ( )( )

E tc

B t=

EM Wave Simulation

(33-5)

Electromagnetic waves are able to transport energy from transmitterto receiver (example: from the Sun to our skin).

The power transported by the wave and itsdirection is quantified by the Poynting vector. John Henry Poynting (1852-1914)

211|| Ec

EBS00

==μμ

The Poynting Vector: The Poynting Vector: Points in Direction of Power FlowPoints in Direction of Power Flow

E

BS

Units: Watt/m2

For a wave, sinceE is perpendicular to B: BES

rrr×=

0μ1

In a wave, the fields change with time. Therefore the Poynting vector changes too!! The direction is constant, but the magnitude changes from 0 to a maximum value.

____________222

___)(sin11 tkxE

cEcSI m ω

μμ−===

00

The average of sin2 overone cycle is ½:

2

21

mEc

I0

=μ

21rmsE

cI

0=

μ

Both fields have the same energy density.

2 22 2

01 1 1 1( )2 2 2 2E B

B Bu E cB uε ε εε μ μ0 00 0 0

= = = = =

or,

EM Wave Intensity, Energy EM Wave Intensity, Energy DensityDensity

A better measure of the amount of energy in an EM wave is obtained by averaging the Poynting vector over one wave cycle. The resulting quantity is called intensity. Units are also Watts/m2.

The total EM energy density is then 022

0 / με BEu ==

Solar EnergySolar EnergyThe light from the sun has an intensity of about 1kW/m2. What would be the total power incident on a roof of dimensions 8m x 20m ?

I = 1kW/m2 is power per unit area.P=IA=(103 W/m2) x 8m x 20m=0.16 MegaWatt!!

The solar panel shown (BP-275) has dimensions 47in x 29in. The incident power is then 880 W. The actual solar panel delivers 75W (4.45A at 17V): less than 10% efficiency….

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The electric meter on a solar home runs backwards — Entergy Pays YOU!

The intensity of a wave is power per unit area. If one has a source that emits isotropically (equally in all directions) the power emitted by the source pierces a larger and larger sphere as the wave travels outwards: 1/r2 Law!

24 rPI s

π=

So the power per unit area decreases as the inverse of distance squared.

EM Spherical WavesEM Spherical Waves

ExampleExampleA radio station transmits a 10 kW signal at a frequency of 100 MHz. At a distance of 1km from the antenna, find the amplitude of the electric and magnetic field strengths, and the energy incident normally on a square plate of side 10cm in 5 minutes.

222 /8.0

)1(410

4mmW

kmkW

rPI s ===

ππ

mVIcEEc

I mm /775.022

1 2 ==⇒= 00

μμ

nTcEB mm 58.2/ ==

mJSAtUA

tUAPS 4.2/

==Δ⇒Δ

==Receivedenergy: