Ch 32 Maxwell’s Equations

1)Gauss’ Law for Electrostatics

2)Gauss’ Law for Magnetostatic

3)Faraday's Law

4)Maxwell-Ampere Law

They predict that light is an E&M Wave propagating at speed =3x108 m/s

1831-1879 (Edinburgh, Scotland)

The first Unification of Physical Laws. The pursuit has never stopped….

Can you ever just get a north pole or a south pole ???

Gauss’s Law for MagnetismGauss’s Law for MagnetismReminder of the definition of magnetic fluxReminder of the definition of magnetic flux

0

qAdB mag

A

B

?????

As of today 4/21/05 there is no evidence of magnetic monopoles

See :Phys.Rev.Lett.85:5292,2000

Gauss’s Law for MagnetismGauss’s Law for MagnetismOnly magnetic dipoles existOnly magnetic dipoles exist

0AdB

A

The Search for magnetic monopoles Goes ON!

Null result sets the lower bound on a monopole mass to be about

m=295-495 GeV

Carl Friedrich Gauss 1777-1855

Eoclosed

surface

qE dA

The total flux passing through a closed The total flux passing through a closed surface is proportional to the charge enclosed surface is proportional to the charge enclosed within that surface.within that surface.

KEY TO USING Gauss’s LawKEY TO USING Gauss’s Law

• The shape of the surrounding surface is one that MIMICS the symmetry of the charge distribution …..

A

E AdE

Faraday: Changing Flux Faraday: Changing Flux due to moving due to moving

permanent magnetpermanent magnet

dt

dN

1791-1867 British Physicist

Induced EMF produced by a changing Magnetic Flux!

dt

d

area

AdB

Induced electric fieldsInduced electric fields

We will assume here that We will assume here that BB is is increasing into the page increasing into the page

area

B AdB

dt

dsdE

sdE

sdEqsdFW

loop

loop

loop

0

so,

loop around chargeunit per work

Electric Field circulates about a changing magnetic field (into the

page here)

dt

dsdE

loop

Maxwell thought is there a symmetry here? Can there be a circulating magnetic field around a changing electric field?

dt

dsdB

dt

dsdB

Eoo

loop

E

loop

????

Maxwell’s Law of Induction

WAIT!!!

Didn’t we already see that

B circulates around a current carrying WIRE ?!!?

Recall Ampere’s Law Recall Ampere’s Law Andre Marie Ampere (1775-1836)Andre Marie Ampere (1775-1836)

encloisdB

For For r>R,r>R, radius of wire the radius of wire the magnetic field must have magnetic field must have cylindrical symmetrycylindrical symmetry

encoisdB

ir2B o

loopAmperian

tor tangent unit vecto t

tr2

iB o

ˆ

ˆ

Long Straight Wire with a constant currentLong Straight Wire with a constant current

enclodo

encloE

oo

loop

ii

idt

dsdB

Ampere-Maxwell Law and Maxwell’s Displacement currentAmpere-Maxwell Law and Maxwell’s Displacement current

Displacement Current

dt

di

dt

EAd

dt

dqi

EAq

dEd

Aq

CVq

Eod

ontdiscplacme

o

o

""

enclodo

encloE

oo

loop

ii

idt

dsdB

Maxwell’s EquationsMaxwell’s Equations

oA

Eq

AdE

Gauss’s LawGauss’s Law

0AdB

A

B

Gauss’s Law for MagnetismGauss’s Law for Magnetism

dt

disdB E

ooo

loopclose

Ampere’s Maxwell’s LawAmpere’s Maxwell’s Law

dt

dsdE B

loopclosed

Faraday’s LawFaraday’s Law

Information AGE, Medicine, Engineering ….

• Based on the physics of E&M waves• T.V. telecommuncations, radio transmission, radar

…• Challenge of today’s engineers to try to envision

what global interconnections will be in 20 years (Halliday and Resnick 04)

• Starting point understanding Electro-mag waves which is summarized by Maxwell’s rainbow!

E&M Waves and Maxwell

• Crowning achievement was to show that light is a traveling E&M wave

• The basis for the study of optics! Phys. 213 and Dr. Romberger Phys. Science 440

• In mid 1800s only infrared, visible and ultraviolet forms of light were only E&M waves know

• Heinrich Hertz later showed that what we call radio waves also ..

m1010 42 to

Maxwell’s Rainbow

Human Eye Sensitivity

Traveling Waves

Consider a transformer: A generating LC circuit which establishes a frequencey

and a secondary coil with a dipole antenna.

You have a changing E and B field coupled together!!!

However, changes don’t appear instantaneously, they happen a the speed of light with the disturbance traveling outward at that speed. Will see

LC

1

sm103cv 8 /

BEMomentum

Looking toward the wave front the amplitudes are related by the cross product

Wave is transverse/perpendicular to the direction of travel

Field vary sinusoidally )sin(,

)sin(,

tkxBtxB

tkxEtxE

m

m

o0

1c

At speed of light!!

Traveling E&M waveNo need for a medium on wave generates the other.

Prediction of Maxwell’s equation and proved by Einstein with a little bit of symmetry 1905

Faraday’s Law of InductionMaxwell’s Law of Induction

dt

dsdE B

surfaceclosed

Faraday’s LawFaraday’s Law

dt

dsdB E

oo

loopclose

Ampere’s Maxwell’s LawAmpere’s Maxwell’s Law

tB

xE

dt

dBdxdE

hdxBdt

dEhhdEE

dt

dsdE B

loopclosed

Faraday’s Law of InductionFaraday’s Law of Induction

Applying Faraday’s law to the loop above

Maxwell’s Law of Induction

tE

εμxB

oo

dt

dEdxdB

hdxEdt

dBhhdBB

dt

dsdB

oo

oo

Eoo

loopclose

Applying Maxwell’s law of inductionto the loop above

The Wave Equation• Source free region ie

no wires and no charges

)sin( ˆ)sin(,

)sin( ˆ)sin(,

tkxBktkxBtxB

tkxEjtkxEtxE

mm

mm

Eliminate

and

and

2

2

2

2

2

22

2

2

2

22

2

2

2

2

oo2

2

xE

ctE

xB

ctB

equation wave251 math xB

εμ1

tB

Get

t

E

x

t

E

xx

B

t

E

x

B

t

B

x

E

tt

B

x

E

oooo

Solution

fast..... pretty though ous!Instantane Nots

ms

m

f

f

k

c

88

00

100.31099792458.2

equation waveinto plugging and sderivative takingfrom 22

equation wavefrom 1

Spherical Wave Energy John Henry Poynting (1852-1914): E&M waves transport energy!

From which we define intensity

rms2

o

m2

o

2m

o

ave2

oave

Ecμ

12

Ecμ

1I

ωt)sin(kxEcμ

1E

cμ1

SI

!!AVE!

1

anglesright at since 1

ˆ 11

2Ec

S

EBS

rEBBES

o

o

oo

20

/1

m

W

area

power

area

timeenergyBES

BEP

Looking toward the wave front the amplitudes are related by the cross product

Wave is transverse/perpendicular to the direction of travel

Field vary sinusoidally )sin(,

)sin(,

tkxBtxB

tkxEtxE

m

m

o0

1c

At speed of light!!