Ch 32 Maxwell’s Equations 1)Gauss’ Law for Electrostatics 2)Gauss’ Law for Magnetostatic...
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Transcript of Ch 32 Maxwell’s Equations 1)Gauss’ Law for Electrostatics 2)Gauss’ Law for Magnetostatic...
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!!