Lecture:IntroductiontoElectrodynamics
5.1Electrodynamics-Electromotive
5
1 El
d
i El
i
Force
MinChen
Mi
Ch
minchen@sjtu.edu.cn
Laboratory for Laser Plasma Physics (LLP)
LaboratoryforLaserPlasmaPhysics(LLP)
DepartmentofPhysicsandAstronomy
ShanghaiJiaoTongUniversity
13-04-15
1
Where are we now?
Wherearewenow?

  E   / 0


B
  E  0
t



E
  B   0 j   0 0
t

B  0
2
Coulomb
Ampere
Gauss
Faraday
Maxwell
El t
ElectromotiveForce
ti F
Oh ’ L
Ohm’sLaw

f
Forceperunitcharge
p
g


J  f
  1/ 
?

J
current
Formostsubstances,thisissatisfied. iscalled
conductivity (电导率) For perfect conductor:
conductivity(电导率).Forperfectconductor:
 
iscalledresistivity.
y
Iftheforceisduetoelectromagneticforce:

  

J   E  v  B   E
Ohm’s Law
Resistorsaremadefrompoorlyconductingmaterials.
4
Example1:
Solution:
E V / L
WeassumeEisuniformhere!Provelater!
I  JA  EA 
5
A
L
V
Example2:
Solution:
l

E r 2rl   E  Er 
0
2r 0

L
I 
 2rdl 
2r 0
0
6
V 
b
a
I


b
dr 
ln 
2r 0
2 0  a 
 2L
ln b / a 
V
V  IR
Resistance
Iftheconductivity is uniform and for steady currents:


1
  E    J /      J  0
Steadycurrent

Conductivityisuniform
The charge is only at surface.
7
Laplace’sequationholdswithinahomogeneousohmic material
carryingasteadycurrent.
Example 3. ProvethefieldinExample1isuniform.
Example3.
Prove the field in Example 1 is uniform.
Solution:WithinthecylinderVobeysLaplace’sequations(uniform
conductivityandsteadycurrent).
Wesetthepotentialatoneendis0,andtheotherendisV
W
h
i l
di 0
d h
h
d i V0.Onthe
O h
surfaceJ.n=0(otherwisethewouldbeleakingoutintothesurrounding
spacewhichwetaketobenonconducting).
ThereforeE.n=0,andthendV/dn=0.
SoVanditsnormalderivativearespecified,fromuniquenesstheoream,V
So
V and its normal derivative are specified from uniqueness theoream V
isuniquelydetermined.ItistoguessthereisansolutionofV(z)=V0z/L
satisfiestheequationandtheboundaryconditions.SoV(z)istheanswer.
And obviously from V(z) we get E is uniform
AndobviouslyfromV(z)wegetEisuniform.
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DiscussiononOhm’slawand
Newton’slaw
l


J  f
Whythisholds?
Why
this holds?
Youhaveaforceonacharge,fromNewton’ssecondlaw
thechargeshouldbeacceleratedandthevelocityis
increasing The current should be increased!
increasing.Thecurrentshouldbeincreased!
Contradiction!
Why???
Actuallywehaveneglected other random forces onthecharges!
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a  f /m
1 2
  at 
2
a
 v  at 

2
J

f
f
L
10
Wrong!
Theelectrondoesnotalwaysstartfrom0velocity.
Ithasahugethermalvelocity!
Theaveragetimebetweencollisions
isnotdeterminedbyf.
 t 
 v 


vthermal
a
a
 t 
 f
2
2vthermal
L
J  nfq
f  v 
11
 nfq 2 
 
 2mvthermal
nfqF
2mvthermal

 E

Iftherearenmoleculesper
unitvolumeandffree
electronspermolecule,
eachwithchargeqand
massm,thecurrent
densityis:
Asaresultofallthecollisions,theworkdonebytheelectricalforceis
As
a result of all the collisions the work done by the electrical force is
convertedintoheatintheresistor.
Thepowerdeliveredis:
P  VI  I R
2
Joule heating law
Joule heating law
12
ElectromotiveForce
ect o ot e o ce
Electricfieldmakesthecurrentthe
sameallthewayaroundtheloop.
f=fs+E
Anykindofdriver!
f=E
 
 E  dl  0
Chargespileup,
Charges
pile up
induceelectricfield,
drivethecharges!
Process is quick enough!
Processisquickenough!
Electromotive force,oremf:Theintegralofaforceperunitcharge.Itis
also called electromotance (电动势).
alsocalledelectromotance
(电动势).
Withinanidealsourceofemf (aresistanceless battery,forexample),the
netforceonthechargesiszero(=infinit):E=-fs
13
 can be interpreted as the work done, per unit charge, by the source.
Motionalemf (Generator)
(
)
Heref isduetomagneticforce
However,whodoesthework?
14
M
Magneticflux
ti fl
ThefluxofBthroughtheloop
15
Flux rule formotionalemf:
Relationshipbetweenemf
and the variation of
andthevariationof
magneticflux!
Proveforanyshapeofacurrentloopthisiscorrect.
v:thevelocityofthewire
u:thevelocityofachargedownthewire
Therealvelocityofthecharge
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Proveforanyshapeofacurrentloopthisiscorrect.

  
d
  B  w  dl
d
dt


 
17


f magg dl
Example4:
Solution:
l
18
Edd
Eddycurrents
t
Alumnidisk
Cut the eddy currents
Magnet
Seehomework7.11
19
Summary
•
•
•
•
•
Ohm slaw
Ohm’s
law
Jouleheatinglaw
Electromotive force (Electronotance)
Electromotiveforce(Electronotance)
Motionalemf,magneticflux
Eddy currents
Eddycurrents
Home work(submissiononceperweek)
(
p
)
7.27.77.87.11
20