lec 10 time varying field and maxwell equations
TRANSCRIPT
ELECTROMAGNETIC PROF. A.M.ALLAM
8/23/2020 LECTURES 1
TIME VARYING FIELD AND
MAXWELL EQUATIONS
EMF
Michael Faraday
(1791–1867
John H Poynting
1884
LEC 10
ELECTROMAGNETIC PROF. A.M.ALLAM
2
Static E&M fields
E &H are independent
We've learned
Now we are going to
Time varying current Electromagnetic waves (E & H)
E &H are interdependent
Time-varying E(t) produces time varying H(t)
Time-varying H(t) produces time varying E(t)
1-Introduction
Stationary charges Electrostatic fields (E)
Steady current Magnetostatic field (H)
ELECTROMAGNETIC PROF. A.M.ALLAM
In 1820 C.H. Oersted demonstrated that an electric current
affected a compass needle
After this, Faraday professed his belief that if a current could
produce a magnetic effect, then the magnetic effect should be
able to produce a current (magnetism)
In 1831, the electric induction phenomenon was discovered as
a results of Faraday’s experiments
If two separate coils are wound on an iron
ring. One of them is connected through a
switch to DC battery
2- Faraday's law of induction
It was observed that whenever the current
was changed, an induced current would
flow in the other coil
•Faraday’s first experiment:
ELECTROMAGNETIC PROF. A.M.ALLAM
4
-If a magnet moves near a coil, an induced
current will be produced in the galvanometer
•Faraday’s second experiment:
-Generally, for any closed path C in space
linked by a changing magnetic field
the induced voltage; electromagnetic force
(emf) around this path is produced and is
equal to the negative time rate of change of
the total magnetic flux through the closed
path
This process is called electromagnetic induction
t
tVfme i n d
)( ..
The negative sign means that the induced voltage is in
such direction that it resists the original change ( Lenz’s
law)
Transformer emf if time varying B(t) links a stationary loop
Motional emf if a moving loop changes its area with time relative to normal B
This is Faraday's law of induction
The change of magnetic flux with time produces an induced EMF
( electric field ) in any closed circuit surrounding that flux=1 +2 +…
different in each
turn
N-turns
= N
same in each turn
(t)
N-turns
ELECTROMAGNETIC PROF. A.M.ALLAM
8/23/2020 LECTURES 5
Faraday’s law in integral form:
tV in d
Faraday’s law in differential form:
SC
SdBt
dE
..
Notes:
The electric field has two sources (charges and time varying magnetic field)
If there is no time variation ( / t =0), gives (Static case)
The induced electric field is not conservative (rotational)
0Eo r 0d.EC
S
SdBt
.
t
BE
)(
Stock’s Th.
C
dE
. S
SdE
).(
Maxwell’s equation in time
varying field, Faraday’s lawE in time varying field is not conservative i.e., the work
done in moving a charge along a closed path is due to
the energy from time varying B
ELECTROMAGNETIC PROF. A.M.ALLAM
Ampere’s law for magnetostatic field says:
There is an identity div-curl =curl-grade =0:
=0The conduction
current
But for the time varying charge:
To satisfy these two conditions we must add another term, such that:
Hence, =0
3-Displacement current
The displacement
current
Ampere’s law for
time varying field
ELECTROMAGNETIC PROF. A.M.ALLAM
Differential form Integral form
)1t
BE
. )3 D
t
DJH
)2
0. )4 B
tJ
.
SC
S.dB .dE
t
VS
d vSdD
.
SSC
SdDt
SdJdH
...
0.S
SdB
VS
dvt
SdJ
.
Constitutive relations:
HB ; EJ ;
ED
where , and are the medium
parameters.
J
Jimp
Jind
Note:
Jcond = E
Jconv = v
4-Maxswell’s equations
Ampere’s circuital law
Gauss flux theorem
Continuity of B lines
Continuity equation
Faraday’s law of induction
ELECTROMAGNETIC PROF. A.M.ALLAM
4-Maxswell’s equations
ELECTROMAGNETIC PROF. A.M.ALLAM
9
In free-space:
[H/m]. 104
]. [F/m 10854.8
7
o
12
o
V/m]. ;er [Volts/met intensity field .Electric.......... E
A/m]. ; [Amperes/m intensity field .Magnetic.......... H
T]. ; Teslaor wb/m; [Webers/m density flux .Magnetic.......... B 22
]C/m ; /m[Coulombsdensity current)ent (Displacemflux .Electric.......... D 22
].A/m ; [Amperes/mdensity current ..Electric.......... J 22
].C/m ; m[Coulombs/ density charge ..Electric.......... 33
H/m]. ; [Henery/mty permeabili .Magnetic..........
F/m]. ; [Farad/my permitivit ic..Dielectr..........
/m].; [Moh/mty conductivi .Electric..........
ELECTROMAGNETIC PROF. A.M.ALLAM
5-Complex representation of field quantities
1. Scalars:
]Re[)( otj
oe
]e Re[ tj
)cos()( oo tt
]e eRe[ tjj
oo
e oj
o
o
o
+1
+j
Complex phasor form which is represented by a point in complex domain
e ˆ ˆ ˆ Re tj
z
j
ozy
j
oyx
j
ox aeEaeEaeE zyx
tj
zzyyxx e a E a E a E Re
zyˆ )cos( ˆ )cos(ˆ )cos(),( atEatEatEtrE zozyoyxxox
2. Vectors:
zzyyxxa E a E a E E
)cos()( oo tt ]e Re[ tj
]e )(- Re[)(
]e )(j Re[)(
tj2
2
2
tj
t
t
t
t
)(
)(
2
2
2
t
jt
3. Derivatives:
The phasor form