ene 325 electromagnetic fields and waves lecture 10 time-varying fields and maxwell’s equations
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ENE 325ENE 325Electromagnetic Electromagnetic Fields and WavesFields and Waves
Lecture 10Lecture 10 Time-Varying Fields Time-Varying Fields and Maxwell’s Equationsand Maxwell’s Equations
Magnetic boundary conditions B
1n = B2n
and Inductance and mutual inductance
self inductance L is defined as the ratio of flux link age to the current generating the flux,
henrys or Wb/A.
mutual inductance M , where M12
= M21
.
ReviewReview
totalNL
I
2 1212
1
1 2121
2
NM
I
NM
I
1 2 12 .nH H a K
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- Time Varying fields and Max- Time Varying fields and Max well’s equations well’s equations
Concept The electric field E is produced by the c
hange in the magnetic field B. The magnetic field B is produced by the
change in the electric field E.
Faraday’s law Faraday’s lawd
emfdt
V
where emf = electromotive force that may establish a current in a suitable closed circuit and is a voltage that arises
from conductors moving in static or changing magnetic fields.
is arisen from
1. the change of flux in a closed path 2. the moving closed path in a stationary magnetic field 3. both 1 and 2 For the N number of loops, v.
d
dt
demf N
dt
d
dt
emf in the closed loop is emf in the closed loop is not zeronot zero
0S
demf E dL B dS
dt
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a) direction of the induced current b) emf
Changing flux in a stationar Changing flux in a stationar y path (transformer y path (transformer emfemf))
From ,S
demf E dL B dS
dt
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apply Stokes’ theorem,
( )S S
BE dS dS
t
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( )B
E dS dSt
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( ) .B
Et
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So we have 1st Maxwell’s equation
TransformerTransformer To transform AC voltages and currents between
a pair of windings in magnetic circuits
1 1 2 2N i N i
With Faraday’s law, we have
1 1 2 2, .d d
v N v Ndt dt
22 1
1
.N
v vN
d
dt
Since the term is the same for both voltages, so we get
Ex1Ex1 Assume , prove the Assume , prove the 11stst Maxwell’s equation. Maxwell’s equation.
ktzoB B e a
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r
z
Changing flux in a moving Changing flux in a moving closed path (1)closed path (1) a conductor moves in a uniform magnetic field.
.
o od dy
emf B w B vwdt dt
The sign of emf determines the direction of the induced current.
0 0
. ���������������������������� y w
z zo oB dS B a dxdya B yw
Changing flux in a moving Changing flux in a moving closed path (2)closed path (2) Examine in a different point of view
So we get V.( )emf v B dL �������������������������� ��
z
y
x
Byva
w
Changing flux in a moving Changing flux in a moving closed path (3)closed path (3) Combing both effects yields
( ) .S
d Bemf E dL dS v B dL
dt t
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Ex2Ex2 Let Let mT, fi mT, findnd
66cos10 sin 0.01 zB t xa��������������
a) flux passing through the surface z = 0, 0 < x < 20 m, and 0 < y < 3 m at t = 1 S.
b) value of closed line integral around the surface specified above at t = 1 S.
3Ex3Ex A moving conductor is located A moving conductor is located on the conducting rail as shown at t on the conducting rail as shown at t
ime ime t = t =0,0,
a) find emf when the conductor is at rest at x = 0.05 m and T.40.3sin10 zB ta
��������������
z
y
x
B
0.05
b) find emf when the conductor is moving with the s peed m/s.150 xv a
Displacement current Displacement current (1)(1)
The next Maxwell’s equation can be found in term - s of time changing electric field
From a steady magnetic field,
H J ����������������������������
0.H J ����������������������������
From the equation of continuity,
vJt
r
��������������
therefore 0.v
t
r
this is impossible!
Displacement current Displacement current (2)(2)
Another term must be added to make the equation valid.
.dH J J ������������������������������������������
2nd Maxwell’s equation
In a non-conductive medium, 0.J ��������������
Displacement current Displacement current (3)(3) We can show the displacement current as
The more general Ampere’s circuital law:
.dds s
DI J dS dS
t
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dH dL I I ����������������������������
Where is the displacement c Where is the displacement c urrent from? urrent from?
Consider a simple current loop, let emf = Vocost
4Ex4Ex Determine the magnitude Determine the magnitude of for the following situation of for the following situation
s:s:a) in the air near the antenna that radiates
V/m.
dJ��������������
880cos(6.277 10 2.092 ) zE t y a ��������������
b) 100a pair of cm2 aaaa aaaaaa aaaaaaaaa aa a 10. mm thick layer of lossy dielectric characteri
zed by r = 50and 10 10-4 aaaaa aaa a/ ol t age across pl at es V(t) 102= . cos( 103ta aa