전기자기학(I)
7. Time-varying Fields and
Maxwell’s Equation
School of Electronic and Electrical Engineering
Prof. Keum Cheol Hwang
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전기자기학(I)
Faraday’s law
 The coupling between the electric and magnetic fields based on
experimental relation.
 This law was formulated in 1831 by Michael Faraday after a series of
experiments.
 Faraday observed that if he moved the magnet while the loop remained
stationary, a current flowed in the loop.
 The current is due to an induced voltage in the
(electromotive force)
loop.
emf = -
¶F
¶
= - ò B ds
¶t
¶t S
(V )
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전기자기학(I)
 The electromotive force of closed current loop is,
emf =V = ò
 E d 
C
ò E d  = emf = C
¶
B ds
ò
S
¶t
Applying Stokes’s theorem
ò E d  = ò (´E )ds = C
 Movement of the loop in the
presence of the stationary magnet
generates an identical current.
S
´E = -
¶B
¶t
¶
B ds
ò
S
¶t
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전기자기학(I)
Field equations for time-dependent electromagnetic field
Difference form
Integral form
¶B
´E = ¶t
ò E d  = -
´H = J
òC H d  = I
D = rv
òS D ds = Q
B = 0
òS B ds = 0
C
¶F
¶t
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전기자기학(I)
A static conductor in a time-varying magnetic field
 Based on the Faraday’s law, the emf is
emf = -
¶F
¶
= - ò B ds
¶t
¶t S
(V )
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전기자기학(I)
Ex) A uniform magnetic field is generated by a time-varying source as
shown in the figure. A stationary square loop is placed such that its
plane is perpendicular to the magnetic flux density B . Assume that the
flux density is sinusoidal and given as B = B0 sin wt with:
B0 = 0.1T, w = 100p rad/s, a = 0.1 m, b = 0.1 m
1) Calculate the induced emf in the loop
¶
B ds
ò
S
¶t
¶
= - BS
¶t
= -S wB0 cos wt
emf = -
= -0.314 cos (314t ) [V ]
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전기자기학(I)
2) What is the induced emf if the loop is at an angle a to the field
B ds = Bds cos (90 - a )
= Bds sin a
¶
emf = - ò B ds
¶t S
¶
= - BS sin a
¶t
= -S wB0 cos wt sin a
= -0.314 cos (314t ) sin a [V ]
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전기자기학(I)
A moving conductor in a static magnetic field
 The force on a charge moving in a constant magnetic field was
calculated
(N )
F = qv ´B
 This force will cause the freely movable electrons in the conductor
to drift toward one end of the conductor and leave the other end
positively charged
d
--
v
v
++
B
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전기자기학(I)
 Electric field intensity generated by motion charge,
F
E = = v ´B
q
 Finally, the emf for a bar is,
d
--
2
emf =V21 = ò E d 
1
v
2
= ò (v ´B )d 
v
1
++
B
 If the moving conductor is a part of
a closed circuit
V =ò
 (v ´B )d 
C
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전기자기학(I)
Ex) A metal bar slides over a pair of conducting rails in a uniform
magnetic field B = aˆz B0 with a constant velocity v
1) Determine the open-circuit voltage V0 that appears across terminals
1 and 2
V0 =V1 -V2 = ò
 (v ´B )d 
C
v
=ò
1¢
2¢
(aˆx v ´aˆz B0 )aˆyd 
= -vB0h
(V )
2) Assuming that a resistance R is connected between the terminals,
find the electric power dissipated in R
Current flow I = uB0h / R ,
2
Power dissipation in R, Pd = I 2R =
(uB0h )
R
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전기자기학(I)
General case - A moving circuit in a time-varying magnetic field
 The electromotive force were obtained from two different situation.
1) The first is generated in a stationary circuit by a change
in flux
2) The second is generated due to circuit motion in a stationary
magnetic field
Total emf =V = -
¶
B ds + ò
v ´ B )d 
(

ò
S
C
¶t
(1)
(2)
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전기자기학(I)
Ex) An h by w rectangular conducting loop is situated in a changing
magnetic field B = aˆy B0 sin wt . The normal of the loop initially makes an
angle a with aˆy , as shown in Fig. 6-6. Find the induced emf in the loop.
v
v
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전기자기학(I)
a.
When the loop is at rest
F = ò B ds
S
= (aˆy B0 sin wt )(aˆn hw )
= B0hw sin wt cos a
\ V1 = -
¶F
= -B0hw w cos wt cos a
¶t
= -B0S w cos wt cos a
b. When the loop rotates with an angular velocity w about the x-axis
Contribution of motional emf is,
V2 = ò
 (v ´B )d 
C
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전기자기학(I)
V2 = ò
 (v ´B )d 
C
=V12 +V23 +V34 +V41
= ò (v ´ B )d  + ò (v ´ B )d  + ò (v ´ B )d  + ò (v ´ B )d 
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34
éæ w ö
ù
÷
ç
ê
= ò çaˆn w ÷÷´(aˆy B0 sin wt )ú (aˆxdx )
úû
2 êç
ëè 2 ø
ù
3 éæ
ö
w
÷
+ ò êçç-aˆn w ÷÷´(aˆy B0 sin wt )ú (aˆxdx )
úû
4 êç
2 ø
ëè
1
H/W
æw
ö÷
ç
= 2 ç wB0 sin wt sin a ÷÷h
çè 2
ø
Therefore, total emf is
\ V = V1 + V2
æw
ö
= -B0S w cos wt cos a + 2 çç wB0 sin wt sin a ÷÷÷h
çè 2
ø
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전기자기학(I)
æw
ö÷
ç
V = -B0S w cos wt cos a + 2 ç wB0 sin wt sin a ÷÷h
çè 2
ø
= -B0S w cos wt cos a + B0S w sin wt sin a
= -B0S w (cos wt cos a - sin wt sin a )
Also, the angle a after time t is, a = wt
\ V = -B0S w (cos wt cos wt - sin wt sin wt )
= -B0S w (cos2 wt - sin 2 wt )
= -B0S w cos 2wt
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전기자기학(I)
Derivation of Maxwell’s equations
Taking divergence of the both sides
´ E = -
¶B
¶t
´H = J (?)
D = rv
B = 0
(´H ) = J
=0
(null identity)
\ J = 0
 However this is generally not true
(in conflict of continuity equation)
¶rv
J = ¶t
 How should Ampere’s law be modified?
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전기자기학(I)
(´H ) = 0 = J +
¶rv
¶t
Using Gauss’s law,
æ
¶D ÷ö
ç
(´ H ) = çJ +
÷
çè
¶t ÷ø
´H = J +
¶D
¶t
 A time varying electric field will give rise to a magnetic field,
even in the absence of a free current flow ( J = 0 ).
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전기자기학(I)
Integral form of Maxwell’s equations
 Applying Stokes’s theorem,
dF
òC E d  = - dt
æ
¶D ö÷
¶D
ç
òC H d  = òS ççèJ + ¶t ÷÷øds = I + òS ¶t ds
 Applying divergence theorem,
òS D ds = òV rvdv = Q
òS B ds = 0
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전기자기학(I)
Electromagnetic field equation – Maxwell’s equation
Difference form
Integral form
Significance
¶B
´ E = ¶t
¶D
´H = J +
¶t
ò E d  = -
¶F
¶t
Faraday’s law
ò H d  = I + ò
¶D
ds
S ¶t
Ampere’s law
D = rv
òS D ds = Q
Gauss’s law
B = 0
ò B ds = 0
No isolated
magnetic charge
C
C
S
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전기자기학(I)
Electromagnetic boundary conditions
Electromagnetic BC for general materials
E1t = E 2t
H 1t - H 2t = J s
D1n - D2n = rs
B1n = B2n
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전기자기학(I)
 Boundary conditions between two perfect dielectrics ( J s = 0, rs = 0 )
E1t = E 2t
H 1t = H 2t
D1n = D2n
B1n = B2n
 Boundary conditions between a perfect dielectric and a perfect
conductor ( E 2 = H 2 = 0 )
E1t = 0
H 1t = J s
D1n = rs
B1n = 0
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전기자기학(I)
Maxwell’s equation in differential equation
¶B(r, t )
 ´ E (r, t ) = ¶t
(Faraday’s law)
¶D(r, t )
(Ampere’s law)
 ´ H (r , t ) =
+ J (r , t )
¶t
Electric current density (A/m2)
  D(r , t ) = re (r , t )
(Gauss’ law)
Electric charge density (C/m3)
  B(r , t ) = 0
(Magnetic Gauss’ law)
전기자기학(I)
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전기자기학(I)
The time-harmonic field representation
 In a time harmonic field, the time dependency is sinusoidal.
E (x , y , z ;t ) = E 0 (x , y , z ) cos (wt + q )
or E 0 (x ,y , z ) sin (wt + q )
E (x , y , z ;t ) = E 0 (x , y , z ) cos (wt + q )
Initial phase angle
= Re {E 0 (x , y , z )e j qe j wt }
 We define the phasor E (x ,y , z ) that contains information on
direction, magnitude, and phase angle (does not contain time).
E (x , y , z ) = E 0 (x , y , z )e j q
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전기자기학(I)
The time-harmonic form of Maxwell’s equation
¶
E (x , y , z ;t ) = j w Re {E 0 (x , y , z )e j qe j wt }
¶t
Time-dependent form
¶B
¶t
¶D
´H = J +
¶t
´E = -
Time-harmonic form
´E = - j wB
´ H = J + j wD
D = rv
D = rv
B = 0
B = 0