UEE 3201 Electromagnetics II
7
Fall, 2014
Time-Varying Fields and Maxwell’s Eqs.
7.1
Introduction and review
1. Electrostatic and magnetostatic fields

 ∇×E = 0
 ∇·D = ρ
(electrosttic field E is conservative)

 ∇·B = 0
 ∇×H = J
(no magnetic monopole)
(ρ is the source of electrostatic field)
(J is the source of magnetostatic field)
2. Constitutional relations (linear isotropic medium)

 D = ϵE
 H = B/µ
3. Continuity relations (conservation of charge)
∇ · J + ∂ρ/∂t = 0
4. Lorentz’s force equation
F = q(E + u × B)
5. Current density
conduction current: J = σE
convection current: J = ρu
6. Potential function

 ∇ × E = 0 → E = −∇V
 ∇·B=0 → B=∇×A
* In static cases, electric and magnetic fields are independent physical
quantities.
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UEE 3201 Electromagnetics II
7.2
Fall, 2014
Faraday’s Law of Electromagnetic Induction
1. Electrostatic field, electrostatic potential, and electromotive force (emf)
(ref: §5-3)
(a) Electrostaic field is a conservative field. The total energy gained
by an electric charge moving along a closed loop in a lossless
medium in an electrostatic field is zero. However, when an electric
energy source (such as a battery) and a load (such as a resistor)
are connected to form a closed loop, the charge will gain a net
energy (which is then lost in the load and leaves the system) from
the source when moving around the closed loop. Therefore, the
’equivalent’ electric field due to the energy source is not conservative.
(b) Electromotive force (emf, V) is used to describe the capability of
the source to provide energy to the charges. It is defined as the
energy gained by a unit charge as it moves across the source.
2. Faraday’s experiment: a current was induced in the circuit when the
magnetic flux linkage changed.
For an open surface S with boundary C, the electromotive force (emf)
V induced in C is defined as the energy gained by a unit charge traveling
around the contour C once:
V =
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UEE 3201 Electromagnetics II
Fall, 2014
The magnetic flux linkage in S is
⇒ Faraday’s law can be expressed as
- one of the fundamental laws (Coulomb’s law, Lorentz’s law, etc.)
- link of E and B through time derivative
- negative sign follows the Lenz’s law
⇒ Fundamental postulates for electromagnetic induction


 integral form:

 differential form:
- if ∂B/∂t ̸= 0 ⇒ ∇ × E ̸= 0, E is not conservative
- if ∂B/∂t = 0 ⇒ electrostatics, conservative field
(a) Lenz’s law
→ induced E and I in the negative direction
→ induced B and Φ (by E and I) in the negative direction
→ induced current and flux are opposing the change of Φ
(b) eddy current and transformer
A time-varying magnetic field induces an emf and a non-conservative
electric field in a conductor such as the ferromagnetic cores in
transformers.
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UEE 3201 Electromagnetics II
Fall, 2014
→ an eddy current J = σE around the magnetic field
→ ohmic loss and local heating for σ ̸= ∞
→ power loss can be reduced with laminated cores
3. Moving circuits in time varying fields (§7-2.4)
In calculating the magnetic flux Φ in a time varying situation, both
field (B) and circuits (S and C) can vary. It can be shown that the
induced emf V = −dΦ/dt can be written as
I
V =
E · dℓ
I
∂B
· ds + (u × B) · dℓ
= −
S ∂t
C
= transformer emf Va
C∫
+ motional (or flux-cutting) emf Va′
Ex 7-4: Rotating loop in changing magnetic field (see textbook)
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UEE 3201 Electromagnetics II
7.3
Fall, 2014
Maxwell’s Equations
1. Elecrtromagnetic postulates including Faraday’s law
2. Continuity equation (conservation of charge) must be satisfied at all
time. Therefore,
⇒ Postulates must be modified
⇒ Macroscopic electromagnetic phenomena can be fully described by
(a) Maxwell’s equations
- Differential
form



∇·D=ρ





 ∇ × E = − ∂B
∂t


∇·B=0





 ∇ × H = J + ∂D
∂t
* ρ, J: free charge and volume current density
* two divergence equations can be derived from two curl
equations
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UEE 3201 Electromagnetics II
Fall, 2014
* constitutive relations (for linear isotropic medium)

 D = ϵE
 H = B/µ
- Integral form
 I


D · ds = Q
(Gauss′ s law)



IS


dΦ


(Faradys′ s law)
E · dℓ = −

dt
IC


B · ds = 0
(No isolated magnetic charge)



S
I
∫


∂D



H · dℓ = I +
· ds
(Ampere′ s law)
∂t
C
S
(b) Continuity equation (conservation of charge)
∇·J+
∂ρ
=0
∂t
(c) Lorentz’s force equation
F = q(E + u × B)
- Conduction current density in a conductor: J = σE
Convection current density in vacuum: J = ρu
- In static cases, electric and magnetic fields are independent physical quantities. In time-varying cases, electric fields and magnetic
fields are coupled.
Ex 7-5: Displacement current in a capacitor
(a) conduction current ic and displacement current id
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UEE 3201 Electromagnetics II
Fall, 2014
(b) Find H around the wire (from Ampere’s law)
C isI the common
I boundary of open
I surfacs S1 and S2 .
⇒
H · dℓ =
∇ × H · ds =
∇ × H · ds
C
S1
S2
i. calculation of H from C and S1
ii. calculation of H from C and S2
⇒
(If displacement current JD were not included in the Maxwell’s
equations, H field calculated from S1 and S2 would be different.)
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UEE 3201 Electromagnetics II
7.5
Fall, 2014
Electromagnetic Boundary Conditions
 I


D · ds = Q



S
∫
I


d


B · ds
E · dℓ = −

dt S
IC


B · ds = 0



S
∫
I


∂D



H · dℓ = (J +
) · ds
∂t
C
S
1. B.C. can be found by applying the integral forms of Maxwell’s Eqs. to
a small loop or box across the boundary of two media. The results are
(same as B.C. in static fields):
- Tangential components

 E =E
1t
2t
 a × (H − H ) = J
n2
1
2
s
- Normal components

 a · (D − D ) = ρ
n2
1
2
s
 B =B
1n
2n
(a)
(b)
(c)
(d)
- In time varying cases, (a) is equivalent to (d), (b) is equivalent to
(c).
2. Interface between two lossless linear media (σ = 0, ρs = 0, Js = 0)

 E =E , H =H
1t
2t
1t
2t
 D =D , B =B
1n
2n
1n
2n
3. Interface between dielectric and perfect conductor
Metal



E2t = 0




 H =0
2t


D2n = 0




 B =0
2n
Dielectric



E1t = 0




 a × H = J , |H | = |H | = |J |
n2
1
s
1
1t
s


an2 · D1 = ρs , |E1 | = |E1n | = ρs /ϵ1




 B =0
1n
(time varying component only)
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Chiu, ECE Dept., NCTU
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UEE 3201 Electromagnetics II
7.6
Fall, 2014
Wave Equations and Solutions
1. Source-free wave equations
In source free regions (ρ = 0, J = 0), the Maxwell’s equations are
reduced to
⇒
⇒
2
=0
Similarly, ∇2 H − 12 ∂ H
u ∂t2
• homogeneous wave equations for E and H
√
• wave propagation speed = 1/ µ0 ϵ0 = C in vacuum
2. In Cartician coordinate,

1 ∂ 2 Hx

∇ Hx − 2
=0 


u ∂t2



2
1 ∂ 2f
1
∂
H
2
y
2
=0
wave
equation:
∇
f
−
∇ Hy − 2
=
0

u2 ∂t2
u ∂t2




1 ∂ 2 Hz

2

∇ Hz − 2
=
0
2
u ∂t
2
1 ∂ 2f
∂2f
−
=0
3. Wave equation in one dimension:
∂x2 u2 ∂t2
The solution of the wave equation (wave function) has the general form
f (x, t) = f (kx ± ωt), where k 2 − ω 2 /u2 = 0. For a sinusoidal solution,
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Chiu, ECE Dept., NCTU
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UEE 3201 Electromagnetics II
Fall, 2014
f (x, t) = f0 sin(kx − ωt), where
4. Phase velocity
f (x, t) = f0 sin(kx − ωt) = sin(ϕ), where ϕ = phase.
At fixed t,
If we look at the velocity of the constant-phase point P ,
5. Propagation and time delay
Consider two points x1 and x2
⇒ the wave function value at x2 is a delayed version of that at x1
6. Waves in 2D and 3D
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UEE 3201 Electromagnetics II
7.7
Fall, 2014
Time-Harmonic (Sinusoidal) Fields
1. In circuit analysis, a sinusoidal time-varying (time harmonic) signal can
be written as
where Is = I0 ejϕ (= phasor) contains both amplitude (I0 ) and phase
(ϕ) information. The phase ϕ represents time delay with respect to a
reference starting time instant.
2. In phasor analysis, Is is used to represent the signal. All time derivative
operation d/dt can be replaced by a multiplication operation by jω.
The real signal in time domain can be obtained by taking the real part
of the product of Is and ejωt , i.e. i(t) = ℜ[Is ejωt ] = ℜ[I0 ej(ωt+ϕ) ].
3. Similarly, for a time harmonic EM field, the field at each point R can
be written as
where
- E(R) is the vector phasor that contains amplitude, direction, and
phase information of the field at R
- Real solution of the fields can be obtained by taking the real part
of E(R)ejωt , i.e. E(R, t) = ℜ[E(R)ejωt ] = ℜ[E0 (R)ejωt+ϕ(R) ],
where ϕ(R) is now a position-dependent phase delay.
- A vector phasor has three scalar phasor components:
where Ex , Ey , Ez are scalar phasors and
⇒ In terms of phasors, the Maxwell’s Eqs. can be written as
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UEE 3201 Electromagnetics II
instantaneous form

∂H


∇ × E = −µ


∂t



∂D

∇×H=J+
∂t
ρ


 ∇·E=


ϵ



∇·H=0
Fall, 2014
phasor form
4. In time harmonic fields, the source free wave equations can be written
2
as (from Eqs. 7-81, 7-82, ∇2 E − µϵ ∂ E
= 0)
∂t2
(homogeneous vector Helmholtz’s equations)
√
where k = ω µϵ = ω/u = 2π/λ = wave number = spatial frequency
5. If the medium is conductive, J = σE ̸= 0,
where
- If permittivity can be a complex number, the same form of Maxwell’s
Eqs. can be applied to conductive and non-conductive materials.
- Imaginary part of ϵc represents energy loss (Ohmic loss, frictional
damping, ...). An equivalent conductivity for all losses can be
defined as σ , ωϵ′′ .
- Similarly, magnetic permeability can be a complex number
µ = µ′ − jµ′′
- For µ′ ≫ µ′′ , µ ∼ µ′ , complex wave number kc in a lossy dielectric
medium becomes
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UEE 3201 Electromagnetics II
Fall, 2014
6. Loss tangent tan δc = ϵ′′ /ϵ′
loss angle δc = tan−1 ϵ′′ /ϵ′
- If ϵ′′ /ϵ′ = σ/ωϵ′ ≫ 1 ⇒ good conductor
If ϵ′′ /ϵ′ = σ/ωϵ′ ≪ 1 ⇒ good insulator
- Dielectric properties may depend on frequency
Ex 7-8 (自己看)
7. Electromagnetic spectrum
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UEE 3201 Electromagnetics II
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Fall, 2014
7-14
UEE 3201 Electromagnetics II
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Fall, 2014
7-15
UEE 3201 Electromagnetics II
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Fall, 2014
7-16