NUS/ECE
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Revision of Maxwell’s Equations
1 Introduction
1.1 Helmholtz’s Theorem
A vector field (vector function) is uniquely determined
if its divergence and curl are specified everywhere
and its boundary conditions are known. Furthermore,
the vector field can be expressed as a sum of a curless
vector field F1 and a divergenceless vector field F2:
F = −∇φ + ∇ × A
with
F1 = −∇φ
and ∇ × F1 = 0
F2 = ∇ × A and ∇ ⋅ F2 = 0
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Revision of Maxwell’s Equations
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1
φ ( x, y , z ) =
4π
1
A ( x, y , z ) =
4π
∫
v'
∫
v'
∇′ ⋅ F ( x ', y ', z ')
R
1
dv '−
4π
∇′ × F ( x ', y ', z ')
R
1
dv '−
4π
v∫
nˆ ⋅ F ( x ', y ', z ')
R
s'
v∫
ds′
nˆ × F ( x ', y ', z ')
R
s'
ds′
where,
R=
( x − x ') + ( y − y ') + ( z − z ')
2
2
2
nˆ = normal unit vector on s'
(for a proof of Helmholtz’s theorem, see ref.[5], Mathematical
Appendix.)
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Revision of Maxwell’s Equations
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1.2 Maxwell’s equations state the divergences and curls
of the electric and magnetic fields. When appropriate
boundary conditions are also known (as well as the
constitutive relations), the electric and magnetic fields
are uniquely determined from Maxwell’s equations.
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Revision of Maxwell’s Equations
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2 Maxwell’s equations in time domain
Differential Form
∇×E = −
∂B
∂t
∇×H = J +
∇⋅D = ρ
∂D
∂t
Integral Form
v∫ E ⋅ dl = − ∫∫
c
s
∂B
⋅ ds
∂t
v∫ H ⋅ dl = I + ∫∫
c
s
Physical Meaning
Farady's law
∂D
⋅ ds Ampere's circuit law
∂t
w
∫∫ D ⋅ ds = Q
Gauss's law
w
∫∫ B ⋅ ds = 0
No magnetic monopoles
s
∇⋅B = 0
s
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Revision of Maxwell’s Equations
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E (V/m)
→
electric field intensity
electric flux density
D (C/m2) →
B (T)
→
magnetic flux density
H (A/m)
→
magnetic field intensity
J (A/m2)
→
electric current density
→
electric charge density
ρ(C/m3)
I (A)
→
electric current
Q(C)
→
electric charge
In the differential form of Maxwell’s equations, we treat J
and ρ as known functions and E, D, B, and H as unknowns.
All variable functions in Maxwell’s equations are functions
of time (t) and space (x, y, z). We also need the constitutive
relations in order to solve Maxwell’s equations.
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Revision of Maxwell’s Equations
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Note that the integral form of Maxwell’s equations are
only derived from the differential form with the
assumption that E, D, B, and H and their first derivatives
are all continuous throughout the space.
3 Constitutive Relations
The unknown functions E, D, B, and H in Maxwell’s
equations cannot be uniquely determined without
specifying the constitutive relations between these
functions.
D=ε E
B = μH
J = σE
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ε (F/m)
μ (H/m)
σ (S/m)
→
→
→
permittivity of the medium
permeability of the medium
conductivity of the medium
In vacuum or air,
ε = ε 0 = 8.854 ×10−12 F/m
μ = μ0 = 4π ×10−7 H/m
σ =0
In other media, we can express ε and μ relative to ε0 and
μ0 and define a relative permittivity εr and a relative
permeability μr so that
ε = ε 0ε r
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μ = μ0 μ r
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Revision of Maxwell’s Equations
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4 Maxwell’s equations in phasor domain
∇ × E = -jωB
∇ × H = J + jωD
∇⋅D = ρ
∇⋅B = 0
Using constitutive parameters,
∇ × E = -jωμH
∇ × H = J + jωεE
∇⋅E = ρ /ε
∇⋅H = 0
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Revision of Maxwell’s Equations
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5 Boundary Conditions for Maxwell’s equations
We need the boundary conditions on the electric and
magnetic fields to solve Maxwell’s equations. When the
region is infinitely large, the boundary conditions on the
electric and magnetic fields are that both the electric and
magnetic fields are zero at infinite distances from the
sources.
When the region is finite and continued by another
different medium, the boundary conditions on the electric
and magnetic fields can be derived from the integral form
of Maxwell’s equations. (See ref. [2], Chapter 4)
The boundary conditions are summarized below.
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Revision of Maxwell’s Equations
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Js ρs
surface current and charge
densities, confined only on
the interface
n̂1
E1, H1
D1 , B1
medium 1
medium 2
E2, H2
D2 , B2
n̂ 2
For tangential components:
nˆ 1 × (E1 − E2 ) = 0
nˆ 1 × (H1 − H 2 ) = J s
For normal components:
nˆ 1 ⋅ (B1 − B 2 ) = 0
nˆ 1 ⋅ (D1 − D2 ) = ρ s
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Revision of Maxwell’s Equations
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To solve a practical problem, the boundary conditions on
the tangential components of E or H are sufficient.
Some special Cases:
1. An infinitely large region (boundary at infinity)
E1 = E 2 = 0
H1 = H 2 = 0
B1 = B 2 = 0
D1 = D2 = 0
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2. Interface between 2 lossless dielectric layers (no
charge, no current at the interface)
nˆ 1 × (E1 − E2 ) = 0
n̂
E ,H
D ,B
dielectric 1
nˆ 1 × (H1 − H 2 ) = J s = 0
J =0
ρ
dielectric 2
nˆ 1 ⋅ (B1 − B 2 ) = 0
E ,H
D ,B
n̂
nˆ 1 ⋅ (D1 − D2 ) = ρ s = 0
1
1
1
1
1
s
s=0
2
2
2
2
2
3. Interface between a dielectric and a perfect conductor
nˆ 1 × E1 = 0
n̂
E ,H
D ,B
dielectric (air)
nˆ 1 × H1 = J s
J
1
1
1
1
1
s
→→→→→→→→→→→→
++++++++++++++++++
ρs
perfect conductor (metal)
E2=H2=0
D2=B2=0
n̂2
nˆ 1 ⋅ B1 = 0
nˆ 1 ⋅ D1 = ρ s
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4. In an infinity large region, in additional to the
vanishing field boundary condition at infinity, the
radiation boundary condition also needs to be
enforced. That is,
lim ⎡⎣∇ × E ( r , θ , φ ) − iω με E ( r ,θ , φ ) ⎤⎦ = 0
r →∞
lim ⎡⎣∇ × H ( r , θ , φ ) − iω με H ( r ,θ , φ ) ⎤⎦ = 0
r →∞
E(r,θ,φ), H (r,θ,φ)
z
Source
r (sufficiently large)
0
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x
y
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Revision of Maxwell’s Equations
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The radiation boundary condition is important for the
fields in those devices such as antennas and waveguides.
When the distance is sufficiently large (not necessary
infinitely large) from the excitation source, the E and H
fields in these devices must satisfy the radiation
boundary condition. See ref. [4], Chapter 3.
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References:
1. David K. Cheng, Field and Wave Electromagnetic, Addison-Wesley
Pub. Co., New York, 1989.
2. Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall, Inc.,
New Jersey, 2007.
3. Matthew N. O. Sadiku, Elements of Electromagnetics, Oxford
University Press, New York, 2001.
4. C. T. Tai, Dyadic Green Functions in Electromagentic Theory,
IEEE Press, New Jersey, 1994.
5. Robert E. Collin, Field theory of guided waves, IEEE Press, New
York, 1991.
6. Joseph A. Edminister, Schaum’s Outline of Theory and Problems of
Electromagnetics, McGraw-Hill, Singapore, 1993.
7. Yung-kuo Lim (Editor), Problems and solutions on
electromagnetism, World Scientific, Singapore, 1993.
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Revision of Maxwell’s Equations