Electromagnetic Wave Propagation
Lecture 1: Maxwell’s equations
Daniel Sjöberg
Department of Electrical and Information Technology
September 3, 2013
Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
2 / 29
Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
3 / 29
Historical notes
Early 1800s: Electricity and magnetism separate phenomena.
1819: Hans Christian Ørsted discovers a linear current
deflects a magnetized needle.
1831: Michael Faraday demonstrates that a changing
magnetic field can induce electric voltage.
1830s: Electrical wire telegraphs come into use.
1864: James Clerk Maxwell publishes his Treatise on
Electricity and Magnetism, joining electricity and
magnetism and predicting the existence of waves.
1887: Heinrich Hertz proves experimentally the existence of
electromagnetic waves.
1897: Marconi founds the Wireless Telegraph & Signal
Company
1905: Albert Einstein publishes his special theory of
relativity, emphasizing the role of the speed of light
in vacuum.
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Historical notes, continued
1930s: The first radio telescopes are built.
Early 1940s: The MIT Radiation Laboratory substantially
advances the knowledge of control of electromagnetic
waves while developing radar technology.
Late 1940s: The Quantum Electrodynamics theory (QED) is
developed by Richard Feynman, Freeman Dyson,
Julian Schwinger, and Sin-Itiro Tomonaga.
1957: Sputnik 1 transmits the first signal from a satellite to
earth.
1970s: Low loss optical fibers are developed.
1981: The NMT system goes online.
2000: Mobile phones connect to internet.
2020: Initial observations by Square Kilometre Array(?)
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Field equations and the electromagnetic fields
Faraday’s law:
Ampère’s law:
Conservation of charge:
∂B(r, t)
∂t
∂D(r, t)
∇ × H(r, t) = J (r, t) +
∂t
∂ρ(r, t)
∇ · J (r, t) +
=0
∂t
∇ × E(r, t) = −
Symbol
Name
Unit
E(r, t)
H(r, t)
D(r, t)
B(r, t)
J (r, t)
ρ(r, t)
Electric field
Magnetic field
Electric flux density
Magnetic flux density
Current density
Charge density
[V/m]
[A/m]
[As/m2 ]
[Vs/m2 ]
[A/m2 ]
[As/m3 ]
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The divergence equations
Often the equations
∇ · D = ρ and ∇ · B = 0
(∗)
are considered as a part of Maxwell’s equations, but they can be
derived as follows. Since ∇ · (∇ × F ) = 0 for any vector function
F , taking the divergence of Faraday’s and Ampère’s laws imply
0=∇·
∂B
∂t
∂D
0=∇· J +
∂t
=−
∂ρ
∂D
+∇·
∂t
∂t
Thus ∇ · B = f1 and ∇ · D − ρ = f2 , where f1 and f2 are
independent of t. Assuming the fields are zero at t = −∞, we
have f1 = f2 = 0 and the divergence equations (∗) follow.
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Materials
Maxwell’s equations give 2 × 3 = 6 equations, but the fields E, B,
H, D represent 4 × 3 = 12 unknowns. The remaining equations
are given by the models of the material behavior. In vacuum we
have
D = 0 E, B = µ0 H
where the speed of light in vacuum is
√
c0 = 1/ 0 µ0 = 299 792 458 m/s (exact) and
0 =
1
c20 µ0
≈ 8.854 · 10−12 As/Vm
µ0 = 4π · 10−7 Vs/Am
(exact)
In a material, there is in addition polarization and magnetization:
1
P = D − 0 E, M =
B−H
µ0
The physics of the material in question determine the fields P and
M as functions of the electromagnetic fields (next lecture!).
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Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
9 / 29
Vector analysis
The vectors have three components, one for each spatial direction:
E = Ex x̂ + Ey ŷ + Ez ẑ
In particular, the position vector is r = xx̂ + yŷ + zẑ. Vector
addition, scalar product and vector product are
r1 × r2
r2
r1
r2
r1 + r2
I
I
I
ϕ
|r 2 | cos ϕ
r2
r1
ϕ
r1
Addition: r 1 + r 2 = (x1 + x2 )x̂ + (y1 + y2 )ŷ + (z1 + z2 )ẑ
Scalar product: r 1 · r 2 = |r 1 | |r 2 | cos ϕ = x1 x2 + y1 y2 + z1 z2 .
Vector product: orthogonal to both vectors, with length
|r 1 × r 2 | = |r 1 | |r 2 | sin ϕ, and r 1 × r 2 = −r 2 × r 1 .
x̂ × ŷ = ẑ,
ŷ × ẑ = x̂,
ẑ × x̂ = ŷ
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Vector analysis, continued
The nabla operator is
∇ = x̂
∂
∂
∂
+ ŷ
+ ẑ
∂x
∂y
∂z
The divergence and curl operations are
∂
∂
∂
Ex +
Ey +
Ez
∂x
∂y
∂z
∂
∂
∂
∇×E =
x̂ × E +
ŷ × E +
ẑ × E
∂x
∂y
∂z
Ex
Cartesian representation: [E] = Ey and
∇·E =
Ez

 

 

 
0 0 0
Ex
0 0 1
Ex
0 −1 0
Ex
∂ 
∂ 
∂ 
0 0 −1 Ey  + ∂y
0 0 0 Ey  + ∂z
1 0 0 Ey 
[∇ × E] = ∂x
0 1 0
Ez
−1 0 0
Ez
0 0 0
Ez
{z
}
{z
}
{z
}
|
|
|
=[x̂×E]
=[ŷ×E]
=[ẑ×E]
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Integral theorems
Generalizations of the fundamental theorem of calculus:
Z b
f 0 (x) dx = f (b) − f (a)
a
Gauss’ theorem:
n̂
ZZZ
V
∇ · F dV =
ZZ
S
V
n̂ · F dS
S
Stokes’ theorem:
ZZ
S
(∇ × F ) · n̂ dS =
Z
C
n̂
S
F · dr
C
dr
Typically: a higher order integral of a derivative can be turned into
a lower order integral of the function.
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Dyadic products
The projection of a vector on the x-direction is defined by
Px · E = x̂Ex
Since Ex = x̂ · E, we can write this
Px · E = x̂Ex = x̂(x̂ · E) = (x̂x̂) · E
Thus Px = x̂x̂, which is a dyadic product. In particular, the
identity operator I · E = E can be written
I = x̂x̂ + ŷ ŷ + ẑẑ
with the Cartesian representation

  
 
 
1 0 0
1
0
0
[I·] = 0 1 0 = 0 1 0 0 + 1 0 1 0 + 0 0 0 1
0 0 1
0
0
1
|
{z
} |
{z
} |
{z
}
=[x̂x̂·]
=[ŷ ŷ·]
=[ẑẑ·]
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Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
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Material interfaces
Consider the interface between two media, 1 and 2:
What are the relations between field quantities at different sides of
the interface?
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Integral form of Maxwell’s equations
Integrate the field equations over the volume V :
ZZZ
ZZZ
∂B
∇ × E dV = −
dV
∂t
ZZZ V
ZZZ V
ZZZ
∂D
dV
∇ × H dV =
J dV +
∂t
V
V
V
ZZZ
∇ · B dV = 0
V
ZZZ
ZZZ
∇ · D dV =
ρ dV
V
V
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Integral form of Maxwell’s equations, continued
Use the integral theorems to find
ZZ
ZZZ
d
n̂ × E dV = −
B dV
dt
V
ZZZ
ZZ S
ZZZ
d
D dV
n̂ × H dV =
J dV +
dt
V
S
V
ZZ
n̂ · B dV = 0
S
ZZ
ZZZ
n̂ · D dV =
ρ dV
S
V
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Pillbox volume
Let the height of the pillbox volume Vh become small, h → 0. On
metal surfaces, the current and charge densities can become very
large. Introduce the surface current and surface charge as
ZZZ
ZZ
lim
J dV =
J S dS
h→0
Vh
S
ZZZ
ZZ
lim
ρ dV =
ρS dS
h→0
Vh
S
The limit of the flux integrals on the other hand is zero,
ZZZ
ZZZ
lim
D dV = lim
B dV = 0
h→0
Vh
h→0
Vh
The limit of E is E 1 from material 1 and E 2 from material 2,
implying
ZZ
ZZ
lim
n̂ × E dS =
(n̂1 × E 1 + n̂2 × E 2 ) dS
{z
}
h→0
Sh
S|
=n̂1 ×(E 1 −E 2 )
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Boundary conditions
We summarize as (when material 2 is a perfect electric conductor
(PEC) the fields with index 2 are zero):
n̂1 × (E 1 − E 2 ) = 0
n̂1 × (H 1 − H 2 ) = J S
n̂1 · (B 1 − B 2 ) = 0
n̂1 · (D 1 − D 2 ) = ρS
In words, this means:
I
The tangential electric field is continuous.
I
The tangential magnetic field is discontinuous if J S 6= 0.
I
I
The normal component of the magnetic flux is continuous.
The normal component of the electric flux is discontinuous if
ρS 6= 0.
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Example
Fields
n̂ · D
ρS
n̂ · B
Material 1
Material 2
Distane to
boundary
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Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
21 / 29
Conservation of charge
The conservation of charge is postulated,
∇·J +
∂ρ
=0
∂t
Alternatively, we could postulate Gauss’ law, ∇ · D = ρ, and derive
the conservation of charge from Ampère’s law ∇ × H = J + ∂D
∂t .
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Energy conservation
Take the scalar product of Faraday’s law with H and the scalar
product of Ampère’s law with E:
H · (∇ × E) = −H ·
∂B
∂t
E · (∇ × H) = E · J + E ·
∂D
∂t
Take the difference of the equations and use the identity
∇ · (a × b) = b · (∇ × a) − a · (∇ × b)
∇ · (E × H) + H ·
∂B
∂D
+E·
+E·J =0
∂t
∂t
This is Poynting’s theorem on differential form.
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Interpretation of Poynting’s theorem
Integrating over a volume V implies (where P = E × H)
ZZ
ZZZ
n̂ · P dS =
∇ · P dV
S
V
ZZZ
ZZZ ∂D
∂B
+E·
dV −
=−
H·
E · J dV
∂t
∂t
V
V
I
The first integral is the total power radiated out of the
bounding surface S.
I
The second integral is the electromagnetic power bounded in
the volume V .
I
The last integral is the work per unit time (the power) that
the field does on charges in V .
This is conservation of power (or energy).
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Momentum conservation
Take the vector product of D with Faraday’s law and B with
Ampère’s law,
D × (∇ × E) = −D × ∂t B
B × (∇ × H) = B × J + B × ∂t D = −J × B − (∂t D) × B
Adding the equations results in
∂t (D × B) + J × B + D × (∇ × E) + B × (∇ × H) = 0
It can be shown that
D × (∇ × E) = (∇ · D)E +
X
i=x,y,z
Di ∇Ei − ∇ · (DE)
implying (using ∇ · D = ρ and ∇ · B = 0)
∂t (D × B) + J × B + ρE = ∇·(DE+BH)−
{z
}
| {z } |
momentum
Lorentz force
X
i=x,y,z
Di ∇Ei +Bi ∇Hi
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A simple material model
Often materials can be described by using an energy potential φ:
∂φ(E, H)
∂φ(E, H)
and B =
∂E
∂H
which is interpreted Di = ∂φ/∂Ei etc. For linear, isotropic media
(
D = E
1
1
2
2
⇔
φ(E, H) = |E| + µ|H|
2
2
B = µH
D=
Nonlinear materials have additional terms like |E|4 , |E|6 etc. For a
material described by potential φ, Poynting’s theorem is
∇ · (E × H) +
∂
(D · E + B · H − φ) = −E · J
∂t
and the conservation of momentum is
∂
(D × B) + ρE + J × B = ∇ · (DE + BH − φI)
∂t
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The EM momentum is still debated!
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Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
28 / 29
Conclusions
I
Maxwell’s equations describe the dynamics of the
electromagnetic fields.
I
Boundary conditions relate the values of the fields on different
sides of material boundaries to each other.
I
Conservation laws can be derived to describe the conservation
of physical entities such as power and momentum; these are
usually products of two fields.
I
Constitutive relations are necessary in order to fully solve
Maxwell’s equations. Topic of next lecture!
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