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Electromagnetic Wave PropagationLecture 1: Maxwell’s equations
Daniel Sjoberg
Department of Electrical and Information Technology
August 2015
Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
2 / 31
Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
3 / 31
International year of light
A few of the equations back then: Nowadays:
p′ = p+df
dt, q′ = q +
dg
dt, r′ = r +
dh
dtJ tot = J +
∂D
∂t
dγ
dy− dβ
dz= 4πp′
dα
dz− dγ
dx= 4πq′
dβ
dx− dα
dy= 4πr′
∇×H = J tot
e+df
dx+dg
dy+dh
dz= 0 ∇ ·D = ρ
de
dt+dp
dx+dq
dy+dr
dz= 0 ∇ · J tot = 0
Note that the original equations are inconsistent.The sign of e needs to change in one of the equations.
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International year of light
A few of the equations back then: Nowadays:
p′ = p+df
dt, q′ = q +
dg
dt, r′ = r +
dh
dtJ tot = J +
∂D
∂t
dγ
dy− dβ
dz= 4πp′
dα
dz− dγ
dx= 4πq′
dβ
dx− dα
dy= 4πr′
∇×H = J tot
e+df
dx+dg
dy+dh
dz= 0 ∇ ·D = ρ
de
dt+dp
dx+dq
dy+dr
dz= 0 ∇ · J tot = 0
Note that the original equations are inconsistent.The sign of e needs to change in one of the equations.
4 / 31
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 presents his paper A Dynamical
Theory of the Electromagnetic Field, joiningelectricity and magnetism and predicting theexistence of waves.
1887: Heinrich Hertz proves experimentally the existence ofelectromagnetic waves.
1897: Marconi founds the Wireless Telegraph & SignalCompany
1905: Albert Einstein publishes his special theory ofrelativity, emphasizing the role of the speed of lightin vacuum.
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Historical notes, continued
1930s: The first radio telescopes are built.
Early 1940s: The MIT Radiation Laboratory substantiallyadvances the knowledge of control of electromagneticwaves while developing radar technology.
Late 1940s: The Quantum Electrodynamics theory (QED) isdeveloped by Richard Feynman, Freeman Dyson,Julian Schwinger, and Sin-Itiro Tomonaga.
1957: Sputnik 1 transmits the first signal from a satellite toearth.
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: ∇×E(r, t) = −∂B(r, t)
∂t
Ampere’s law: ∇×H(r, t) = J(r, t) +∂D(r, t)
∂t
Conservation of charge: ∇ · J(r, t) + ∂ρ(r, t)
∂t= 0
Symbol Name Unit
E(r, t) Electric field [V/m]H(r, t) Magnetic field [A/m]D(r, t) Electric flux density [As/m2]B(r, t) Magnetic flux density [Vs/m2]J(r, t) Current density [A/m2]ρ(r, t) Charge density [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 bederived as follows. Since ∇ · (∇× F ) = 0 for any vector functionF , taking the divergence of Faraday’s and Ampere’s laws imply
0 = ∇ · ∂B∂t
0 = ∇ ·(J +
∂D
∂t
)= −∂ρ
∂t+∇ · ∂D
∂t
Thus ∇ ·B = f1 and ∇ ·D − ρ = f2, where f1 and f2 areindependent of t. Assuming the fields are zero at t = −∞, wehave 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 equationsare given by the models of the material behavior. In vacuum wehave
D = ε0E, B = µ0H
where the speed of light in vacuum isc0 = 1/
√ε0µ0 = 299 792 458m/s (exact) and
ε0 =1
c20µ0≈ 8.854 · 10−12As/Vm
µ0 = 4π · 10−7Vs/Am (exact)
In a material, there is in addition polarization and magnetization:
P = D − ε0E, M =1
µ0B −H
The physics of the material in question determine the fields P andM 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
10 / 31
Literature
Here, we give a brief overview of vector analysis used in the course.
A summary of vector relations can be found in Orfanidis,Appendices C–E. If you want more in-depth, you can consult atypical first text book on electromagnetism, like Griffiths or Cheng.
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Vector analysis
The vectors have three components, one for each spatial direction:
E = Exx+ Eyy + Ezz
In particular, the position vector is r = xx+ yy + zz. Vectoraddition, scalar product and vector product are
r1 + r2
r1
r2r1
r2
ϕ
|r2| cosϕr1
r2
ϕ
r1 × r2
I Addition: r1 + r2 = (x1 + x2)x+ (y1 + y2)y + (z1 + z2)zI Scalar product: r1 · r2 = |r1| |r2| cosϕ = x1x2 + y1y2 + z1z2.I Vector product: orthogonal to both vectors, with length|r1 × r2| = |r1| |r2| sinϕ, and r1 × r2 = −r2 × r1.
x× y = z, y × z = x, z × x = y
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Vector analysis, continued
The nabla operator is
∇ = x∂
∂x+ y
∂
∂y+ z
∂
∂z
The divergence and curl operations are
∇ ·E =∂
∂xEx +
∂
∂yEy +
∂
∂zEz
∇×E =∂
∂xx×E +
∂
∂yy ×E +
∂
∂zz ×E
Cartesian representation: [E] =
(ExEy
Ez
)and
[∇×E] = ∂∂x
0 0 00 0 −10 1 0
Ex
Ey
Ez
︸ ︷︷ ︸
=[x×E]
+ ∂∂y
0 0 10 0 0−1 0 0
Ex
Ey
Ez
︸ ︷︷ ︸
=[y×E]
+ ∂∂z
0 −1 01 0 00 0 0
Ex
Ey
Ez
︸ ︷︷ ︸
=[z×E]
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Integral theorems
Generalizations of the fundamental theorem of calculus:∫ b
af ′(x) dx = f(b)− f(a)
Gauss’ theorem:
∫∫∫V∇ · F dV =
∫∫Sn · F dS
n
V
S
Stokes’ theorem:
∫∫S(∇× F ) · ndS =
∫CF · dr
nS
C dr
Typically: a higher order integral of a derivative can be turned intoa 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 = xEx
Since Ex = x ·E, we can write this
Px ·E = xEx = x(x ·E) = (xx) ·E
Thus Px = xx, which is a dyadic product. In particular, theidentity operator I ·E = E can be written
I = xx+ yy + zz
with the Cartesian representation
[I·] =
1 0 00 1 00 0 1
=
100
(1 0 0)
︸ ︷︷ ︸=[xx·]
+
010
(0 1 0)
︸ ︷︷ ︸=[yy·]
+
001
(0 0 1)
︸ ︷︷ ︸=[zz·]
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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:
n
h
S
1
2
What are the relations between field quantities at different sides ofthe interface?
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Integral form of Maxwell’s equations
Integrate the field equations over the volume V (red and green):∫∫∫V∇×E dV = −
∫∫∫V
∂B
∂tdV∫∫∫
V∇×H dV =
∫∫∫VJ dV +
∫∫∫V
∂D
∂tdV∫∫∫
V∇ ·B dV = 0∫∫∫
V∇ ·D dV =
∫∫∫VρdV
n
h
S
1
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Integral form of Maxwell’s equations, continued
Use the integral theorems to find∫∫Sn×E dV = − d
dt
∫∫∫VB dV∫∫
Sn×H dV =
∫∫∫VJ dV +
d
dt
∫∫∫VD dV∫∫
Sn ·B dV = 0∫∫
Sn ·D dV =
∫∫∫VρdV
n
h
S
1
2 19 / 31
Pillbox volume
Let the height of the pillbox volume Vh become small, h→ 0. Onmetal surfaces, the current and charge densities can become verylarge. Introduce the surface current and surface charge as
n
h
S
1
2
limh→0
∫∫∫Vh
J dV =
∫∫SJS dS
limh→0
∫∫∫Vh
ρ dV =
∫∫SρS dS
The limit of the flux integrals on the other hand is zero,
limh→0
∫∫∫Vh
D dV = limh→0
∫∫∫Vh
B dV = 0
The limit of E is E1 from material 1 and E2 from material 2,implying
limh→0
∫∫Sh
n×E dS =
∫∫S(n1 ×E1 + n2 ×E2)︸ ︷︷ ︸
=n1×(E1−E2)
dS
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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):
n1 × (E1 −E2) = 0
n1 × (H1 −H2) = JS
n1 · (B1 −B2) = 0
n1 · (D1 −D2) = ρS
In words, this means:
I The tangential electric field is continuous.
I The tangential magnetic field is discontinuous if JS 6= 0.
I The normal component of the magnetic flux is continuous.
I The normal component of the electric flux is discontinuous ifρS 6= 0.
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Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
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Conservation of charge
The conservation of charge is postulated,
∇ · J +∂ρ
∂t= 0
Alternatively, we could postulate Gauss’ law, ∇ ·D = ρ, and derivethe conservation of charge from Ampere’s law ∇×H = J + ∂D
∂t .
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Energy conservation
Take the scalar product of Faraday’s law with H and the scalarproduct of Ampere’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∂t
+E · ∂D∂t
+E · J = 0
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)∫∫Sn ·P dS =
∫∫∫V∇ ·P dV
= −∫∫∫
V
(H · ∂B
∂t+E · ∂D
∂t
)dV −
∫∫∫VE · J dV
I The first integral is the total power radiated out of thebounding surface S.
I The second integral is the rate of change of electromagneticenergy stored in the volume V (not obvious at this point).
I The last integral is the work per unit time (the power) thatthe 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 withAmpere’s law,
D × (∇×E) = −D × ∂tBB × (∇×H) = B × J +B × ∂tD = −J ×B − (∂tD)×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 +∑
i=x,y,z
Di∇Ei −∇ · (DE)
implying (using ∇ ·D = ρ and ∇ ·B = 0)
∂t (D ×B)︸ ︷︷ ︸momentum
+J ×B + ρE︸ ︷︷ ︸Lorentz force
= ∇·(DE+BH)−∑
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:
D =∂φ(E,H)
∂Eand B =
∂φ(E,H)
∂H
which is interpreted Di = ∂φ/∂Ei etc. For linear, isotropic media
φ(E,H) =1
2ε|E|2 + 1
2µ|H|2 ⇔
{D = εE
B = µH
Nonlinear materials have additional terms like |E|4, |E|6 etc. For amaterial described by potential φ, Poynting’s theorem is
∇ · (E ×H) +∂
∂t(D ·E +B ·H − φ) = −E · J
and the conservation of momentum is
∂
∂t(D ×B) + ρE + J ×B = ∇ · (DE +BH − φI)
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Outline
1 Maxwell’s equations
2 Vector analysis
3 Boundary conditions
4 Conservation laws
5 Conclusions
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Conclusions
I Maxwell’s equations describe the dynamics of theelectromagnetic fields.
I Boundary conditions relate the values of the fields on differentsides of material boundaries to each other.
I Conservation laws can be derived to describe the conservationof physical entities such as power and momentum; these areusually products of two fields.
I Constitutive relations are necessary in order to fully solveMaxwell’s equations. Topic of next lecture!
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