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Adapted from notes by Prof. Stuart A. Long 1 Notes 4 Maxwell’s Equations ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018
Adapted from notes by Prof. Stuart A. Long
1
Notes 4 Maxwell’s Equations
ECE 3317Applied Electromagnetic Waves
Prof. David R. JacksonFall 2018
2
Here we present an overview of Maxwell’s equations. A much more thorough discussion of Maxwell’s equations may be found in the class notes for ECE 3318:
http://courses.egr.uh.edu/ECE/ECE3318
Notes 10: Electric Gauss’s lawNotes 18: Faraday’s lawNotes 28: Ampere’s lawNotes 28: Magnetic Gauss law
D. Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, 2008.
Overview
Electromagnetic Fields
Four vector quantities
E electric field strength [Volt/meter]
D electric flux density [Coulomb/meter2]
H magnetic field strength [Amp/meter]
B magnetic flux density [Weber/meter2] or [Tesla]
Each are functions of space and timee.g. E(x,y,z,t)
J electric current density [Amp/meter2]
ρv electric charge density [Coulomb/meter3]
3
MKS units
length – meter [m]mass – kilogram [kg]time – second [sec]
Some common prefixes and the power of ten each represent are listed below
femto - f - 10-15
pico - p - 10-12
nano - n - 10-9
micro - μ - 10-6
milli - m - 10-3
mega - M - 106
giga - G - 109
tera - T - 1012
peta - P - 1015
centi - c - 10-2
deci - d - 10-1
deka - da - 101
hecto - h - 102
kilo - k - 103
4
0
v
t
t
ρ
∂∇× =−
∂∂
∇× = +∂
∇⋅ =∇ ⋅ =
BE
DH J
BD
Maxwell’s Equations
(Time-varying, differential form)
5
MaxwellJames Clerk Maxwell (1831–1879)
James Clerk Maxwell was a Scottish mathematician and theoretical physicist. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. His set of equations—Maxwell's equations—demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon: the electromagnetic field. From that moment on, all other classical laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the "second great unification in physics", after the first one carried out by Isaac Newton.
Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. Finally, in 1864 Maxwell wrote A Dynamical Theory of the Electromagnetic Field where he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is considered to be one of the greatest advances in physics.
(Wikipedia)6
Maxwell’s Equations (cont.)
0
v
t
t
ρ
∂∇× =−
∂∂
∇× = +∂
∇⋅ =∇ ⋅ =
BE
DH J
BD
Faraday’s law
Ampere’s law
Magnetic Gauss law
Electric Gauss law
7
Current Density Vector
8
σ=J E Ohm’s law
I S∆ = ∆J
Eσ
J+++
S∆
I∆
Medium
9
σ=J E Ohm’s law
( )ˆI n S∆ = ⋅ ∆J
Current Density Vector (cont.)
Eσ
J+++
S∆
n̂
Medium
Current Density Vector (cont.)
10
ˆS
I n dS= ⋅∫ J
Note:The direction of the unit normal vector
determines whether the current is measured going in or out.
n̂ J
S
( )ˆI n S∆ = ⋅ ∆J
( )
( ) 0
t
t
t
∂∇× = +
∂∂ ∇ ⋅ ∇× = ∇⋅ +∇ ⋅ ∂
∂= ∇ ⋅ + ∇ ⋅
∂
DH J
DH J
J D
Law of Conservation of Electric Charge (Continuity Equation)
v
tρ∂
∇ ⋅ = −∂
JFlow of electric
current out of volume (per unit volume)
Rate of decrease of electric charge (per unit volume)
11
Continuity Equation (cont.)
Apply the divergence theorem:
Integrate both sides over an arbitrary volume V:
v
V V
dVtρ∂
∇ ⋅ = −∂∫ ∫J
ˆ v
S V
n dVtρ∂
⋅ = −∂∫ ∫
J
V
S
n̂
12
ˆV S
n∇⋅ = ⋅∫ ∫J J
Hence:
v
tρ∂
∇ ⋅ = −∂
J
Continuity Equation (cont.)
Physical interpretation:
V
S
n̂
ˆ v
S V
n dVtρ∂
⋅ = −∂∫ ∫
J
vout v
V V
i dV dVt tρ ρ∂ ∂
= − = −∂ ∂∫ ∫
enclout
Qit
∂= −
∂
(This assumes that the surface is stationary.)
enclin
Qit
∂=
∂or
13
Continuity Equation (cont.)
enclin
Qit
∂=
∂
14
This implies that charge is never created or destroyed. It only moves from one place to another!
J
enclQ
0
0
0
v
v
E
t t
E D H J B
ρ
ρ
∂ ∂∇× = − ∇× = + ∇⋅ = ∇ ⋅ =
∂ ∂
∇× = ∇⋅ = ∇× = ∇⋅ =
B DE H J B D
Decouples
Time -Dependent
Time -Independent (Static s)
and vH E H Jρ⇒ comes from and comes from
15
Maxwell’s Equations (cont.)
Note: Regular (not script) font is used for statics, just as it is for phasors.
0
v
E j BH J j D
BD
ωω
ρ
∇× = −∇× = +∇ ⋅ =∇ ⋅ =
Time-harmonic (phasor) domain jt
ω∂→
∂
16
Maxwell’s Equations (cont.)
Constitutive Relations
The characteristics of the media relate D to E and H to B
0
0
0
0
( = )
(
= )µµ
ε ε=
=
D E
B H
permittivity
permeability
-120
-70
8.8541878 10 [F/m]
= 4 10 [H/m] ( )µ
ε
π
×
×
exact
0 0
1cµ ε
= (exact value that is defined)
Free Space
17
8 2.99792458 10 [m/s]c ≡ ×
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[ ] [ ]72 2 10 N/m when 1 mxF d−= × =
Definition of I =1 Amp:
I
I
d
Fx2x
# 1
# 2
Two infinite wires carrying DC currents
Definition of the Amp:
20
2 2xIF
dµπ
= 70 4 10 [A/m]µ π −= ×From ECE 3318:
Constitutive Relations (cont.)
Constitutive Relations (cont.)
Free space, in the phasor domain:
This follows from the fact that
( )a t aV⇔V
(where a is a real number)19
0
0 0
0 (
( )
D E
B µ H µ
ε ε ==
= =
permittivity)
permeability
In a material medium:
(
( )
D E
B µ µH
ε ε ==
= =
permittivity)
permeability
0
0
= r
rµ µ
ε ε ε
µ=
εr = relative permittivity
µr = relative permittivity
20
Constitutive Relations (cont.)
21
Constitutive Relations (cont.)
+
++
-
+
-
+-
--
-+
Where does permittivity come from?
0D E Pε≡ +
1i
VP p
V ∆
≡∆ ∑ iip p
p qd
=
=
i
+
-
d
q+
q−
+-
+-
+-
+-
xE
22
Constitutive Relations (cont.)
Linear material: 0 eP Eε χ=
Define:
0 rD Eε ε=Then
Note: χe > 0 for most materials
The term χe is called the “electric susceptibility.”
( )0 0
0 1e
e
D E EE
ε ε χε χ
= +
= +
1r eε χ≡ +
0D E Pε≡ +
so
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Constitutive Relations (cont.)
Teflon
Water
Styrofoam
Quartz
2.2811.035
r
r
r
r
εεεε
====
(a very polar molecule, fairly free to rotate)
Note: εr > 1 for most materials: 1 , 0r e eε χ χ≡ + >
24
Constitutive Relations (cont.)
Where does permeability come from?
Because of electron spin, atoms tend to acts as little current loops, and hence as electromagnetics, or bar magnets. When a magnetic field is applied, the little atomic magnets tend to line up.
0
1µ
≡ −H B M
1i
VV ∆
≡∆ ∑M m ( )ˆi in iA=m
a
ˆini
2A aπ=
B
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Constitutive Relations (cont.)
0 0µ µ= +B H M
mχ=M H
so
( )0 0
0 1m
m
µ µ χµ χ
= +
= +
B H H
H
( )1r mµ χ= +Define:
0 rµ µ=B HThen
The term χm is called the “magnetic susceptibility.”
Linear material:
Note: χm > 0 for most materials
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Constitutive Relations (cont.)
Note: Values can often vary depending on purity and processing.http://en.wikipedia.org/wiki/Permeability_(electromagnetism)
Material Relative Permeability µr
Vacuum 1Air 1.0000004
Water 0.999992Copper 0.999994
Aluminum 1.00002Silver 0.99998Nickel 600Iron 5000
Carbon Steel 100Transformer Steel 2000
Mumetal 50,000Supermalloy 1,000,000
Variation Independent of Dependent on
Space Homogenous Inhomogeneous
Frequency Non-dispersive Dispersive
Time Stationary Non-stationary
Field strength Linear Non-linear
Direction of Isotropic AnisotropicE or H
Terminology
27
Properties of ε or µ
Isotropic Materials
ε (and μ) are scalar quantities,which means that D || E (and B || H )
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D EB µH
ε==
E
x
y
D
x
y
B
H
ε (or μ) is a tensor (can be written as a matrix)
This results in E and D being NOT proportional to each other.
0 00 00 0
x x x
y y y
x x x
z z z
y y y
z z z
D ED E
D ED ED E D E
εεε
εε
ε
=
= =
=
Anisotropic Materials
29
Example:
D Eε= ⋅or
0 00 00 0
x h x
y h y
z v z
D ED ED E
εε
ε
=
Anisotropic Materials (cont.)
30
Practical example: uniaxial substrate material
Teflon substrate
Fibers (horizontal)
There are twodifferent
permittivity values, a horizontal one
and a vertical one.
Anisotropic Materials (cont.)
31
This column indicates that εv is being measured.
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