Prof. D. R. Wilton
Adapted from notes by Prof. Stuart A. Long
Notes 13
Maxwell’s Equations
ECE 3317
Electromagnetic Fields
Four vector quantities
E
electric field strength [Volt/meter] = [kg

m/sec
3
]
D
electric flux density [Coul/meter
2
] = [Amp

sec/m
2
]
H
magnetic field strength [Amp/meter] = [Amp/m]
B
magnetic flux density [Weber/meter
2
] or [Tesla] = [kg/Amp

sec
2
]
each are functions of space and time
e.g.
E
(
x
,
y
,
z
,
t
)
J
electric current density [Amp/meter
2
]
ρ
v
electric charge density [Coul/meter
3
] = [Amp

sec/m
3
]
Sources generating
electromagnetic fields
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

10
6
giga

G

10
9
tera

T

10
12
peta

P

10
15
centi

c

10

2
deci

d

10

1
deka

da

10
1
hecto

h

10
2
kilo

k

10
3
0
v
B
E
t
D
H J
t
B
D
ρ
Maxwell’s Equations
(time

varying, differential form)
Maxwell’s Equations
James 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)
Maxwell’s Equations (cont.)
ˆ
ˆ ˆ
ˆ
0
ˆ
C S
C S S
S
v
S V
d
E dr B ndS
dt
d
H dr J ndS D ndS
dt
B ndS
D ndS
ρ dV
Faraday’s law
Ampere’s law
Magnetic Gauss law
Electric Gauss law
(Time

varying, integral form)
The above are the most fundamental form of Maxwell’s equations since
all differential forms, all boundary forms, and all frequency domain forms
derive from them!
Maxwell’s Equations (cont.)
0
v
B
E
t
D
H J
t
B
D
ρ
Faraday’s law
Ampere’s law
Magnetic Gauss law
Electric Gauss law
(Time

varying, differential form)
0
D
H J
t
D
H J
t
J D
t
Law of Conservation of Electric
Charge (Continuity Equation)
v
J
t
Flow of electric
current out of volume
(per unit volume)
Rate of decrease of electric
charge (per unit volume)
[2.20]
Continuity Equation (cont.)
v
J
t
Apply the divergence theorem:
Integrate both sides over an arbitrary volume
V
:
v
V V
J dV dV
t
ˆ
v
S V
J ndS dV
t
V
S
ˆ
n
Continuity Equation (cont.)
Physical interpretation:
V
S
ˆ
n
ˆ
v
S V
J ndS dV
t
v
out v
V V
i dV dV
t t
encl
out
Q
i
t
(This assumes that the
surface is stationary.)
encl
in
Q
i
t
or
Maxwell’s Equations
Decouples
0
0 0
v
v
E
B D
E H J B D
t t
E D H J B
TimeDependent
TimeIndependent (Static
s)
and is a function of and is a functio
n of
v
H E H J
E B
H J D
B 0
D
v
j
j
Maxwell’s Equations
Time

harmonic (phasor) domain
j
t
Constitutive Relations
Characteristics of media relate
D
to
E
and
H
to
B
0
0
0
0
( = permittivity )
(
= permeability)
D E
B µ H
µ
12
0
7
0
[F/m]
8.8541878 10
= 4 10
H/m] ( )
[
µ
exact
[2.24]
[2.25]
[p. 35]
0 0
1
c
c = 2.99792458
10
8
[m/s]
(exact value that is defined)
Free Space
Constitutive Relations (cont.)
Free space, in the
phasor domain
:
0
0
0
0
( = permittivity )
(
D
=
= E
B
pe
=
rmeability
)
H
µ
µ
This follows from the fact that
V
aV t a
(where
a
is a real number)
Constitutive Relations (cont.)
In a
material medium
:
( = permittivity )
( = permeability
D = E
B=
)
H
µ
µ
0
0
=
r
r
µ µ
r
= relative permittivity
r
= relative permittivity
μ
or
ε
Independent of
Dependent on
space
homogenous
inhomogeneous
frequency non

dispersive dispersive
time
stationary
non

stationary
field strength linear
non

linear
direction of isotropic
anisotropic
E
or
H
Terminology
Isotropic Materials
ε
(or
μ
) is a scalar quantity,
which means that
E

D
(and
H

B
)
x x
B = H
y y
B = H
y
H
x
H
x
y
x x
D = E
y y
D = E
y
E
x
E
x
y
ε
(or
μ
) is a tensor (can be written as a matrix)
This results in
E
and
D
being
NOT
parallel to each other; they are
in different directions.
0 0
0 0
0 0
x x x
y y y
x x x
z z z
y y y
z z z
D E
D E
D E
D E
D E
D E
Anisotropic Materials
D
E
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