ECE 329 Lecture Notes — Sp09, Erhan Kudeki
1 Vector felds and Forces
•
Fundamental building blocks of matter,
electrons
and
protons
, are
electri
cally charged
. Interactions between these “charge carriers” are described in
terms of
electric
and
magnetic felds
, much as
gravitational
interactions
are described in terms of
gravitational felds
produced by massive bodies.
Maxwell’s equations
:
I
S
E
·
d
S
=
Z
V
ρ
±
o
dV
I
S
B
·
d
S
=0
I
C
E
·
d
l
=

Z
S
∂
B
∂t
·
d
S
I
C
B
·
d
l
=
μ
o
Z
S
(
J
+
±
o
∂
E
)
·
d
S
with
μ
o
≡
4
π
×
10

7
H
m
,
and
±
o
=
1
μ
o
c
2
≈
1
36
π
×
10
9
F
m
,
in mksA units, where
c
=
1
√
μ
o
±
o
≈
3
×
10
8
m
s
is the speed of light in free space.
(In Gaussiancgs units
B
c
is used in
place of
B
, while
±
o
=
1
4
π
and
μ
o
=
1
±
o
c
2
=
4
π
c
2
.)
–
In general, charge carrier interactions and electromagnetic ±eld vari
ations account for all electric and magnetic phenomena observed in
nature.
•
We think of electric and magnetic ±elds
E
and
B
produced
by individual
charge carriers permeating all space (with proper time delays).
–
That is, we associate with each space location
(
x,y,z
)
≡
r
,
a pair of timedependent
vectors
E
(
r
,t
)=(
E
x
(
r
)
,E
y
(
r
)
z
(
r
))
and
B
(
r
) = (
B
x
(
r
)
,B
y
(
r
)
z
(
r
))
that we refer to as
E
and
B
for brevity (dependence on position
r
and
time
t
is
implied
).
–
Field vectors
E
and
B
and electric charge and current densities
ρ
and
J
(describing the distribution and motions of charge carriers) satisfy a set
of coupled linear constraints known as
Maxwell’s equations,
shown
in the margin.
◦
Maxwell’s equations are expressed in terms of closed surface and
line integrals of the ±eld quantities enclosing arbitrary volumes
V
and surfaces
S
in 3D space — recall MATH 241 and PHYS 212
— or in equivalent di²erential forms (see the margin on the next
page).
◦
Maxwell’s equations were “discovered” as a consequence of experi
mental and theoretical studies led by 19th century scientists includ
ing Gauss, Ampere, Faraday, and Maxwell. They remain intact as
the valid description of electromagnetic ±elds despite the scienti±c
upheavals (paradigm shifts) of 20th century: relativity and quan
tum mechanics
1
.
–
Given the charge and current densities
ρ
and
J
, Maxwell’s equations can
be
solved
for the ±elds
E
and
B
, which, in turn, specify the interaction
force exerted on a “test particle” with a charge
q
, position
r
, and velocity
v
≡
˙
r
=
d
r
dt
according to
Lorentz Force
formula
Lorentz
Force
F
=
q
(
E
+
v
×
B
)
.
As such,
1
Quantum mechanics — which you were exposed to in PHYS 214 — teaches us that electro
magnetic felds are
quantized
and interact with systems oF bound charge carriers (i.e., atoms
and molecules) one quantum at a time, each quantum (photon) having a fnite amount oF
energy proportional to the Frequency oF feld variations. In quantum electrodynamics, feld
strength provides a measure oF likelihood For the detection (i.e., an atomic interaction) oF a
feld quantum at a given time and locale — classical concepts oF position and trajectory are
not used For feld quanta.