ECE 329 Lecture Notes — Summer 09, Erhan Kudeki
1 Vector felds and Lorentz Force
•
Interactions between charged particles can be described and modeled
1
in terms of
electric
and
magnetic
felds
just like gravity can be
formulated in terms of
gravitational felds
of massive bodies.
–
In general, charge carrier dynamics and electromagnetic Feld vari
ations
2
account for all electric and magnetic phenomena observed
in nature and engineering applications.
•
Electric and magnetic Felds
E
and
B
generated by charge carriers —
electrons
and
protons
at microscopic scales — permeate all space with
proper time delays, and combine additively.
±
2
±
1
1
2
x
±
2
±
1
1
2
y
–
Consequently we associate with each location of space having Carte
sian coordinates
(
x, y, z
)
≡
r
a pair of timedependent
vectors
E
(
r
,t
)=(
E
x
(
r
)
,E
y
(
r
)
z
(
r
))
1
Interactions can also be formulated in terms of
past locations
(i.e., trajectories) of charge carriers.
Unless the charge carriers are stationary — i.e., their past and present locations are the same — this
formulation becomes impractically complicated compared to Feld based descriptions.
2
Timevarying Felds can exist even in the absence of charge carriers as we will Fnd out in this course
— light propagation in vacuum is a familiar example of this.
1
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B
(
r
,t
)=(
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
).
Maxwell’s equations
:
∇·
E
=
ρ
±
o
B
=0
∇×
E
=

∂
B
∂t
B
=
μ
o
J
+
μ
o
±
o
∂
E
.
such that
F
=
q
(
E
+
v
×
B
)
,
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
above, while
±
o
=
1
4
π
and
μ
o
=
1
±
o
c
2
=
4
π
c
2
.)
•
Field vectors
E
and
B
and electric charge and current densities
ρ
and
J
— describing the distribution and motions of charge carriers — are
related by (i.e., satisfy) a coupled set of linear constraints known as
Maxwell’s equations,
shown in the margin.
–
Maxwell’s equations are expressed in terms of divergence and curl
of ±eld vectors — recall MATH 241 — or, equivalently, in terms
of closed surface and line integrals of the ±elds enclosing arbitrary
volumes
V
and surfaces
S
in 3D space, as you have ±rst seen in
PHYS 212.
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 Spring '08
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 Dot Product, Force, Electromagnet, Maxwell’s equations

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