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Swartz 06/5/20
ECE 303 –
Electromagnetic Fields and Waves
– Summer 2006
Lecture 2:
ECE 303
Electromagnetic Fields and Waves
Instructor:
Dr. Wesley E. Swartz
Fall 2008 Lecture 4
2008/9/5
Electric Potential
Electric Scalar Potential
Laplace’s and Poisson’s Equations
Potentials of Some Simple Charge Distributions
3
Swartz 06/5/20
ECE 303 –
Electromagnetic Fields and Waves
– Summer 2006
Lecture 2:
Conservative or Irrotational Fields
•
Irrotational or Conservative Fields:
–
Vector fields
for which
are called “irrotational” or
“conservative” fields.
–
This implies that the line integral of
around any closed loop is
zero.
•
Equations of Electrostatics:
–
Recall the equations of electrostatics from a previous lecture:
In
electrostatics or electroquasistatics
, the Efield is conservative or irrotational.
0
=
×
∇
F
F
0
s
d
F
=
•
ρ
ε
=
•
∇
E
o
0
=
×
∇
E
F
4
Swartz 06/5/20
ECE 303 –
Electromagnetic Fields and Waves
– Summer 2006
Lecture 2:
Conservative or Irrotational Fields
•
More on Irrotational or Conservative Fields:
–
If the line integral of
around any closed loop is zero …
–
… then the line integral of
between any two points is independent of any
specific path (i.e. the line integral is the same for all possible paths
between the two points):
0
s
d
F
=
•
B
r
r
A
r
r
B
r
r
A
r
r
B
r
r
A
r
r
2
1
2
1
2
1
2
1
1
2
2
1
s
d
F
s
d
F
0
s
d
F
s
d
F
0
s
d
F
s
d
F
path
path
path
path
path
path
•
=
•
=
•
−
•
=
•
+
•
F
1
r
2
r
path A
path B
0
s
d
F
=
•
F
5
Swartz 06/5/20
ECE 303 –
Electromagnetic Fields and Waves
– Summer 2006
Lecture 2:
•
The scalar potential:
–
Any conservative field can always be written as the gradient of some scalar
quantity. This is because the curl of a gradient is always zero.
•
For the conservative Efield one writes:
–
(The negative sign is just a convention)
–
Where
is the
scalar electric potential
.
•
The scalar potential is defined only up to a constant:
–
If the scalar potential
gives a certain electric field,
–
then the scalar potential
will also give the same electric field
(where
c
is a constant).
–
Remember that the derivative of a constant is zero.
•
The absolute value of potential in a problem is generally fixed by some
physical reasoning that essentially fixes the value of the constant
c
.
The Electric Scalar Potential (1)
Φ
−∇
=
E
(
)
()
0
F
=
∇
×
∇
=
×
∇
then
∇
=
F
If
c
)
r
(
+
)
r
(
6
Swartz 06/5/20
ECE 303 –
Electromagnetic Fields and Waves
– Summer 2006
Lecture 2:
The Electric Scalar Potential (2)
•
We now know that:
•
This immediately suggests that:
–
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 Fall '06
 RANA
 Electromagnet

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