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Electromagnetics
School of Electrical and Computer Engineering
University of Tehran
Homework 1
Fall Semester 1384
Due 84/7/16
Problem 1
The scalar field
()
)
2
exp(
)
2
sin(
,
y
x
y
x
π
−
π
=
Φ
is defined in the region
1
0
≤
≤
x
and
0
≥
y
.
a)
Evaluate
Φ
∇
=
r
r
F
, and show that the resulting
F
r
is a divergenceless vector field.
b)
What is the maximum and minimum value of
)
,
(
y
x
Φ
in the region
1
0
≤
≤
x
and
0
≥
y
?
c)
Obtain the mathematical expression of the equipotential curves corresponding to
5
.
0
=
Φ
and
5
.
0
−
=
Φ
, and draw these curves in the region
1
0
≤
≤
x
and
0
≥
y
.
d)
Consider a circle of radius 0.0005 whose center is at
)
1
,
8
1
(
)
,
(
=
y
x
. Determine the point
on this circle for which the function
)
,
(
y
x
Φ
attains a maximum.
Problem 2
The volume
v
of a threedimensional region
V
should be determined. Assume that both the
position vector
r
r
and the unit normal vector
n
ˆ
are known at every point on
V
∂
. Obtain a
mathematical expression for
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 Spring '11
 Dolatabadi
 Electromagnet

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