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School of Electrical and Computer Engineering, Cornell University
ECE 303: Electromagnetic Fields and Waves
Fall 2005
Homework 4
Due on Sep. 23, 2005 by 5:00 PM
Reading Assignments:
i) Review the lecture notes.
ii) Relevant sections of the online
Haus and Melcher
book for this week are 8.08.2, 8.6, 11.011.2. Note
that the book contains more material than you are responsible for in this course. Determine relevance by
what is covered in the lectures and the recitations. The book is meant for those of you who are looking for
more depth and details.
Solutions of Laplace’s Equation for ECE303
Spherical Coordinate System
Cylindrical Coordinate System
()
A
r
=
φ
Constant potential
( )
A
r
=
Constant potential
()
r
A
r
=
Spherically symmetric potential
( ) ( )
r
A
r
ln
=
Cylindrically symmetric potential
() ()
θ
cos
r
A
r
=
Potential for uniform z
directed EField in spherical coordinates
( ) ( )
cos
r
A
r
=
Potential for uniform x
directed EField in cylindrical coordinates
()
()
2
cos
r
A
r
=
Potential for pointcharge
dipolelike solution with the dipole pointing in the
+zdirection
()
( )
r
A
r
cos
=
Potential for a linecharge
dipolelike solution with the dipole pointing in
the
+xdirection
Problem 4.1: (A linechargedipole)
Consider a linecharge dipole consisting of two infinite line charges, carrying
λ
+
and
−
Coulombs/m
of charge, and separated by a distance
d
as shown.
−
+
x
y
r
d
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From lecture notes you already know the expression for the potential in cylindrical coordinates. Here you
need to show that for distances
r
>>
d
, the potential takes the following form (excuse the bad notation
where the same symbol is being used for the potential and the angle):
()
( )
r
d
r
o
φ
ε
π
λ
cos
2
=
r
Problem 4.2: (A perfect metal rod in a uniform electric field)
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 Fall '06
 RANA
 Electromagnet

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