PHY 505
Classical Electrodynamics I
Fall 2010
Midterm Exam 1 with solutions
Problem M.1
(To be graded of 150 points). Calculate the spatial distributions of
and
E,
created by a
long, round cylinder of radius
R
, with the electric charge uniformly spread over its volume. Compare the
result with that of for the uniformly charged sphere. Can you calculate the electrostatic energy of the
system (per unit length)? If not, can you estimate the total energy of such cylinder of a finite length
L
>>
R
?
Solution
: Due to the axial symmetry of the problem, we may write
(
r
) =
(
),
E
(
r
) =
n
E
(
), where
is
the distance from the cylinder’s axis. Applying the Gauss law to a long cylinder of radius
, coaxial with
the charged cylinder, we get
,
for
,
/
1
,
for
,
/
2
)
(
2
0
R
R
R
E
(*)
where
is the electric charge per unit length. Now integrating the relation
E
(
) = 
d
(
)/
d
which
follows, for our symmetry, from the general expression for gradient in cylindrical coordinates,
1
we get
.
for
,
/
ln
,
for
,
2
/
2
)
(
2
1
2
2
0
R
c
R
R
c
R
Since the electrostatic potential has to be continuous at
=
R
, the integration constants have to be
related as
2
1
1
2
c
c
.
Comparison of Eq. (*) with Eq. (1.19) and (1.22) of the lecture notes shows that while the field
inside the cylinder changes similarly to that inside the uniformly charged sphere,
outside the cylinder it
changes much slower – as l/
rather than 1/
r
2
. Due the this change, integral (1.67) for the electrostatic
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 Fall '08
 Stephens,P
 Charge, Electrostatics, Electric charge, 0 L, unit length

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