Problem Set 4:
(Assignment 5)
For a given system the electric field and current density are given by
Assignment 5
E
r
,
t
E
0
exp
z
ct
2
2
L
2
cos
k
0
z
ct
e
1
J
r,
t
J
0
exp
z
ct
2
2
L
2
cos
k
0
z
ct
0
e
1
(a)
Calculate the magnetic flux density,
B
r
,
t
.
(b)
Calculate the frequency dependent fields and current densities
e
r
,
,
b
r
,
, and
j
r
,
.
(c)
What are the units of these three quantities in the gaussian units used by Jackson , in
the SI (MKSC) units?
(d)
Starting with the results of part (b) evaluate
e
r
,
b
r
, and
j
r
defined in the notes.
(e)
Let
0
/6 and
k
0
L
2
10
12
and calculate the time average of
E
r
,
t
J
r
,
t
over the period from
t
T
/2 to
t
T
/2 at
z
L
and at
z
0. (Note, a tabulated
integral).
(f)
Plot the results of part (e) for 0
cT
/
L
1.
Solution
(a):
The electric and magnetic fields are related by Faraday’s law
E
r
,
t
1
c
t
B
r
,
t
.
The given electric field has the form
E
r
,
t
f
z
ct
e
1
therefore the curl of the electric field
has the form
E
r
,
t
f
z
ct
e
2
. It follows that
1
c
t
B
r
,
t
f
z
ct
e
2
or
B
r
,
t
f
z
ct
e
2
c
E
0
exp
z
ct
2
2
L
2
cos
k
0
z
ct
e
2
The bracketed
c
should be added for SI units
Solution (b)
: The frequency dependent electric field amplitude is
e
r,
E
0
exp
z
ct
2
2
L
2
cos
k
0
z
ct
exp
i
t
dt
e
1
e
1
c
E
0
exp
i
z
c
exp
u
2
2
L
2
i
c
u
cos
k
0
u
du
e
1
2
c
E
0
exp
i
z
c
exp
u
2
2
L
2
i
c
k
0
u
du
e
1
2
c
E
0
exp
i
z
c
exp
u
2
2
L
2
i
c
k
0
u
du
e
1
c
L
2
E
0
exp
i
z
c
exp
L
2
2
c
k
0
2
exp
L
2
2
c
k
0
2
Similarly the frequency dependent amplitudes of the magnetic flux density are
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
b
r
,
c
e
2
c
L
2
E
0
exp
i
z
c
exp
L
2
2
c
k
0
2
exp
L
2
2
c
k
0
2
while the frequency dependent amplitudes of the current density are
j
r
,
e
1
c
L
2
J
0
exp
i
z
c
exp
L
2
2
c
k
0
2
i
0
exp
L
2
2
c
k
0
2
i
0
Solution (c):
The gaussian units for
E, B,
and
J
are statvolt/cm, gauss, and statampere/cm
2
respectively. The frequency dependent amplitudes have corresponding units statvolt
s/cm,
gauss
s, and statcoulomb/cm
2
.In this set of units
E
and
B
have the same dimensions,
charge/length
2
. The SI units are volt/m, Tesla (including the factor of c in the brackets), and
ampere/m
2
for the fields and current density and volt
s
/m, Tesla
s
(including the factor of c in
the brackets), and coulomb/m
2
for their frequency dependent amplitudes.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 G.
 Current, Magnetic Field

Click to edit the document details