329
1. ECE Verify that vector identity
Homework 7
H
Due: March 9, 2010, 5PM
EE
H=
(E H)
holds for E = 4e and H = 3e by expanding both sides of the identity. Treat as a real
x
y
constant.
You should download the table of vector identities from the ECE 329 web site and examine the list
to familiarize yourself with the listed identities they are widely employed in electromagnetics as
well as in other branches of engineering such as uid dynamics.
2. Charge conservation states that the net outward ux of current density from a volume V through its
bounding surface S equals the time rate of decrease of net charge contained within the volume:
z
z
J dS =
S
d
dt
dV
V
Note that applying the Divergence theorem yields the dierential form:
J=
d
dt
a) Calculate J dS for the surface of a cubic volume V = 8 m centered at the origin if the
current density is J = 2xx + 3x y + 4z (y 1) z A/m .
b) Is the total charge contained in the cube increasing, decreasing, or neither?
c) What are the physical units of the coecients 2, 3, and 4 used in dening J.
d) Find the charge density at the origin, (0, 0, 0, t) as a function of time, if (0, 0, 0, 0) = 0.
3. Consider a homogeneous conductor where J = E.
a) Use Gauss's law E = and the continuity equation J = to derive the dierential
equation:
3
S
2
2
2
2
t
o
t
+
=0
for the charge density .
b) Find the solution of the dierential equation above for t > 0 if at t = 0 the charge density is
(x, y, z, 0) = sin(100x) C/m over all space, where is positive.
c) Find the current density J(x, y, z, t) by using the continuity equation, considering the problem's
geometry, and assuming that there are no external applied E elds.
d) According to solution the found in part (b), how long would it take for to reduce to 0.01 sin(100x)
C/m ? Assume that = 5.8 10 S/m.
e) Explain why the magnitude of the net charge decreases everywhere, by using the results of part
(c) to discuss specically the ow of current at x = 0 and x = .
0
3
o
0
0
3
7
100
1
4. In this problem, we will study the propagation of waves in free space. In a source-free region where
J = 0 and = 0, Maxwell's equations reduce to:
E=0
B
t
B=0
E
B = 0 0
t
E=
For E = xE (z, t), where E (z, t) is an arbitrary function of coordinates z and t only,
a) Show that Gauss' Law is satised.
b) Show that E = y .
c) Show that ( E) = x .
d) Take the curl of Faraday's law and substitue in Ampere's law to show that:
x.
( E) =
e) Combine (c) and (d) to derive the wave equation:
x
x
Ex
z
2 Ex
z 2
2 Ex
0 0 t2
2 Ex
2 Ex
0 0
=0
z 2
t2
5. Let E = xE (z, t) as in problem 4 above, where E (z, t) = E cos(t z + ) and E (amplitude),
(radian frequency), (wavenumber), and (phase shift) are positive real scalars. These elds
describe waves polarized in the x direction and propagating in the z directions, respectively.
a) Show that the specied E satises the wave equation derived above if:
x
x
0
0
x
= 0
0
=
c
where c 3 10 m/s is the speed of light in free space.
b) Substitute the specied E into Faraday's law and solve for B(z, t) assuming that B(0, 0) = 0.
What is a possible value for ? Show that your answer can be expressed as B = y (be
careful about maintaining the correct order for the upper and lower signs).
c) Use B = H to show that H = y in terms of an appropriately dened constant . What
is in terms of and ?
8
x
Ex
c
Ex
0
0
0
0
0
0
2