, 2002

1) Resistance

Determine the resistance of a hollow copper sphere. The inner surface is located at r = a
and the outer surface is located at r = b. Assume current flows from the outer surface to
the inner surface and that it is uniform on an arbitrary cross-section perpendicular to
current flow.

2) Amperes Law

The following current distributions can be used to approximate a coaxial cable
transmission line, where a < b

]
/
[
]
/
[
)
,
,
(
2
3
m
A
m
A
b
r
z
J
a
r
z
r
J
z
r
J
s
o



= <
= r


A current density exists in the region r < a and a surface current density exists at the
location r = b. Furthermore, the total current in the direction is equal and opposite to
the total current in the
direction. You may assume the entire domain is free space.
z
z

1) What are the regions of the problem?

2) What coordinate system applies to this problem (this better be easy).

3) What direction are the field lines? What direction is the current?

4) Given the answer to part 3, what surface would be appropriate to apply Amperes
Law? What would is r ? What is r that bounds the surface?
S
d
l
d

5) Set up, (do not solve) the integral, = l
S
S
d
J
l
d
H
r
r
r
r
. Indicate the limits and
differentials.

6) For each region, determine the magnetic field. Remember, when you are considering
an arbitrary surface inside a given region, the current passing through that surface
includes any regions inside the region you are currently considering. Specifically, pay
attention to where the current exists. When you have finished determining the field, your
solution should include all information: the field in that region, the units, and the
specification of the region (similar to the format of the charge distribution given at the
top of the page).

7) Verify your solution is correct for each region by using the differential form of
Maxwells Laws as applied to electrostatics,
r
r
and
r
.
J
H
=
× 0
= B
K. A. Connor and J. Braunstein Revised: 10/27/02
Rensselaer Polytechnic Institute Troy, New York, USA
1 Fields and Waves I
Name _______________ ECSE-2100 Fall 2002 Section ____________
3) Amperes Law

Repeat all the steps in problem 2 for the following current distribution
r
]
/
[ )
,
,
(
2
m
A
a
r
r
J
z
r
J
o
<
=




d
1





small
h
loop



h


o
vr




Jr




a


d
2

r
]
/
[ )
,
,
(
2
m
A
a
r
z
r
J
z
r
J
o
<
=

The above figure represents a lengthwise view of a current density in a wire with radius
a.


4) Approximations

If d
1
>> a, what is the total field passing through the small circular loop on the right
in the above figure? The center of the loop is a distance d
Br
1
from the origin and the loop
has radius b. The surface of the loop is in the same plane as the wire.

5) Faradays Law

A square loop is placed a distance from the wire and oriented such that the wire lays in
the same plane as the surface of the loop. The loop moves a constant velocity away from
the wire. Determine the EMF induced across the leads of the loop for t > 0. At t = 0, the
left edge of the loop is a distance d
2
from the origin.
K. A. Connor and J. Braunstein Revised: 10/27/02
Rensselaer Polytechnic Institute Troy, New York, USA
2 Fields and Waves I
Name _______________ ECSE-2100 Fall 2002 Section ____________

K. A. Connor and J. Braunstein Revised: 10/27/02
Rensselaer Polytechnic Institute Troy, New York, USA
3