8.02 Exam Three Spring 2007 Solutions
PART I (25 points)
Question A (5 points).
A parallel-plate capacitor with circular plates of radius
R
and
separated by a distance
d
is charged through a straight wire carrying current
I
, as
shown in the figure below:
During the time interval that the plates are charging, the direction of the magnetic field
at the point
P
on the figure above
1.
points into the page
2.
points out of the page
3.
points radially outward away from the axis of the capacitor.
4.
points radially inward toward the axis of the capacitor.
5.
points from bottom to top
6.
points from top to bottom
Solution: 2
. A time changing electric field acts like a current with regards to creating
magnetic field. The direction of the magnetic field depends on two factors , which
direction the electric field is pointing and whether it is increasing or decreasing. The
generalized Ampere’s Law is
0
0
0
through
Amperian
open
loop
surface
d
d
I
d
dt
µ
µ ε
⋅
=
+
⋅
∫
∫∫
B
s
E
a
G
G
G
G
v
.
(1.1)

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If we choose a circle centered on the axis in between the plates as our Amperian loop,
and the disk defined by the circle as the open surface then there is no conduction current
flowing through the disk, only changing electric flux. Then Eq
(1.1) becomes
0
0
circle
disk
d
d
d
dt
µ ε
⋅
=
⋅
∫
∫∫
B
s
E
a
G
G
G
G
v
.
(1.2)
(This is case is similar to determining the direction of the electric field when magnetic
flux is changing, however Faraday’s Law has a minus sign which will give directions
opposite to our case).
We orient the disk, so that the electric flux is positive, this means the circle is oriented
counterclockwise as seen from above. Since the plates are charging, the electric field is
increasing, hence the electric flux is increasing. Therefore, the line integral is positive so
that the direction of the magnetic field must point in the direction that we circulated
around the line integral which was counterclockwise (as seen from above). Thus at the
point
P
on the left, the magnetic field points out of the plane of the paper.