Chapter 30
Maxwell’s Equations and Electromagnetic Waves
19
••
[SSM]
There is a current of 10 A in a resistor that is connected in series with a parallel plate
capacitor. The plates of the capacitor have an area of 0.50 m
2
, and no dielectric exists between the plates. (
a
)
What is the displacement current between the plates? (
b
) What is the rate of change of the electric field strength
between the plates? (
c
) Find the value of the line integral
∫
⋅
C
d
B
, where the integration path
C
is a 10cm
radius circle that lies in a plane that is parallel with the plates and is completely within the region between them.
Picture the Problem
We can use the conservation of charge to find
I
d
, the
definitions of the displacement current and electric flux to find
dE
/
dt
, and
Ampere’s law to evaluate
d
⋅
B
around the given path.
(
a
) From conservation of charge we
know that:
A
10
d
=
=
I
I
(
b
) Express the displacement current
I
d
:
[
]
dt
dE
A
EA
dt
d
dt
d
I
0
0
e
0
d
∈
∈
φ
∈
=
=
=
Substituting for
dE
/
dt
yields:
A
I
dt
dE
0
d
∈
=
Substitute numerical values and
evaluate
dE
/
dt
:
(
29
s
m
V
10
3
.
2
m
50
.
0
m
N
C
10
85
.
8
A
10
12
2
2
2
12
⋅
×
=
⋅
×
=

dt
dE
(
c
) Apply Ampere’s law to a circular
path of radius
r
between the plates
and parallel to their surfaces to
obtain:
enclosed
0
C
I
d
μ
=
⋅
∫
B
Assuming that the displacement
current is uniformly distributed and
letting
A
represent the area of the
circular plates yields:
A
I
r
I
d
2
enclosed
=
π
⇒
d
2
enclosed
I
A
r
I
π
=
Substitute for
enclosed
I
to obtain:
d
2
0
C
I
A
r
d
π
μ
=
⋅
∫
B
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Substitute numerical values and evaluate
∫
⋅
C
d
B
:
(
29
(
29
(
29
m
T
79
.
0
m
50
.
0
A
10
m
10
.
0
A
/
N
10
4
2
2
2
7
C
⋅
=
×
=
⋅

∫
μ
π
π
d
B
19
••
[SSM]
There is a current of 10 A in a resistor that is connected in series with a parallel plate
capacitor. The plates of the capacitor have an area of 0.50 m
2
, and no dielectric exists between the plates. (
a
)
What is the displacement current between the plates? (
b
) What is the rate of change of the electric field strength
between the plates? (
c
) Find the value of the line integral
∫
⋅
C
d
B
, where the integration path
C
is a 10cm
radius circle that lies in a plane that is parallel with the plates and is completely within the region between them.
Picture the Problem
We can use the conservation of charge to find
I
d
, the
definitions of the displacement current and electric flux to find
dE
/
dt
, and
Ampere’s law to evaluate
d
⋅
B
around the given path.
(
a
) From conservation of charge we
know that:
A
10
d
=
=
I
I
(
b
) Express the displacement current
I
d
:
[
]
dt
dE
A
EA
dt
d
dt
d
I
0
0
e
0
d
∈
∈
φ
∈
=
=
=
Substituting for
dE
/
dt
yields:
A
I
dt
dE
0
d
∈
=
Substitute numerical values and
evaluate
dE
/
dt
:
(
29
s
m
V
10
3
.
2
m
50
.
0
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 Spring '08
 Turner
 Physics, Current, Magnetic Field, Substitute numerical values, Brms

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