CHAPTER
32
Maxwell’s Equations and
Electromagnetic Waves
1* ·
A parallelplate capacitor in air has circular plates of radius 2.3 cm separated by 1.1 mm. Charge is flowing
onto the upper plate and off the lower plate at a rate of 5 A. (
a
) Find the time rate of change of the electric field
between the plates. (
b
) Compute the displacement current between the plates and show that it equals 5 A.
(
a
) Use Equ. 2325:
E
=
Q
/
e
0
A
;
dE
/
dt
= (
dQ
/
dt
)/
e
0
A
(
b
) Use Equ. 323:
f
e
=
EA
dE
/
dt
=
I
/
e
0
A
= 3.40
×
10
14
V/m
.
s
I
d
=
e
0
A
(
dE
/
dt
) =
I
= 5 A
2
·
In a region of space, the electric field varies according to
t
2000
sin
)
C
/
N
(0.05
=
E
0, where
t
is in
seconds. Find the maximum displacement current through a 1m
2
area perpendicular to
E
.
Use Equs. 2314 and 33
I
d
= (8.85
×
10
–12
×
0.05
×
2000) A = 8.85
×
10
–10
A
3
··
For Problem 1, show that at a distance
r
from the axis of the plates the magnetic field between the plates is
given by
B
= (1.89
×
10
–3
T/m)
r
if
r
is less than the radius of the plates.
1. Use Equ. 324;
I
=
I
d
; apply cylindrical symmetry
2. Evaluate
B
(
r
)
2
p
rB
=
μ
0
I
d
(
r
2
/
R
2
);
B
=
μ
0
I
d
r
/2
p
R
2
B
(
r
) = (1.89
×
10
–3
T/m)
r
4
··
(
a
) Show that for a parallelplate capacitor the displacement current is given by
I
d
=
C dV
/
dt
, where
C
is the
capacitance and
V
the voltage across the capacitor. (
b
) A parallel plate capacitor
C
= 5 nF is connected to an
emf
E
=
E
0
cos
?
t
, where
E
0
= 3 V and
?
= 500
p
. Find the displacement current between the plates as a
function of time. Neglect any resistance in the circuit.
(
a
) Use Equs. 2510 and 323;
E
=
V
/
d
(
b
)
dV
/
dt
= –
E
0
sin
?
t
I
d
=
e
0
A
(
dE
/
dt
) = (
e
0
A
/
d
)(
dV
/
dt
) =
C dV
/
dt
I
d
= –(23.6
μ
A) sin 500
p
t
5* ··
Current of 10 A flows into a capacitor having plates with areas of 0.5 m
2
. (
a
) What is the displacement
current between the plates? (
b
) What is
dE
/
dt
between the plates for this current? (
c
) What is the line integral
of
l
d
⋅
B
around a circle of radius 10 cm that lies within and parallel to the plates?
(
a
) See Problem 1
(
b
)
dE
/
dt
=
I
d
/
e
0
A
(see Problem 1)
(
c
) Use Equ. 324;
I
d
enclosed =
I
d
(
p
r
2
/
A
)
I
d
= 10 A
dE
/
dt
= 2.26
×
10
12
V/m
.
s
∫
B
.
d
l
=
μ
0
I
d
(
p
r
2
/
A
) = 7.90
×
10
–7
T
.
m
6
··
A parallelplate capacitor with circular plates is given a charge
Q
0
. Between the plates is a leaky dielectric
having a dielectric constant of
?
and a resistivity
?
. (
a
) Find the conduction current between the plates as a
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Maxwell’s Equations and Electromagnetic Waves
function of time. (
b
) Find the displacement current between the plates as a function of time. What is the total
(conduction plus displacement) current? (
c
) Find the magnetic field produced between the plates by the leakage
discharge current as a function of time. (
d
) Find the magnetic field between the plates produced by the
displacement current as a function of time. (
e
) What is the total magnetic field between the plates during
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 Spring '08
 MILLAN/THORSTENSEN
 Physics, Charge, Magnetic Field, electromagnetic wave

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