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8.02 Sample Final C (Modified from Spring 1993)
p. 1/4
1.
Displacement Current
A parallelplate capacitor consists of two circular
plates, each with radius
R
, separated by a distance
d
.
The electric field
E
between the plates is directed
upwards (see sketch).
R
r
d
P
E
(a) What is the total energy stored in the electric field o
the capacitor, in terms of
E
,
R
,
d
, and appropriate
physical constants? Assume that the electric field is
uniform between the plates and zero outside of the
plates (i.e., neglect fringing fields).
f
(b) Now, suppose that the electric field is
increasing
with time (
dE/dt > 0
). The point
P
is
located between the plates at radius
r < R
(see sketch)
. Derive
an expression for the
magnitude of the magnetic field
B
at point
P
in terms of
dE/dt
,
r
, and appropriate
physical constants.
Indicate the direction of
B
at
P
on the sketch.
(c) What is the Poynting flux at point
P
in terms of the quantities given? Give both
direction and magnitude.
(d)
Derive
an expression for the total electromagnetic energy flowing into the capacitor
per unit time across
r = R,
in terms of
E
,
dE/dt
, and given constants. Write down an
equation relating this quantity to the electric energy contained in the capacitor (see part
(a)).
2.
Waves
For the case of electromagnetic fields in a vacuum with the form
E
= E(x,t)
;
B
= B(x,t)
where
are unit vectors in the
x
,
y
, and
z
directions, respectively, Faraday's and Ampere's Laws may be written as
y
ˆ
z
ˆ
z
y
x
ˆ
and
,
ˆ
,
ˆ
t
B
x
E
∂
∂
−
=
∂
∂
(Faraday’s Law);
t
E
x
B
∂
∂
−
=
∂
∂
0
0
ε
µ
(Ampere’s Law)
(a) Using these equations,
derive
a secondorder differential equation for
E(x,t)
alone.
(b) Consider the function
E(x,t) = E
0
(axbt)
2
, where
a
and
b
are constants. What
condition must hold for this function to be a solution to the differential equation you
obtained in part (a)?
(c) Again consider the function
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This note was uploaded on 04/27/2009 for the course 8 8.02 taught by Professor Hudson during the Spring '07 term at MIT.
 Spring '07
 Hudson

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