The Electric Dance: Oscillating Electric Fields
Between the plates of a charged capacitor there is a more or less uniform electric field. The field
points from the positive plate to the negative one and
has a magnitude of E = V/d, where V is the potential
difference between the plates and d is the plate
separation. The battery had to do work to cause a
charge separation on the capacitor plates. This work
can be thought to be stored in the electric field, so that
there is a an energy spread throughout the volume
between the capacitor plates. This energy density (i.e.
energy per cubic meter of volume between the plates)
fig.1 A capacitor charged
is expressed mathematically by the equation
(1) uE =
by a voltage source
1
2
ε0 E 2
where 0 is the permittivity constant, 0 = 8.854×10-12 J/V2m. Just as a battery produces a fixed
potential difference between its terminals, an alternating current generator produces a
sinusoidally varying potential between its terminals.
fig.2 An ac generator connected to
a load resistor
fig. 3 The potential difference across the
resistor in the circuit at the left
The generator is assumed to be an ideal voltage source which produces a potential difference
across its terminals given by V ( t ) = V0 cos(ω t ) . This is the potential difference that appears
across the resistor. Potentials with this sort of time dependence (varying as a sine or cosine) are
said to be time harmonic.
If an ac generator is connected to a capacitor, the electric field between the plates is still given by
E = V/d and will also vary sinusoidally. This is shown at the top of the following page.
3.1
fig.4 An ac generator connected
to a capacitor
fig. 5 The electric field between the
capacitor plates in the circuit on the left
Two graphical representations of the oscillating
electric field are shown at the left. The field can be
represented mathematically as
E ( t ) = E0 cos(ω t )
Remember that the electric field is a vector that
points from the positive plate to the negative one.
It changes direction every half cycle as the plates
change polarity.
The quantity is called the angular frequency and
is measured in units of radians per second.
The energy density of this electric field is still
given by
fig. 6 How the electric field vector
varies with time in the circuit of fig. 4
uE ( t ) = 12 ε0 E 2( t ) = 12 ε0 E0 cos2 (ω t )
This is plotted in figure 7. If the oscillations are very rapid we are often interested only in the
average value of the energy density. The average value of cos2(x), when averaged over a full
period, is ½. Averages are indicated with angled brackets.
(2) cos2( t) = ½
So that
(3)
uE ( t ) =
1
2
1
2
ε0 E02 cos2 (ω t ) =
ε0 E02 cos2 (ω t ) = 14 ε0 E02
3.2
fig. 7 The time dependent electric field energy
density for the circuit of figure 4 and its average
value
When two voltage sources are connected in series, the net potential difference is the sum of the
voltages of each. In figure 8, two batteries are
shown connected to a capacitor. The potential
difference across the capacitor plates is V1 + V2
and the electric field is given by E = (V1 + V2) / d
= E1 + E2 where E1 is the field that would have
been produced if only battery #1 were in the
circuit, and similarly for E2. This is an example of
the important principle of superposition: the net
electric field produced by two charge distributions
acting together is the vector sum of the electric
fields produced by each acting alone.
Superposition lies at the heart of many optical
phenomena.
We want to use the superposition principle to study the behavior of time harmonic electric fields.
To reduce that study to its basic elements, we will discuss synchronized time harmonic ideal
electric generators. For the purposes of the present section we’ll assume that all generators operate
at the same circular frequency and that the generators differ only in the amplitude and the phase
of their voltage oscillations. These values
will be indicated for each generator.
3.3
fig. 11 A generator producing voltage
oscillations of amplitude V0 and phase
.
fig. 12 The potential difference produced by the
generator of fig. 11 between the points A and B.
The delay time is t = / . The amplitude of the
oscillations is V0.
A generator like the one shown in figure 11 produces a potential difference across its terminals of
V0 cos( t + ). Another way to write this is V0 cos[ (t + / )] V0 cos( t + t), where t = /
is the (fixed) delay time.
When two ideal generators act in series they produce a net voltage which is the sum of their
individual voltages.
fig. 13 Two generators
connected in series
The potential difference between points A and C in figure 13 is
VC - VA = (VB - VA) + (VC - VB) = V0 cos( t + 1) + V0 cos( t +
identity (A.6) from Appendix A, which says that
cos(α ) + cos( β ) = 2 cos(
(4) VC - VA = 2 V0 cos((
1
-
α−β
2
2
) cos(
α +β
)/2) cos( t + (
2
1
+
). This can be simplified using
2
) so that
)/2).
2
When two voltages, oscillating at a common frequency, combine as in figure 13 they are said to
interfere. The resulting voltage, expressed in equation (4), oscillates with the same frequency as
the interfering voltages but with an amplitude that depends on the phase difference between the
two voltages. We will see many examples in optics of this sort of interference.
3.4
Problem 1: Find the amplitude of the voltage oscillations in eq. (4) in terms of V0 when the phase
difference is:
C. /2
D. /4
A. 0
B.
Quote your answer to three significant figures
When the two generators of figure 13 are connected to a capacitor, the electric field is
proportional to the voltage difference between the plates.
Figure 17 shows a situation where the electric field is E(t) = 2 E0 cos(
where E0 = V0 /d and
= 1 - 2 . The
instantaneous energy density between the plates is
uE ( t ) =
=
/2) cos( t + (
1
+
)/2)
2
ε0 2
E (t )
2
ε0
φ +φ
∆φ
( 4 E02 cos2 ( 2 ))cos2 (ω t + 1 2 2 )
2
and the average energy density is
(5)
uE =
ε0
φ +φ
∆φ
∆φ
( 4 E02 cos2 ( 2 )) cos2 (ωt + 2 ) = ε0 E02 cos2 ( 2 )
2
1
2
Problem 2: Plot the ratio uE / 0 E02 in equation (5) as a function of the phase difference
each maximum and minimum in this plot, comment physically on why the two interfering
oscillations produce the average energy shown in the plot.
3.5
. For
Problem 3: Analyze the electric field produced by the following arrangement.
Use the expressions in Appendix A to deduce that
E(t ) =
[
]
1/ 2
V0
3 + 2 (cos(φ1 − φ2 ) + cos(φ1 − φ3 ) + cos(φ2 − φ3 ) ) cos(ωt + γ )
d
where
tan(γ ) =
sin(φ1 ) + sin(φ2 ) + sin(φ3 )
cos(φ1 ) + cos(φ2 ) + cos(φ3 )
What is the average electric field energy density?
3.6
3.7