Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave
We have known the close connection between changing eletric fields
and magnetic fields. They can create each other and form a system of
electromagnetic fields.
Electromagnetic fields can propagate in the space (vacuum or material
environment). We call them electromagnetic waves. They play a very
important role in science and technology.
GENERAL PHYSICS II
Electromagnetism
&
Thermal Physics
4/8/2008 1
Chapter XIII
Electromagnetic Oscilation,
Eletromagnetic Field and Wave
§1. Oscillating circuits
§2. System of Maxwell’s equations
§3. Maxwell’s equations and electromagnetic waves
4/8/2008 2
We have known the close connection between changing eletric fields
and magnetic fields. They can create each other and form a system of
electromagnetic fields.
Electromagnetic fields can propagate in the space (vacuum or material
environment). We call them electromagnetic waves. They play a very
important role in science and technology.
In this chapter we will study how can describe electromagnetic fields,
what are their properties (in comparison with mechanical waves).
First we consider the oscillating circuits in which there exist oscillating
currents and voltages. They are sources for electromagnetic fields
4/8/2008 3
§1. Oscillating circuits:
1.1 L-C circuits and electrical oscillations:
• Consider the RC and LC
series circuits shown:
++++ ++++
• Suppose that the circuits are C R C
formed at t=0 with the capacitor ---- ---- L
charged to value Q.
There is a qualitative difference in the time development of the
currents produced in these two cases. Why??
1.1.1 Consider from point of view of energy (qualitatively):
• In the RC circuit, any current developed will cause energy to be
dissipated in the resistor.
• In the LC circuit, there is NO mechanism for energy dissipation;
energy can be stored both in the capacitor and the inductor!
4/8/2008 4
I
I
Q
+++
Q +++ C L
C R ---
---
RC: LC:
current decays exponentially current oscillates
I
I
0
0
0 0
t t
4/8/2008 5
Recall: Energy in the Electric and Magnetic Fields
Energy stored in a capacitor ... +++ +++
E
1 --- -- -
U CV 2
2
1 2
… energy density ...
uelectric 0E
2
Energy stored in an inductor …. B
1 2
U LI
2
1 B2
… energy density ... umagnetic
2 0
4/8/2008 6
Energy is stored in the capacitor Energy is stored in the inductor
I0 I 0
I
+ +
C L C L
- -
Q Q0
Q 0
I 0
I 0
I
- -
C L
C L + +
Q0 Q Q0
Energy is stored in the inductor Energy is stored in the capacitor
4/8/2008 7
1.1.2 Equations for Q(t) and I(t): 1 2 I
At t = 0 the switch is transfered from
the position 1 to the position 2:
L
2 C
Q dI d Q
V C V L L L
C dt dt 2
d 2Q Q
L 2
0 remember: d 2x
dt C m 2 x 0
k
dt
The solution Q(t) has the form analogue
to SHM (simple hamonic motion):
Q 0 cos(t )
Q 0
where and Q0 determined from initial conditions
• Differentiate above form for Q(t) and substitute into the differential
equation we can find
4/8/2008 8
dQ d 2Q
0Q 0 sin( 0 t )
2
02 Q 0 cos( 0 t )
dt dt
L Q0 cos( t ) 0 cos( t ) 0 0 L 0
1 1
02
0 Q 0 2
C C
1
Therefore,
0
LC
The oscillation of the current in LC circuit is determined by the
following equation:
dQ
I 0 Q0 sin( t )
0
dt
I I m sin( t ),
0
I m 0Q0
4/8/2008 9
1.2 LCR circuit and damped oscillation: R
1 2
When the swicth is transferred to 2: I
Q dI
V C R V L
V RI L
C dt C L
d 2Q dQ Q
L 2
R 0
dt dt C
The solution Q(t) has the form of a damped oscillation:
t
β
Q Q 0 e cos( 'o t )
R 1 R2
where and ' o 2
2L
LC 4L
The frequency of oscilation
Damping constant
4/8/2008
(In an LRC circuit, depends also on R) 10
Some remarks: R R 0
Q
• The amplitude of oscillation is damping,
0
since energy is dissipated in the resistor.
The role of resistance in oscillations of the
current in LCR circuit is analogue to friction
in mechanical oscillations ! 1
t
• The resistance R increases → the amplitude
of oscillation decreases faster. Oscillations Q R0
R
will occure as long as 0 is real, so there
’
4
exists a critical resistance
0
1 R c2 L
2 Rc 4 1
LC 4L C
If R R c the circuit is called underdamped; t
R R c : critically damped; R R c : overdamped.
4/8/2008 11
1.3 LRC circuit with alternating current (AC) source:
This is the case analogue to the mechanical R
driven oscillations with a periodic force.
Suppose that the emf of the source has C L
the following form:
sint
= m
The equation for Q(t) must be modified
by adding AC emf in the right hand side:
d 2Q dQ Q
L 2 R m sin t
dt dt C
First we consider the particular cases: the circuit or
R,
C,
or only.
L
4/8/2008 12
1.3.1 Circuit:
R R
The above equation leads to
dQ
R sin t I
dt
m
The formulas for the voltage and current
across R are as follows
V R R I R s i n t IR m s i n t
m
R
m R
m
VR IR
0 0
-
m
m 1
1
0 R 0
t t
4/8/2008 13
• Remark: Both the current and the voltage vary with time as sin .
t
We say they are “in phase” .
Phasor diagram:
• Vectors represent the maximum
voltage and the maximum current
• Both vectors make an angle t
with the positive x-axis
• With time both vectors rotate
counter-clockwise
• The vertical component of each
vector represents the instantaneous
value of voltage or current.
Impedance: The ratio of the maximum voltage to
the maximum current
For a resistor
Impedance of a resistor
4/8/2008 don’t depend on frequency. 14
1.3.2 Circuit:
C
Q
VC m sin
t Q C sin
m t C
IC
C
IC
dQ
C c o s
m t
dt
In this case the voltage on C and the current through C are not
in phase, we say that they are ”out of phase”.
The current has peaks at an earlier time than the voltage. The
current leads the voltage by one-quarter cycle or 90 .
m
C m
VC IC
0 0
m1 C 1
m
0 t 0
t
4/8/2008 15
Phasor diagram:
The vectors which represent the
current and the voltage are
perpendicular each to other,
as shown in the picture.
Impedance: We can calculate the impedance for capacitor
Note that the impedance of a capacitor depends on, beside C, also
the frequency. The impedance will be large at low frequencies.
The capacitor can play a role as a filter which stops low frequencies
and passes high frequencies.
4/8/2008 16
1.3.3 Circuit:
L
IL L
VL L L m s i n d I s i n t d t
dI
t m
dt L
L
I L dI L cos m sin t / 2
m
t
L L
In this case the voltage across L leads the current through L
by one-quarter cycle (90
).
m
m
VL L
IL
0 0
m
1
m 1
4/8/2008 0 t L 0 17
t
Phasor diagram: Two vectors representing
voltage and current are perpendicular each to
other.
(Comparison: For circuits, the current
C
leads the voltage, but for circuits, the voltage
L
leads the current).
Calculate the impedance for inductor:
The impedance of an inductor depends on L and frequency .
The impedance will be large at high frequencies.
The inductor can play a role as a filter which stops high frequencies
and passes low frequencies.
4/8/2008 18
1.3.4 Driven LCR circuit:
R
Using the obtained results, we can consider
circuits LCR with AC source.
C L
The current at any time is equal in all the circuit
element:
Phasor diagram for the circuit
is shown in the picture.
According to the phasor diagram,
we have
Using the definition of impedance:
4/8/2008 19
Factoring out imax
we can calculate the total impedance of the circuit:
Usually, the symbol Z is used for the total impedance of a circuit:
The phasor diagram gives also the formula for phasor angle:
4/8/2008 20
