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GENERAL PHYSICS II
Electromagnetism
&
Thermal Physics
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Chapter XIII
Electromagnetic Oscilation,
Eletromagnetic Field and Wave
§1. Oscillating circuits
§2. System of Maxwell’s equations
§3. Maxwell’s equations and electromagnetic waves
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 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
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§1. Oscillating circuits:
1.1 L-C circuits and electrical oscillations:
• Consider the RC and LC
series circuits shown:
• Suppose that the circuits are
formed at t=0 with the capacitor
charged to value Q.

LC:
current oscillates
I
0
0
t
I
Q
+++
- - -
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Recall: Energy in the Electric and Magnetic Fields
2
1
2
U LI
2
magnetic
0
1
2
B
u


… energy density ...
Energy stored in an inductor ….
B
Energy stored in a capacitor ...
2

-
-
0I
0
QQ 
L
C
0
II 
0Q

L
C
0
II 
0Q
Energy is stored in the capacitor Energy is stored in the inductor
Energy is stored in the capacitorEnergy is stored in the inductor
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where

and Q
0
determined from initial conditions
• Differentiate above form for Q(t) and substitute into the differential
equation we can find


L
C

the position 1 to the position 2:
0
2
2

C
Q
dt
Qd
L
The solution Q(t) has the form analogue
to SHM (simple hamonic motion):
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)sin(
000

 tQ
dt
dQ
)cos(
00
2
0
2
2

 tQ
dt
Qd


000

 tQ
dt
dQ
I
00
QI
m

),sin(
0

 tII
m
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I
1
2
L
C
R
1.2 LCR circuit and damped oscillation:
dt
dI
LRI
C
Q
VVV
LRC










2
2
4
1
L
R
LC
'
o

and
The frequency of oscilation
(In an LRC circuit, depends also on R)
Damping constant
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t
0
Q
0
Q
t

4
1
2
2

c
RR 
c
RR 
c
RR 
If
the circuit is called underdamped;
: critically damped;
: overdamped.
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1.3 LRC circuit with alternating current (AC) source:
 This is the case analogue to the mechanical
driven oscillations with a periodic force.
L
C

R

2
2
sin
m
d Q dQ Q
L R t


s i n
R R m
V R I t
 
 
1
0
0
t
I
R
R
m

R
m


1
0
0
V
R
t
The formulas for the voltage and current
across R are as follows
s i n
m
R

• 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
don’t depend on frequency.
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1.3.2

C Circuit:



C
I
C

 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.
t
C
Q
V
mC



m

m


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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.
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1.3.3

L Circuit:


In this case the voltage across L leads the current through L
by one-quarter cycle (90).
I
L





 t
L
m
t
0
0
1
V
L
t
0
0
1
L
m


L
m



I
L
m

m


Phasor diagram for the circuit
is shown in the picture.
According to the phasor diagram,
we have
Using the definition of impedance: