Chapter 31Electromagnetic Oscillations and Alternating Current In this chapter we will cover the following topics: -Electromagnetic oscillations in an LC circuit -Alternating current (AC) circuits with capacitors -Resonance in RCL circuits -Power in AC-circuits -Transformers, AC power transmission (31 - 1)
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8/2/2019 Lecture 1410
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Chapter 31
Electromagnetic Oscillations and AlternatingCurrent
In this chapter we will cover the following topics:
-Electromagnetic oscillations in an LC circuit
-Alternating current (AC) circuits with capacitors
-Resonance in RCL circuits
-Power in AC-circuits-Transformers, AC power transmission
(31 - 1)
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Suppose this page is perpendicular to a uniform magnetic field
and the magnetic flux through it is 5Wb. If the page is turned by
30 around an edge the flux through it will be:
A. 2.5Wb
B. 4.3Wb
C. 5Wb
D. 5.8Wb
E. 10Wb
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A car travels northward at 75 km/h along a straight road in a
region where Earth’s magnetic field has a vertical component of
0.50 × 10−4 T. The emf induced between the left and right
side, separated by 1.7m, is:
A. 0
B. 1.8mV
C. 3.6mVD. 6.4mV
E. 13mV
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L
C The circuit shown in the figure consists of a capacitor
and an inductor . We give the capacitor an initial
chanrge and then abserve what happens. The capacitor will discharge th
C
L
Q
LC Oscillations
rough the inductor resulting in a time
dependent current .i
We will show that the charge on the capacitor plates as well as the current
1in the inductor oscillate with constant amplitude at an angular frequency
The total energy in the circuit is t
q i
LC
U
ω =
2 2
he sum of the energy stored in the electric field
of the capacitor and the magnetic field of the inductor. .2 2
The total energy of the circuit does not change with time. Thus
E B
q LiU U U
C dU
= + = +
2
2
2
2
0
0.1
0
dt
dU q dq di dq di d q Li i
dt C d
d q L
t dt dt q
dt dt dt C +
=
= + = = → → == (31 - 2)
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L
C
2
2
2
2
10 ( )
This is a homogeneous, second order, linear differential equation
which we have encountered previously. We used it to d
10
escribe
the simple harmonic oscillat
o
d q
L qdt C
d qq
dt LC
+ == →
+
eqs.1
22
20
with sol
r (SHO)
( )
ution: ( ) cos( )
d x x
dt
x t X t
ω
ω φ
+ =
= +
eqs.2
( )
If we compare eqs.1 with eqs.2 we find that the solution to the differential
equation that describes the LC-circuit (eqs.1) is:1
( ) cos where , and is the phase angle.
The current
q t Q t LC
ω φ ω φ = + =
( )sindq
i Q t dt
ω ω φ = = − +
( )( ) cosq t Q t ω φ = +
1 LC
ω =
(31 - 3)
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L
C
( )
( ) ( )
2 22
2 2 2 22 2
2
The energy stored in the electric field of the capacitor
cos2 2
The energy stored in the magnetic field of the inductor
sin sin2 2 2
The total energy
2
E
B
E B
q QU t
C C
Li L Q QU t t
C
U U U
QU
ω φ
ω ω φ ω φ
= = +
= = + = +
= +
= ( ) ( )2
2 2cos sin2
The total energy is constant;
Qt t C C
ω φ ω φ + + + =
energy is conserved
2
2
3The energy of the has a value of at 0, , , , ...2 2 2
3 5The energy of the has a value of at , , , ...
2 4 4 4
When is maximum is ze E B
Q T T t T C
Q T T T t
C
U U
=
=
electric field maximum
magnetic field maximum
Note : ro, and vice versa
(31 - 4)
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0t =
1
2
/ 8t T =
3
/ 4t T =
4
3 / 8t T =
5
5/ 2t T =
432
1
6
6
5 / 8t T =
3 / 4t T =
7 / 8t T =
7
8
7
8
(31 - 5)
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2
2
If we add a resistor in an RL cicuit (see figure) we must
modify the energy equation because now energy is
being dissipated on the resistor.
2 E B
dU i Rdt
qU U U
C
= −
= + = +
Damped oscillations in an RCL circuit
22
2
Li dU q dq di Li i R
dt C dt dt → = + = −
( )
2
2
2
/ 2
2
2 2
1 0 This is the same equation as that
of the damped harmonics o 0 which hscillator:
The a
as the solution
( ) co ngul r f s a
:
bt mm
dq di d q d q dqi L R qdt dt dt dt d
d x dxm b kx
dt dt
x t x e t
t C
ω φ −
+ + =
′= +
= → = → + + =
( )
2
2
2
2
/ 2 1( )
requency
For the damped RCL circuit the solut
cos
ion is:
The angular fre que
4
ncy
4
Rt L Rq
k bm m
t Qe t
LC L
ω
ω φ ω − ′ ′= + = −
′ = −
(31 - 6)
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/ 2 Rt LQe
−
/ 2 Rt LQe
−
( )q t Q
Q−
( )q t ( )/ 2( ) cos Rt Lq t Qe t ω φ
− ′= +
2
2
1
4
R
LC Lω ′ = −
/ 2
2
2
The equations above describe a harmonic oscillator with an exponetially decaying
amplitude . The angular frequency of the damped oscillator
1is always smaller than the angular f
4
Rt LQe
R
LC Lω
−
′ = −
2
2
1requency of the
1undamped oscillator. If the term we can use the approximation