Electromagnetic Oscillations and
Currents
March 13, 2014
Chapter 30
1
LC Circuits
!  Previous chapters introduced three circuit element
•  Capacitors
•  Resistors
•  Inductors
!  We have examined simple single-loop circuits containing
resistors and capacitors (RC circuits) or resistors and
inductors (RL circuits)
!  Now we’ll consider simple single-loop circuits containing
inductors and capacitors: LC circuits
!  We’ll see that LC circuits have currents and voltages that
vary sinusoidally with time, rather than increasing or
decreasing exponentially with time, as in RC and RL circuits
March 13, 2014
Chapter 30
2
LC Circuits
!  These variations of voltage and current in LC circuits are
called electromagnetic oscillations
!  Consider a simple single-loop circuit consisting of an
inductor and a capacitor
!  The energy stored in the electric field of a capacitor with
capacitance
C is given by
2
UE =
1q
2C
!  The energy stored in the magnetic field of an inductor with
inductance L is given by
1 2
U B = Li
2
March 13, 2014
Chapter 30
3
LC Circuit
!  Let look at how these energies vary with time
!  We start with a capacitor that is initially fully charged and
then connect it to the circuit
!  The energy in the circuit resides solely in the electric field of
the capacitor
!  The current is zero
!  Now let’s follow the evolution with time of the current,
charge, magnetic energy, and electric energy in the circuit
March 13, 2014
Chapter 30
4
LC Circuit Time
Evolution
March 13, 2014
Physics for Scientists&Engineers 2
5
LC Circuits
!  The charge on the capacitor varies with time
•  Max positive to zero to max negative to zero back to max positive
!  The current in the inductor varies with time
•  Zero to max negative to zero to max positive back to zero
!  The energy in the inductor varies with the square of the
current and the energy in the capacitor varies with the
square of the charge
•  The energies vary between zero and a maximum value
March 13, 2014
Chapter 30
8
LC Circuits
March 13, 2014
Chapter 30
9
Analysis of LC Oscillations
!  Now that we have a good intuitive feel for LC oscillations,
let’s describe them quantitatively
!  We assume a single loop circuit containing a capacitor C
and an inductor L and that there is no resistance in the
circuit
!  We can write the energy in the circuit U as the sum of the
electric energy in the capacitor and the magnetic energy in
the inductor
U = U E +U B
!  We can re-write the electric and magnetic energies in terms
of the charge q and current i
1 q2 1 2
U = U E +U B =
+ Li
2C 2
March 13, 2014
Chapter 30
10
Analysis of LC Oscillations
!  Because we have assumed that there is no resistance, the
energy in the circuit will be constant
!  Thus the derivative of the energy in the circuit with respect
to time will be zero
!  We can then write
dU d ⎛ 1 q 2 1 2 ⎞ q dq
di
= ⎜
+ Li ⎟ =
+ Li = 0
dt dt ⎝ 2 C 2 ⎠ C dt
dt
!  Remembering that i = dq/dt we can write
di d ⎛ dq ⎞ d 2q
= ⎜ ⎟= 2
dt dt ⎝ dt ⎠ dt
March 13, 2014
Math reminder
Chapter 30
11
Analysis of LC Oscillations
!  We can then write
2
⎛ d 2q ⎞ ⎞
dU q dq
⎛ dq ⎞ ⎛ d q ⎞ dq ⎛ q
=
+ L⎜ ⎟ ⎜ 2 ⎟ = ⎜ + L⎜ 2 ⎟ ⎟ = 0
⎝ dt ⎠ ⎝ dt ⎠ dt ⎝ C
dt C dt
⎝ dt ⎠ ⎠
!  Or, finally:
⎛ d 2q ⎞
q
d 2q q
+ L⎜ 2 ⎟ = 0 ⇒
+
=0
2
C
dt
LC
⎝ dt ⎠
!  This differential equation has the same form as that of simple
harmonic motion describing the position x of an object with
mass m connected to a spring with spring constant k
d 2x k
+ x =0
2
dt
m
March 13, 2014
Chapter 30
12
Analysis of LC Oscillations
!  The solution to the differential equation describing
simple harmonic motion was
x = x max cos (ω 0t + φ )
!  Where ϕ is a phase constant and ω0 is the angular frequency
k
ω0 =
m
!  By analogy we can get the charge as a function of time
q = qmax cos (ω 0t − φ ) ( Note the − sign )
!  ϕ is a phase constant and ω0 is the angular frequency
ω0 =
March 13, 2014
1
=
LC
1
LC
Chapter 30
13
Analysis of LC Oscillations
!  The current is then given by
dq d
i=
= ( qmax cos (ω 0t − φ )) = −ω 0qmax sin (ω 0t − φ )
dt dt
!  Realizing that the magnitude of the maximum current in the
circuit is given by imax = ω0qmax , we get
i = −imax sin(ω 0t − φ )
!  Having expression for the charge as a function of time we can
write an expression for the electric energy
2
1 q 1 ( qmax cos (ω 0t − φ )) qmax
UE =
=
=
cos 2 (ω 0t − φ )
2C 2
C
2C
2
March 13, 2014
2
Chapter 30
14
Analysis of LC Oscillations
!  Having expression for the current as a function of time we
can write and expression for the magnetic energy
1 2 L
L 2
2
U B = Li = ( −imax sin(ω 0t − φ )) = imax sin 2 (ω 0t − φ )
2
2
2
!  Remembering that
imax = ω 0qmax and ω 0 =
1
LC
!  We then see that
2
L 2
L 2 2
qmax
imax = ω 0 qmax =
2
2
2C
!  The maximum possible magnetic energy in the circuit is
exactly the same as the maximum possible electric energy
March 13, 2014
Chapter 30
15
Analysis of LC Oscillations
!  The magnetic energy as a function of time is
2
qmax
UB =
sin 2 (ω 0t − φ )
2C
!  We can write an expression for the total energy in the circuit
by adding the electric energy and the magnetic energy
2
2
qmax
q
U = U E +U B =
cos 2 (ω 0t − φ ) + max sin 2 (ω 0t − φ )
2C
2C
2
2
qmax
q
U=
sin 2 (ω 0t − φ )+ cos 2 (ω 0t − φ )) = max
(
2C
2C
!  Thus the total energy in the circuit remains constant with
time and is proportional to the square of the original charge
put on the capacitor
March 13, 2014
Chapter 30
16
Characteristics of an LC Circuit
!  Consider a circuit containing a capacitor
C = 1.50 μF and an inductor L = 3.50 mH.
The capacitor is fully charged using a
12.0 V battery and then connected to
the circuit.
PROBLEMS
!  What is the angular frequency of the circuit?
!  What is the total energy in the circuit?
!  What is the charge on the capacitor after t = 0.25 ms?
SOLUTIONS
!  The angular frequency is
1
ω0 =
==
LC
March 13, 2014
(3.50⋅10
1
−3
H )(1.50⋅10−6 F )
Chapter 30
= 1.38⋅104 Hz
17
Characteristics of an LC Circuit
!  The total energy in the circuit is
2
qmax
U=
2C
!  The max charge on the capacitor is
qmax = CVemf = (1.50⋅10−6 F )(12.0 V )
qmax = 1.80⋅10−5 C
!  The initial energy stored in the electric field is the same as
the total energy in the circuit
1.80⋅10 C )
(
q
−4
U=
=
=
1.08⋅10
J
−6
2C
2⋅1.50⋅10 F
2
max
March 13, 2014
−5
2
Chapter 30
18
Characteristics of an LC Circuit
!  The charge on the capacitor as a
function of time is given by
q = qmax cos (ω 0t − φ )
at t = 0, q = qmax ⇒ φ = 0
q = qmax cos (ω 0t )
qmax = 1.80⋅10−5 C
ω 0 = 1.38⋅104 Hz
at t = 0.25 ms =2.5⋅10−4 s, we have
(
q = (1.80⋅10−5 C )cos ⎡⎣1.38⋅104 Hz ⎤⎦ ⎡⎣ 2.5⋅10−4 s ⎤⎦
)
q = 1.797⋅10-5 C
March 13, 2014
Chapter 30
19
Damped Oscillations in an RLC Circuit
!  Now let’s consider a single loop
circuit that has a capacitor C and
an inductance L with an added
resistance R
!  We observed that the energy of a
circuit with a capacitor and an inductor
remains constant and that the energy translated from
electric to magnetic and back gain with no losses
!  If there is a resistance in the circuit, the current flow in the
circuit will produce ohmic losses to heat
!  Thus the energy of the circuit will decrease because of these
losses
March 13, 2014
Chapter 30
20
Damped Oscillations in an RLC Circuit
!  The rate of energy loss is given by
dU
= −i 2 R
dt
!  We can rewrite the change in energy of the circuit as a
function time as
dU d
q dq
di
= (U E +U B ) =
+ Li = −i 2 R
dt dt
C dt
dt
!  Remembering that i = dq/dt and di/dt = d 2q/dt2 we can
write
2
q dq
di 2
q dq
dq d 2q ⎛ dq ⎞
+ Li + i R =
+L
+⎜ ⎟ R = 0
2
C dt
dt
C dt
dt dt ⎝ dt ⎠
March 13, 2014
Chapter 30
21
Damped Oscillations in an RLC Circuit
!  We can then write the differential equation
d 2q dq R 1
+
+
q=0
2
dt
dt L LC
!  The solution of this differential equation is
q = qmax e
−
Rt
2L
cos (ω t )
(damped harmonic oscillation!), where
⎛ R⎞
ω = ω −⎜ ⎟
⎝ 2L ⎠
2
0
March 13, 2014
2
ω0 =
1
LC
Chapter 30
22
Damped Oscillations in an RLC Circuit
!  Now consider a single loop circuit that contains a
capacitor, an inductor and a resistor
!  If we charge the capacitor then hook it up to the circuit, we
observe a charge in the circuit that varies sinusoidally with
time and while at the same time decreasing in amplitude
!  This behavior with time is illustrated below
March 13, 2014
Chapter 30
23
Damped Oscillations in an RLC Circuit
!  The charge varies sinusoidally with time but the amplitude
is damped out with time
!  After some time, no charge remains in the circuit
!  We can study the energy in the circuit as a function of time
by calculating the energy stored in the electric field of the
capacitor
⎛
qmax e
⎜
2
1⎝
1q
=
UE =
2C 2
Rt
−
2L
2
⎞
cos (ω t )⎟
Rt
2
−
⎠
qmax
=
e L cos 2 (ω t )
C
2C
!  We can see that the energy stored in the capacitor decreases
exponentially and oscillates in time
March 13, 2014
Chapter 30
24
Driven AC circuits
!  Now we consider a single loop circuit
containing a capacitor, an inductor,
a resistor, and a source of emf
!  This source of emf is capable producing
a time varying voltage as opposed the
sources of emf we have studied in previous chapters
!  We will assume that this source of emf provides a sinusoidal voltage as
a function of time given by
vemf = Vmax sin ωt
!  Where ω is the
angular frequency
and Vmax is the
amplitude or maximum
value of the emf
March 13, 2014
Physics for Scientists & Engineers 2
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