March 13, 2014 Chapter 30 1
Electromagnetic Oscillations and Currents
March 13, 2014 Chapter 30 2
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 3
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
! The energy stored in the magnetic field of an inductor with inductance L is given by
UE =
12
q2
C
UB =
12
Li2
March 13, 2014 Chapter 30 4
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 Physics for Scientists&Engineers 2 5
LC Circuit Time Evolution
March 13, 2014 Chapter 30 8
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 9
LC Circuits
March 13, 2014 Chapter 30 10
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
! We can re-write the electric and magnetic energies in terms
of the charge q and current i
U =UE +UB
U =UE +UB =
12
q2
C+ 1
2Li2
March 13, 2014 Chapter 30 11
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
! Remembering that i = dq/dt we can write
dUdt
= ddt
12
q2
C+ 1
2Li2⎛
⎝⎜⎞⎠⎟= q
Cdqdt
+ Li didt
= 0
didt
= ddt
dqdt
⎛⎝⎜
⎞⎠⎟ =
d2qdt 2
Math reminder
March 13, 2014 Chapter 30 12
Analysis of LC Oscillations ! We can then write
! Or, finally:
! 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
d2xdt 2 + k
mx = 0
dUdt
= qC
dqdt
+ L dqdt
⎛⎝⎜
⎞⎠⎟
d2qdt 2
⎛⎝⎜
⎞⎠⎟= dq
dtqC+ L d2q
dt 2⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟= 0
qC+ L d2q
dt 2⎛⎝⎜
⎞⎠⎟= 0 ⇒ d2q
dt 2 +q
LC= 0
March 13, 2014 Chapter 30 13
Analysis of LC Oscillations ! The solution to the differential equation describing
simple harmonic motion was
! Where ϕ is a phase constant and ω0 is the angular frequency
! By analogy we can get the charge as a function of time
! ϕ is a phase constant and ω0 is the angular frequency
x = xmax cos ω0t +φ( )
ω0 =
km
q = qmax cos ω0t −φ( ) Note the − sign( )
ω0 =
1LC
= 1LC
March 13, 2014 Chapter 30 14
Analysis of LC Oscillations ! The current is then given by
! Realizing that the magnitude of the maximum current in the circuit is given by imax = ω0qmax , we get
! Having expression for the charge as a function of time we can write an expression for the electric energy
i = dq
dt= d
dtqmax cos ω0t −φ( )( ) = −ω0qmax sin ω0t −φ( )
UE =
12
q2
C= 1
2qmax cos ω0t −φ( )( )2
C= qmax
2
2Ccos2 ω0t −φ( )
i = −imax sin(ω0t −φ)
March 13, 2014 Chapter 30 15
Analysis of LC Oscillations
! Having expression for the current as a function of time we can write and expression for the magnetic energy
! Remembering that
! We then see that
! The maximum possible magnetic energy in the circuit is exactly the same as the maximum possible electric energy
UB =
12
Li2 = L2
−imax sin(ω0t −φ)( )2 = L2
imax2 sin2(ω0t −φ)
imax =ω0qmax and ω0 =
1LC
L2
imax2 = L
2ω0
2qmax2 = qmax
2
2C
March 13, 2014 Chapter 30 16
Analysis of LC Oscillations
! The magnetic energy as a function of time is
! We can write an expression for the total energy in the circuit by adding the electric energy and the magnetic energy
! 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
UB =
qmax2
2Csin2(ω0t −φ)
U =UE +UB =
qmax2
2Ccos2 ω0t −φ( )+ qmax
2
2Csin2(ω0t −φ)
U = qmax
2
2Csin2(ω0t −φ)+ cos2 ω0t −φ( )( )
= qmax
2
2C
March 13, 2014 Chapter 30 17
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
ω0 =
1LC
== 13.50 ⋅10−3 H( ) 1.50 ⋅10−6 F( )
=1.38 ⋅104 Hz
March 13, 2014 Chapter 30 18
Characteristics of an LC Circuit
! The total energy in the circuit is
! The max charge on the capacitor is
! The initial energy stored in the electric field is the same as the total energy in the circuit
U = qmax
2
2C
qmax =CVemf = 1.50 ⋅10−6 F( ) 12.0 V( )
qmax =1.80 ⋅10−5 C
U = qmax
2
2C=
1.80 ⋅10−5 C( )2
2 ⋅1.50 ⋅10−6 F=1.08 ⋅10−4 J
March 13, 2014 Chapter 30 19
Characteristics of an LC Circuit
! The charge on the capacitor as a function of time is given by
q = qmax cos ω0t −φ( )
q =1.797 ⋅10-5 C
at t = 0, q = qmax ⇒ φ = 0
q = qmax cos ω0t( )
qmax =1.80 ⋅10−5 C
ω0 =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⎡⎣ ⎤⎦( )
March 13, 2014 Chapter 30 20
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 21
Damped Oscillations in an RLC Circuit ! The rate of energy loss is given by
! We can rewrite the change in energy of the circuit as a
function time as
! Remembering that i = dq/dt and di/dt = d 2q/dt2 we can write
dUdt
= −i2R
dUdt
= ddt
UE +UB( ) = qC
dqdt
+ Li didt
= −i2R
qC
dqdt
+ Li didt
+ i2R = qC
dqdt
+ L dqdt
d2qdt 2 +
dqdt
⎛⎝⎜
⎞⎠⎟
2
R = 0
March 13, 2014 Chapter 30 22
Damped Oscillations in an RLC Circuit ! We can then write the differential equation
! The solution of this differential equation is
(damped harmonic oscillation!), where
d2qdt 2 +
dqdt
RL+ 1
LCq = 0
q = qmaxe−
Rt2L cos ωt( )
ω = ω0
2 − R2L
⎛⎝⎜
⎞⎠⎟
2
ω0 =1LC
March 13, 2014 Chapter 30 23
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 24
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
! We can see that the energy stored in the capacitor decreases exponentially and oscillates in time
= qmax
2
2Ce−
RtL cos2 ωt( )= 1
2
qmaxe− Rt2L cos ωt( )⎛
⎝⎜⎞⎠⎟
2
CUE =
12q2
C
Physics for Scientists & Engineers 2 25
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
! Where ω is the angular frequency and Vmax is the amplitude or maximum value of the emf
max sinemfv V tω=
March 13, 2014
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