electromagnetic oscillations and currents€¦ · march 13, 2014 chapter 30 3 lc circuits ! !ese...
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![Page 1: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/1.jpg)
March 13, 2014 Chapter 30 1
Electromagnetic Oscillations and Currents
![Page 2: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/2.jpg)
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
![Page 3: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/3.jpg)
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
![Page 4: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/4.jpg)
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
![Page 5: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/5.jpg)
March 13, 2014 Physics for Scientists&Engineers 2 5
LC Circuit Time Evolution
![Page 6: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/6.jpg)
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
![Page 7: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/7.jpg)
March 13, 2014 Chapter 30 9
LC Circuits
![Page 8: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/8.jpg)
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
![Page 9: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/9.jpg)
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
![Page 10: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/10.jpg)
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
![Page 11: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/11.jpg)
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
![Page 12: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/12.jpg)
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)
![Page 13: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/13.jpg)
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
![Page 14: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/14.jpg)
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
![Page 15: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/15.jpg)
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
![Page 16: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/16.jpg)
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
![Page 17: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/17.jpg)
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⎡⎣ ⎤⎦( )
![Page 18: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/18.jpg)
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
![Page 19: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/19.jpg)
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
![Page 20: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/20.jpg)
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
![Page 21: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/21.jpg)
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
![Page 22: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/22.jpg)
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
![Page 23: Electromagnetic Oscillations and Currents€¦ · March 13, 2014 Chapter 30 3 LC Circuits ! !ese variations of voltage and current in LC circuits are called electromagnetic oscillations](https://reader035.vdocument.in/reader035/viewer/2022071301/609d13506e8eac7d73083fc2/html5/thumbnails/23.jpg)
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