electromagnetic oscillations and currents€¦ · march 13, 2014 chapter 30 3 lc circuits ! !ese...

March 13, 2014 Chapter 30 1

Electromagnetic Oscillations and Currents


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


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


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


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


LC Circuits


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



!  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


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


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


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)



!  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



!  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


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


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


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⎡⎣ ⎤⎦( )


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


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


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


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


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

electromagnetic oscillations and currents€¦ · march 13, 2014 chapter 30 3 lc circuits ! !ese...

Documents