Tài liệu Chapter XIII Electromagnetic Oscilation, Eletromagnetic Field and Wave doc

We say they are “in phase” .Phasor diagram: • Vectors represent the maximum voltage and the maximum current • Both vectors make an angle t with the positive x-axis • With time both vec

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Chapter XIII

Electromagnetic Oscilation, Eletromagnetic Field and Wave

§1 Oscillating circuits

§2 System of Maxwell’s equations

§3 Maxwell’s equations and electromagnetic waves

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 We have known the close connection between changing eletric fieldsand magnetic fields They can create each other and form a system ofelectromagnetic fields

 Electromagnetic fields can propagate in the space (vacuum or materialenvironment) We call them electromagnetic waves They play a veryimportant role in science and technology

In this chapter we will study how can describe electromagnetic fields,what are their properties (in comparison with mechanical waves)

 First we consider the oscillating circuits in which there exist oscillatingcurrents and voltages They are sources for electromagnetic fields

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§1 Oscillating circuits:

1.1 L-C circuits and electrical oscillations:

• Consider the RC and LC

series circuits shown:

• Suppose that the circuits are

formed at t=0 with the capacitor

charged to value Q.

There is a qualitative difference in the time development of thecurrents produced in these two cases Why??

L C

-1.1.1 Consider from point of view of energy (qualitatively):

• In the RC circuit, any current developed will cause energy to be

dissipated in the resistor

• In the LC circuit, there is NO mechanism for energy dissipation;

energy can be stored both in the capacitor and the inductor!

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Recall: Energy in the Electric and Magnetic Fields

2

1 2

2 magnetic

0

1 2

B u

U  C V

2 electric 0

1 2

… energy density

+++ +++

- - - - E

Energy is stored in the capacitor Energy is stored in the inductor

Energy is stored in the capacitorEnergy is stored in the inductor

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where and Q0 determined from initial conditions

• Differentiate above form for Q(t) and substitute into the differential

equation we can find 

L C

dI L C

Q V

VC  L     

) cos( 0

d x

1 2

At t = 0 the switch is transfered from

the position 1 to the position 2:

02

Q d L

The solution Q(t) has the form analogue

to SHM (simple hamonic motion):

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) sin( 0

dQ

)cos( 0

0

2 0 2



  02 0 cos( 0   )   1  Q0 cos( 0t   )   0

C

t Q

L  02  1  0

C L

0

0Q

Im   ),

sin( 0  

 I t

I m

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I

1 2

L C

R

1.2 LCR circuit and damped oscillation:

dt

dI L

RI C

Q V

dQ R

dt

Q d

L

When the swicth is transferred to 2:

The solution Q(t) has the form of a damped oscillation:

cos(

e Q

• The amplitude of oscillation is damping,

since energy is dissipated in the resistor

The role of resistance in oscillations of the

current in LCR circuit is analogue to friction

in mechanical oscillations !

• The resistance R increases → the amplitude

of oscillation decreases faster Oscillations

will occure as long as ’

0 is real, so thereexists a critical resistance

C

L R

If the circuit is called underdamped;

: critically damped; : overdamped

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1.3 LRC circuit with alternating current (AC) source:

 This is the case analogue to the mechanical

driven oscillations with a periodic force

Suppose that the emf of the source has

the following form:

= m sint

The equation for Q(t) must be modified

by adding AC emf in the right hand side:

First we consider the particular cases: the circuit R, or C,

or L only.

R m

The formulas for the voltage and current

across R are as follows

s i n

m R

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• Remark: Both the current and the voltage vary with time as sint

We say they are “in phase” Phasor diagram:

• Vectors represent the maximum

voltage and the maximum current

• Both vectors make an angle t

with the positive x-axis

• With time both vectors rotate

counter-clockwise

• The vertical component of each

vector represents the instantaneous

value of voltage or current

Impedance: The ratio of the maximum voltage to

the maximum currentFor a resistor

Impedance of a resistordon’t depend on frequency

 In this case the voltage on C and the current through C are not

in phase, we say that they are ”out of phase”

 The current has peaks at an earlier time than the voltage The

current leads the voltage by one-quarter cycle or 90

t C

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Phasor diagram:

The vectors which represent the

current and the voltage are

perpendicular each to other,

as shown in the picture

Impedance: We can calculate the impedance for capacitor

Note that the impedance of a capacitor depends on, beside C, also

the frequency The impedance will be large at low frequencies

The capacitor can play a role as a filter which stops low frequencies

and passes high frequencies.

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Phasor diagram: Two vectors representing

voltage and current are perpendicular each to

other

(Comparison: For C circuits, the current

leads the voltage, but for L circuits, the voltage

leads the current)

Calculate the impedance for inductor:

The impedance of an inductor depends on L and frequency 

The impedance will be large at high frequencies.

The inductor can play a role as a filter which stops high frequencies

and passes low frequencies.

Using the obtained results, we can consider

circuits LCR with AC source.

The current at any time is equal in all the circuit

element:

Phasor diagram for the circuit

is shown in the picture

According to the phasor diagram,

we have

Using the definition of impedance:

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Factoring out imax

we can calculate the total impedance of the circuit:

Usually, the symbol Z is used for the total impedance of a circuit:

The phasor diagram gives also the formula for phasor angle:

m m

) sin(   

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Resonance in a driven LCR circuit:

We have had before

It means that the current which passes through the circuit depends onthe frequency This dependence Is represented by the graphics shown

in the picture

The peak current occurs when

that is, when = 0 , where

This is called

resonance frequency

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The resonance phenomenon in LCR circuits has important applications.

One example is the tuning circuit in radio

By varying the capacitance we can

choose the appropriate resonance

frequency to receive the corresponding

radio signal

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1.3.5 Power in LRC Circuit:

 The power supplied by the emf in a series LRC circuit

depends on the frequency  Not surprisingly, the maximum

power is supplied at the resonant frequency 0. This occurs

when the current and drive voltage are in phase, i.e., = 0

• The instantaneous power (for some frequency, ) delivered at

time t is given by:

• The most useful quantity to consider here is not the instantaneous

power but rather the average power delivered in a cycle:

• To evaluate the average on the right, we expand the sin(t- )term:

cos(sin

sin)

sin(

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0 cos

2

1sin

sint cost

t

0 +1

• Taking the averages



 cos 2

1 )

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§2 System of Maxwell’s equations:

 In the previous chapters we have known the basis laws concerning toelectric and magnetic phenomena:

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2.1 Maxwell’s statement on displacement current and

Maxwel-Amper’s equation:

• Consider applying Ampere’s

Law to the current shown in the

diagram

• If the surface is chosen as

1, 2 or 4, the enclosed current

= I

• If the surface is chosen as

3, the enclosed current = 0!

(i.e., there is no current

between the plates of the

capacitor)

circuit

) (

0 I ID

d

    

Big Idea: In order to have

for surface 1 = for surface 3Maxwell proposed there was an extra “displacement current” in theregion between the plates, equal to the current in the wire

This idea leads to the modified Ampere’s law:

Bd 

Bd 

enclosed

I d

B

      0

Recall def’n of flux:

 But what is the “displacement current” and where does come from?!

 We know that current is a flow of moving charges

 Although there is no actual charge moving between the plates,

something is changing – the electric field between them → a

changing electric field is equivalent to a fictitious current

 We can calculate quantitively this current:

 The electric field E between the plates of the capacitor is

determined by the charge Q on the plate of area A:

E = Q/(A 0) → Q = E A 0

 Because there is current flowing through the wire, there must be a

change in the charge on the plates:

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Modified Ampere’s Law:

dt

d I

d

0 0

• Conduction current is a flow of moving charges

• Displacement current is a fictitious current which is proportional

to the rate of changing electric flux

Owing to the concept of displacement current we can explain why

an alternating current (AC) can pass through a capacitor (DC does

not pass): the displacement current connects induction currents

which come into one plate of capacitor and come out the other plate

d E

Gauss’s law for :

 B A d 0

Gauss’s law for B :

there are nomagnetic chargesAmper’s law including displacement current:

encl

E

dt

d I

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Remark on the symmetry between electric and magnetic fields in theMaxwell’s equations:

• In empty space (where there is no charge) the Gauss’s laws for

and have the same form

• In empty space (the charge Q=0 and the induction current I = 0

we can write Amper’s and Faraday’s laws in the symmetrical form:

B E

   E d A

dt

d l

d

E

(Amper’s law)(Faraday’s law)

The Maxwell’s equations can be considered as foundation of theory

of electromagnetism These equations predict the existence of

electromagnetic disturbances consisting of time-varying electric and

magnetic fields that can propagate from one region of space to another

electromagnetic waves

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§3 Maxwell’s equations and electromagnetic waves:

First we review of what we have known about “waves”:

) (

) , ( x t h1 x vt h2 x vt

has a general solution of the form:

where h1 represents a wave traveling in the +x direction and h2

represents a wave traveling in the -x direction

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) sin(

b e

s ˆ  ˆ  ˆ

s b

s ˆ  ˆ  ˆ

The direction of propagation sˆis given by the cross product

3.1 Plane Harmonic Wave:

An important specific case of electromagnetic waves is the

plane harmonic wave in which E and B have the following form:

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Some remarks:

• The electric field E has only x-component and the magnetic field B

has only y-component These components vary in t and z sinusoidally.

• The fields E and B must be perpendicular to each other and to the

direction of propagation

Later we will show that this specific solution is consistent with Maxwell’sequations

Why do we call this solution “plane wave”?

For any given value of z, the magnitude of the electric field is uniformeverywhere in the x-y plane with that z value In other words, the wave

front is plane.

The similar situation

is true for the

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3.2 Derivation of the electromagnetic wave equation:

Now we show that

• The components of electric and magnetic fields obey waveequations which are derived from the Maxwell’s equations

• Harmonic plane waves described before are really consistentwith the Maxwell’s equations

Assume we have a plane wave propagating in z (i.e., E, B arefunctions of z and t, do not depend on x or y):

) sin(

0 kz t E

Ex   

3.2.1 Derivation:

E B

0 0

Now use M Eqn:

Combine obtained results to eliminate B y :

The obtained equation has the form of wave equation !!

2

t

E z

2

1

t

E c

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Now use M Eqn:

We can also combine obtained results to eliminate E x :

It has the same form as the equation for E x !!

2

2 2

t

B z

2

1

t

B c

2

t

B z

) cos( 0

0 kz t dt k E kz t kE

0 kz t E

Ex   

A definite ratiobetween magnitudes !

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3.2.3 The meaning of the constant c:

• We have had before:

0 0 2

c

s

m A

N m

N C

c

/10

3)

/10

4)(

./10

85.8(

1

1

8 2

7 2

2 12

0 0

• In the wave equation, c is the speed of wave, that is the speed

that the wave front moves We say that electromagnetic waves

propagate with the speed c (in vacuum).

• This is also the speed of light in vacuum By the experiments of

Foucault (also Fizeau) in 1860, the light speed is c ≈2.98∙108 m/s

• Maxwell proposed that light were in nature likely to be electromagneticwaves

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 How are Maxwell’s eqns in matter different?

   ≡ 

    ≈ (for most materials)

3.3 Electromagnetic waves in matter:

 Eletromagnetic waves can propagate not only in vacuum, but in matter

We will extend our analysis to e-m waves in nonconducting materials

where n = “index of refraction” of the material:

Therefore, the speed of light in matter is

related to the speed of light in vacuum by:

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3.4 Energy and momentum in electromagnetic waves:

 Electromagnetic waves contain energy We already know

expressions for the energy density stored in static E and B fields

These also hold for electromagnetic waves :

2 0

2

2 0

u

u  E  B  

 Therefore, the total energy density in an e-m wave = u , where

 This is true for the spatially and time-varying electric field E

3.4.1 Energy flow and the Poynting vector:

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 Usually we are interested in the average energy density:

• Calculate the amount of energy transfered per unit time through unit

cross-sectional area, which is denoted by S :

Adz u

1



 cE EB dt

dU A

The vector is called the Poynting vector :

 The direction of is the direction of propagation of the wave

 The magnitude of is equal to the energy being transported

by the wave in unit time through unit cross-sectional area:

 We define also the concept of intensity of a wave is the

spatial-and time-average of S:

2

0 0

EB E S

c

E t

kz c

E c

E S

I

0

2 max 2

0

2 max 0

2

2

1 )

( sin

2 max 0

0

2

1 2

1

cE E



 



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3.4.2 Electromagnetic momentum flow and radiation pressure:

Recall the case of a nonrelativistic motion of

particle:

A particle with kinetic energy

absorbed by a surface will transfer to surface

an amount of momentum

For electromagnetic wave the formula is as follows

(from the theory of relatvity)

Forcearea

I

Using the following relations:

we can conclude that the radiation pressure on a surface is the momentumtransfer from electromagnetic wave to surface

Substituting the formula for I we have for the radiation pressure p r

2 max 0

2

1

E c

I

pr   

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Summary

The Maxwell’s equations accept the plane electromagnetic wave

solutions which hav the following properties:

• They travel in empty space with the constant speed c

• The electric field E is perpendicular to the magnetic field B , and

both are perpendicular to the direction of wave propagation

• The magnitudes of E, B obey the wave equation, and depend

sinusoidally on time and spacial distance

• The peak of E and the peak of B differ by the factor c:

• Electromagnetic waves transfer energy and momentum when

they travel in space:

- The transer of energy is characterized by the Poynting vector

-The transfer of momentum to a surface causes the radiationpressure where I is the wave intensity