7time-varing fields and maxwell equations

8/12/2019 7Time-Varing Fields and Maxwell Equations

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M1

7 Time-Varing Fields and Maxwells Equations

Faradays Law

Displacement Current

Maxwells Equatioins

t

= B

E

dsD

s t

0==

+=

=

B

D

DJH

B

E

v

t

t


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M2


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4/30M4

Static Electric Fields

energy used for moving an electric charge around aclosed loop is equal to zero

the electric flux density emerging from a point equals tothe volume charge density

0= E

v= D

Magnetic sources exists in pair (North and South pole)

magnetic field around a closed path equals to thecurrent inside

JH=

0= B


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Faradays Law

From Amperes Law, current can produce a magnetic field.

From Faradays Law, magnetic fields can produce an

electric current in a loop, but only if the magnetic fluxlinking the surface area of the loop change with time

B(t)

C =C S dtd SBlE

where S is a surface bounded by

the closed line C.

In differential form,t

=

BE

=C IdlH


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Fundamental postulate for time-varying EM fields


7/30M7

Example: Stationary Loop in a Time-varying Magnetic Field

The induced emf is

=

Semf d

t

V SB


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Demonstration: D6.1 Circular Loop in Time-varying Magnetic Field


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M9


Consider a capacitor connected to a voltage source

Applying Amperes Law,

if is chosen, RHS is equal to Ic

if is chosen, RHS is equal to 0

2C

1c

= encIdlH

2c


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M10

To resolve the conflict, Maxwell proposed the followingAmperes law in time-varying fields:

t

+=

DJH:formPoint

dsD

lH

+=

+=

scond

Dispcond

tI

IId

:DispI

currentConduction:cond

I

Displacement current



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M12

Displacement current density


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M13

In summary, Maxwells equation for time-varying fields:In differential form:

0=

=+=

=

B

D

DJH

BE

v

t

t

In integral form:

=

=

+=

=

S

S

C S

C S

d

Qd

dt

Id

dt

d

0SB

SD

SD

lH

SB

lE

Maxwells equations


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M14

Maxwells equations


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M15

Suppose the Electric field in a source free (i.e. v=0) region is givenby a wave travelling in the z-direction

xaE )sin( ztEo =

Find the value of the magnetic field present. What must be the

value of so that both fields satisfy Maxwells equations?

Solution: Substituting E into Faradays law,

( )( ) sin( )

sin( )

x z o

o

t

E t zx y z

E t zz

=

= + +

=

y x

y

BE

a a a a

a ya)cos( ztEo =

Example


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M16

Integrating with time gives:

cos( )o

E t z dt = yB a

yy aa Czt

Eo

+= )sin(

In time-varying fields, we can ignore the DC term, so that

yaB

H )sin( ztE

o

o

o

==

This shows that an associated time-varying H-field must co-

exist.

Example


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M17

To find the value of , lets substitute the above expression of H

into Amperes law:

xaHE

==

)sin( zt

E

zt o

oo

xa)cos(2

ztE

o

o

=

Integrating with time, we find E to be

xaE )sin(2

2

ztE

oo

o

=

Comparing with the given expression of E

8phase velocity / 1 / 3 10 /o o p o ou m s = = = =

Example


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M18

Potential functions


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M19

Potential functions


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M20

Solution of Wave Equations for Potentials


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M21

Solution of Wave Equations for Potentials


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M22

Wave Equations of E & H in Source-Free Region


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M23

Time Harmonic Electromagnetics


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M24

Time Harmonic Electromagnetics

Ti H i Fi ld


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Time-Harmonic Fields

In this chapter, both electric and magnetic fields areexpressed as functions of time and position.

In real applications, time signals can be expressed as

sum of sinusoidal waveforms. So it is convenient to usethe phasor notation to express fields in the frequencydomain.

tjezyxtzyx ),,(Re),,,( EE

{ }tj

tj

ezyxj

tezyx

ttzyx

),,(Re

),,(Re),,,(

E

EE

=

Ti H i Fi ld


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M2626


Therefore, the differential form of Maxwells equationsbecomes the differential form time-harmonic Maxwellsequations

00

====

+=

+=

=

=

BB

DD

DJHD

JH

BE

B

E

vv

jt

jt

Attenti

on:A

loto

fequa

tions

D,E,B,H,J,v are functions of x,y,z,t

D,E,B,H,J,v are functions of x,y,z

Ti H i Fi ld


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M2727


Using the constitutive relations and ,the equations becomes

00

/

==

==

+=+= ==

HB

ED

EJHDJHHEBE

vv

jjjj

EDHB ==

W ti i S F M di


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M2828

Wave equations in Source-Free Media

In a source-free media (i.e. charge density v=0),Maxwells equations become:

The equation can be written as

where and it is called complexpermittivity.

0

0)(

=

= =+=

=

H

EEJEH

HE

j

j

EH )( j+=

EH cj=

/''' jjc

Wave equations in Source Free Media


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M2929


We will derive a differential equation involving E or Halone. First take the curl of both sides of the 1st equation:

using the vector identity .

is an operator and called Laplacian

0

0

==

=

=

H

E

EH

HE

cj

j

EEE

HE

)(2 cjj

j

=

=

AAA

2

2

2

2

2

2

22

zyx +

+

Wave equations in Source Free Media


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M3030


where and

is called propagation constant

The equation is called the homogeneous wave equation forE.

Similarly, we can obtain, (Try yourself)

0

0

==

=

=

H

E

EH

HE

cj

j

0

0

)(

22

22

2

=

=+=

EE

EE

EEE

c

cjj

c22 =

022 = HH

7time-varing fields and maxwell equations

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