Field and Wave Electromagnetic - Seoul National...

Field and Wave Electromagnetic

Chapter7Chapter7

The time varying fields and Maxwell’s equationThe time varying fields and Maxwell’s equation

Introduction (1)Introduction (1)Time static fields

0, ,E D D Eρ ε∇× = ∇ = =i

1) Electrostatic

2) Magnetostatic

10, ,B H J H B

E D B H

μ∇ = ∇× = =i

not

2) Magnetostatic

) and are not related to and for time ste atic casesE D B Hnot ) and are not related to and for time ste atic cases Example)

( )E J E

E

σ⇒ =

⇒

A static field in a conducting medium steady current.

give rises to a static magnetic field:Ampere's law. But field can be completely determined from the static electric charge or potential distributions

⇒ magnetic field is a consequence

Electromagnetic Theory 2 2

Introduction (2)Introduction (2)Time varying fieldsTime varying fields

and are properly related to and

1) modify equation fundamental postulate leading to Faraday's law

E D B H

E∇× →

2) then modify the equation to be consisteH∇×

) 0

nt with the equation of continuity

for static. but for time varyingcf J Jtρ∂

∇ = ∇ = −∂

i i

0 3) and never changes.D Bρ∇ = ∇ =i i


Faraday’s LawFaraday s LawMichael Faraday 1831, experimental law postulate⇒ ⇒Michael Faraday 1831, experimental law postulate

Definition : the quantitative relationship between the induced emf and the rate of change of flux linkage

⇒ ⇒

Fundamental postulate for Electromagnetic Induction

Non-conservative field cannot be expressed

as the gradient of a scalar potential

BEt

∂∇× = − ⇒

∂ as the gradient of a scalar potential

C S

BE dl dSt

∂= −

∂∫ ∫i i


A Stationary Circuit in a Time Varying Magnetic Field (1)A Stationary Circuit in a Time Varying Magnetic Field (1)

d d∂ ∂∫ ∫ ( , 0) since stationary C S

d dE dl B ds dsdt t dt t

∂ ∂= − → =

∂ ∂∫ ∫i i ∵

11

2

Right hand rule (counter clock wise)2

1, , 0d demf v E dl

dt dtΦ Φ

= = − >∫ i Assume

1

0dt dt

v⇒ < ⇒

∫

driving current to flow in the direction of clock wise potential difference of gap between terminal 1 and2


12 2 120V E dl V V= − < >∫ i ∵ assume


: emf induced in circuit with contour CDefine v E dl∫ : emf induced in circuit with contour CCDefine v E dl= ∫ i

1

2 ~2

1

l t ti f d i i t i th di ti f i ht h d l

2 1( 0)

v

E dl E dl E E dl V V V= = = = =∫ ∫ ∫i i ∵ i

: electromagnetive force driving current in the direction of right hand rule Meaning of contour integral

inside contour

Field between the terminal in the gap

1 2 121 2( 0)right

handE dl E dl E E dl V V V= = = − = − =

⇒

∫ ∫ ∫i i ∵ i inside contour ca 12v V=n be replaced with voltage source. But polarity of depends on

the change of the flux linkage


the change of the flux linkage


B∂120 . 0

[ ]

B v i e Vt

B ds S Wb

∂> <

∂

Φ = ∫ i

e.g) then (current is in the direction of left hand rule)

Define : magnetic flux crossing surface [ ]SB ds S Wb

dvdt

Φ =

Φ= − ⇒

∫ iDefine : magnetic flux crossing surface

then This is valid even in the absense of a physical closed circuit

note The emf induced in a stationary loop caused by a time-varying magnetic field is a transformer emf


Ex 7 1) A Circular Loop of N Turns of Conducting WireEx 7-1) A Circular Loop of N Turns of Conducting Wire

( ) irπA i l l f N t 0

2

cos( )sin2

i ) ( 2 ) ) 2

rB zB wtb

rB d B t d f dπ

π

π φ

=

Φ ∫ ∫ ∫b

A circular loop of N turns,

Find the emf induced in the loop

sol) each turn ( 0 02

0

cos sin ) ( 2 ) ) 22

8 1) sin

SB ds zB wt z rdr cf d

bb B wt

π φ π

ππ

Φ = =

= −

∫ ∫ ∫i i0sol) each turn = (

(2

z

2

08 1) cos

Nd Nbv N B wtdt

ππ

∴ ⇒ Φ

Φ∴ = − = − −

N-turns

( [V]2

yb


x

Transformers (1)Transformers (1)

mmf

j j k kj k

N I = ℜ Φ∑ ∑

mmf

1 2 1 2, , , number of turns and the currents : the reluctance of the magnetic circuit

N N i i ⇒ℜ

N N ℜΦ1 1 2 21 1 2 2 :

(where : mmf in the positive direction, mmf in the negative direction)

N i N iN i N i

l

∴ − = ℜΦ

ℜ =

1 1 2 2

SlN i N iS

μ

μ

ℜ =

∴ − = Φ


μ

Transformers (2)Transformers (2))a Ideal transformer

1 21 1 2 2

2 1

)

)

ai NN i N ii N

cf

μ →∞ = ⇒ =

Ideal transformer

,

Faraday's law

1 1

)cfdv Ndtd

Φ= Φ

Φ

Faraday s law

( No negative sign, careful of sign of flux )

v N2 2

dv NdtΦ

= (But flux is in the reverse direction) 1 12 2

v Nv N

N

∴ =

⎛ ⎞

effective load seen by the source connected to primary winding

122

21 11

1 222

( )eff L

N vNv NR R

i NN i

⎛ ⎞⎜ ⎟ ⎛ ⎞⎝ ⎠= = = ⎜ ⎟⎛ ⎞ ⎝ ⎠⎜ ⎟

21

2

11

2

( )eff L

N

NZ ZN

⎜ ⎟⎝ ⎠

⎛ ⎞∴ = ⎜ ⎟

⎝ ⎠Impedance transformation


2⎝ ⎠

Transformers (3)Transformers (3)) Real transformerb

1 1 2 2

) Real transformer

blN i N is

s sμμ μ

− = Φ

2 21 1 1 1 2 2 2 2 1 2 1 2 2

1 2 1 21 1 12 2 12 2

( ), ( )

,

1

s sN N i N N i N N N i N il l

di di di div L L v L L

dt dt dt dt

μ μ⇒ Λ = Φ = − Λ = Φ = −

= − = −∵

2 21 1 2 2 12 1 2, , ) (where

For an ideal tran

dt dt dt dts s sL N L N L N N

l l lμ μ μ

= = =

sformer No leakage flux L L L⇒ ∴ For an ideal tran 12 1 2

12 1 2 , 1 :

sformer No leakage flux

For a real transformer ( coefficient of coupling)

L L L

L k L L k k

⇒ ∴ =

∴ = <


Transformers (4)Transformers (4)

Equivalent ci trcuiEquivalent ci trcui

1 2, ::winding resistanceleakage inductive reactance

R RX X1 2, :

::

leakage inductive reactancepower loss due to hysteresis and eddy currentnonlinear inductive reactance due to the nonlinear magnetization behaviorof the ferro

c

c

X XRX

magnetic core


of the ferromagnetic core

A Moving Conductor in a Static Magnetic FieldA Moving Conductor in a Static Magnetic Field

F qu B= ×mF qu B

F F

×→→

→

Charge Seperation Coulomb force of an attraction

and will balance each other to be in equilibrium.m eF F

F F

→

∴

∫

and will balance each other to be in equilibrium.

Magnetic force per unit charge

2

21 1

, ,

( )

m mF Fu B V E dl Eq q

V u B dl

= × = − ⋅ = −

∴ = × ⋅

∫

∫

The emf g

' ( )C

V u B dl= × ⋅ →∫enerated around the closed loop is

flux cutting emf


Ex 6 5) A Metal Bar Sliding Over Conducting RailsEx 6-5) A Metal Bar Sliding Over Conducting Rails

ˆB zB u= constant0

0 1 2

,

( )C

B zB u

V V V u B dl

=

= − = × ⋅∫

constant

a) 1'

02'

0

ˆ ˆˆ( ) ( )xu zB ydl

uB h

= × ⋅

= −∫

220 0( ), l

V uB hI P I RR R

= = = b)

1'

0ˆmF I dl B xIB h= × = −∫ c) mechanical power

02'm

I∫

( : n2 2 2

0

)

m m

dlu B hP F u F u

R∴ = ⋅ = − ⋅ =

egative direction to


m m R

A Moving Circuit in a Time Varying Magnetic Field (1)A Moving Circuit in a Time Varying Magnetic Field (1)

( )F q E u B= + ×( )

','

mF q E u B

EE E u B

+ ×

= + ×

To an observer moving with C,

the force on q can be interpreted as caused by an electric field

' ( )C S C

BE dl ds u B dlt

∂⋅ = − + × ⋅ →

∂∫ ∫ ∫ General form of Faraday law

the emf induced

B

motional emf in the moving due to the motion

frame of reference of the circuit in transformer emf due to the time

i i variation


A Moving Circuit in a Time Varying Magnetic Field (2)A Moving Circuit in a Time Varying Magnetic Field (2)

The time rate of chage of magnetic flux,

2 12 10

1lim ( ) ( )

S

S St

d d B dsdt dt

B t t ds B t dstΔ →

Φ= ⋅

⎡ ⎤+ Δ ⋅ − ⋅⎢ ⎥⎣ ⎦Δ

∫

∫ ∫

=

( )

2 1

( )( ) . . .

tB tB t t B t t H OT

td B ds

⎣ ⎦Δ∂

+ Δ = + Δ +∂

∴ ⋅ =

cf) Taylor's series

2 11lim . . .B ds B ds B ds H OT∂ ⎡ ⎤⋅ + ⋅ − ⋅ +⎢ ⎥⎣ ⎦∫ ∫ ∫ ∫B dsdt∴ 2 12 10

3

lim . . .S S S St

ds B ds B ds H OTt t

dS dl u t

Δ →+ +⎢ ⎥⎣ ⎦∂ Δ

Δ

= × Δ

∫ ∫ ∫ ∫3assuming side surface S as the area swept out by the conductor in time t

f om di e gence theo em

2 12 1S S

B dv B ds B ds B ds∇⋅ = ⋅ − ⋅ + ⋅∫ ∫ ∫V

from divergence theorem

3

3

2 1 ( )

S

B ds B ds t u B dl∴ ⋅ − ⋅ = −Δ × ⋅

∫

∫ ∫ ∫2 1

2 1 ( )

( )

' '

S S C

S S C

B ds B ds t u B dl

d BB ds ds u B dldt t

d dV E dl B d

∴ Δ ×

∂∴ ⋅ = ⋅ − × ⋅

∂Φ

∫ ∫ ∫

∫ ∫ ∫

∫ ∫


' 'C S

d dV E dl B dsdt dt

Φ∴ = = − ⋅ = −∫ ∫i

Maxwell’s Equation (1)Maxwell s Equation (1)

static Time varying

0E∇× = BE ∂∇× = −∂

, D D Eρ ε∇ = =it

D ρ∂

∇ =i

0,

H J

B B Hμ

∇× =

∇ = =i0

DH Jt

B

∂∇× = +

∂∇ 0B∇ =i


Maxwell’s Equation (2)Maxwell s Equation (2)

①Note Continuity equation

0J

J ρ∇ =

∂∇

i

①Note Continuity equation

: for steady state current

: time varying currentJtρ

ρ

∇ = −∂

∂

i

②

: time varying current

Vector identity

( ) 0H J Jtρ∂

∇ ∇× = = ∇ ⇒ ∇ = −∂

∴

i i i contradiction

( ) 0 DH J J Dρ ρ⎛ ⎞∂ ∂

∇ ∇× = = ∇ + = ∇ + = ∇⎜ ⎟i i i iwhere

Displacement current density. [A/m2]Time varying electric field and induced magnetic field→coupling

∴ ( ) 0H J J Dt t

DH J

ρ∇ ∇× = = ∇ + = ∇ + = ∇⎜ ⎟∂ ∂⎝ ⎠

∂∇× = +

∂

i i i i, where

Time varying electric field and induced magnetic field→coupling

( )t

F q E u B∂

= + ×Cf) Lorentz force equation,


Integral Form of Maxwell’s EquationIntegral Form of Maxwell s Equation→Cf) Differential form Point function

BE S Ct

→

∂∇× = − ⇒

∂

Cf) Differential form Point function

Apply stokes's theorem over open surface with contour

( )S S

BE ds dst

∂∇× = −

∂∫ ∫i i

c s

B dE dl dst dt

∂ Φ= − = −

∂∫ ∫i i① : Faraday's lawD∂H②

c s

s

Ddl I dst

D D ds Qρ

∂= +

∂∇ = ⇒ =

∫ ∫∫

i i

i i③

: Ampere's circuital law

: Gauss law

0 0s

sB B ds∇ = ⇒ =

∫∫i i④ : No isolated magnetic charge


Ex 7 5Ex. 7-5(a) Displacement current = conduction current

dv

→(a) Displacement current conduction current 1 conduction current current on the wire. Apply circuit theorem

1 1 0 coscCdvi C C V tdt

ω ω= =

2 Displacement cur

D∂

rent. Reminding

1

DH Jt

ACd

μ ε

∂∇× = +

∂

=

Assuming the area A, plate separation d, permitivity , then

E Assume is uniform in the dielectric (ignori

0 icv VE D E t

ng fringing effects) then

0

0 1 0

, sin

cos cos

c

D CA

E D E td d

D Ai ds V t C V t it d

ε ε ω

ε ω ω ω ω

= = =

∂= = = =

∂∫ i


Ex 7 5Ex. 7-5

(b) Magnetic field intensity reminding Ampere's law

,C S

D DH J H dl I dst t

∂ ∂∇× = + = +

∂ ∂∫ ∫i i

(b) Magnetic field intensity reminding Ampere s law

0 2D H dl rπ= =∫ i

①

②

①

surface S1 with ring C surface S2 with ring C

H0, 2C

D H dl rπ= =∫ i①

1 0 cosC

H

H

I J ds i C V t

φ

φ

ω ω

⇒

= = =∫ i ( Symmetry around the wire along the contour C) constant

1 0

1 0

cos

, cos

CS

d

J ds i C V t

C VI i H tφ

ω ω

ω ω= ∴ =

∫② no conduction current, but displacement current

, cos2d

I i H trφω ω

π∴


Potential Functions (1)Potential Functions (1)

0 ( ) 0

A

B A B

B A

= ∇×

∇ ∇ ∇×i i

Vector magnetic potential,

(Solenoidal nature of )

Vector identity 0, ( ) 0

( )

B A

B AE E A E

∇ = ∇ ∇× =

⎛ ⎞∂ ∂ ∂∇× ∴∇× ∇× ⇒∇× +⎜

i i Vector identity Recall Faraday's law▷

0⎟

Curl free

( )E E A Et t t

∇× = − ∴∇× = − ∇× ⇒∇× +⎜∂ ∂ ∂⎝ ⎠

0

( ) 0V

=⎟

∇× ∇ =Vector identity

d d f l

▷

[ / ]

E V

A AE V E V V mt t

= −∇

∂ ∂+ = −∇ = −∇ −∂ ∂

and reminding for electromagnetics

for time varying i.e) t t∂ ∂



A∂ 0A E Vt

E ρ

∂→ = ∴ = −∇

∂Cf) Static

Time varying is induced by charge distribution and time varying

J

B A E B

→ magnetic field time varying current,

l d d l d▷ ,

1

B A E B

V

∴

=

also depends on are coupled

▷

▷ 0' '

0

', '4 4v v

Jdv A dvR R

μρπε π

=∫ ∫ : From the static condition0

2 20V A J

ρ μ∇ = − ∇ = −

These are solution of poisson equation

and 00ε

The time-retardation effects associated with the finite velocity of ▷ propagation is neglected



JR

ρ

Quasi-static fields

- and vary slowly with time- the range of interest is small compared to the wavelengthR

R - the range of interest is small compared to the wavelength cf) Frequency is high, and is large compared to wavelength : time-retardation effect must be included.

, , ( , )A DB A E V H J B H D Et t

μ ε∂ ∂= ∇× = −∇ − ∇× = + = =∂ ∂

From the equations

t t

AA J Vt t

μ με

∂ ∂

⎛ ⎞∂ ∂∇×∇× = + −∇ −⎜ ⎟

∂ ∂⎝ ⎠

⎝ ⎠

Recalling vec2( )A A A∇×∇× = ∇ ∇ −∇i

tor identity


Potential Functions (4)Potential Functions (4)2V A∂ ∂⎛ ⎞2

2

22

( ) V AA A Jt t

A VA J A

μ με με

με μ με

∂ ∂⎛ ⎞∴∇ ∇ −∇ = −∇ −⎜ ⎟∂ ∂⎝ ⎠

∂ ∂⎛ ⎞∇ +∇ ∇ +⎜ ⎟

i

2A J At t

A B A

με μ με∇ − = − +∇ ∇ +⎜ ⎟∂ ∂⎝ ⎠∇× = ∇

i

i

- we only designated but we are free to choose

A - vector will b A A

V V

∇× ∇

∂ ∂

ie specified by giving and

0 , 0 0V VA At t

με ∂ ∂∇ + = = ∴∇ =∂ ∂

i i - let for static

Lorentz gauge for potentialsg g p



0, 0

0

V At t

A

∂ ∂= =

∂ ∂∴∇ =i

cf) For static,

20A Jμ∇ = −

then vector poisson equation

2 A∂

- Then nonhomogeneous wave equation for vector potential becomes

22

AAt

με ∂∇ − = −∂

:Jμ Vector potential wave equation

Potential Functions(6)Potential Functions(6)

,A AE V D Vt t

ρ ε ρ⎛ ⎞∂ ∂

= −∇ − ∇ = ⇒ −∇ ∇ + =⎜ ⎟∂ ∂⎝ ⎠

i i

Scalar potential wave equation

∨

2 ( ) ,

t t

VV A At t

ρ μεε

∂ ∂⎝ ⎠∂ ∂

∴∇ + ∇ = − ∇ = −∂ ∂

i i

22

2

VVt

ρμεε

∂∴ ∇ − = −

∂


Boundary Condition (1)Boundary Condition (1)Electric field's boundary condition⊙

c s s v

BE dl ds D ds dvt

ρ∂= − =∂∫ ∫ ∫ ∫i i i

Electric field s boundary condition

... ...

⊙

0 0 0B ds h S∂ → Δ → →∫ i

From equation

when since area0 0, 0s

ds h St

→ Δ → →∂∫ when since area

1 2 1 2 0t t t tE E E w E w∴ = Δ − Δ = ( )


Boundary Condition (2)Boundary Condition (2)

From equation

1 2 2 1 2 1 2( ) ( )

( )

ssD ds D n D n S n D D S S

n D D D D

ρ

ρ ρ

= + Δ = − Δ = Δ

∴ − = − =

∫ i i i i

i

From equation

2 1 2 2 1( ) ,s n n sn D D D D

H

ρ ρ∴ = =i

Magnetic field's boundary conditions

Ddl J ds

⎛ ⎞∂= +⎜ ⎟∫ ∫i i

1 2 1 2

2 1 2

( )

. ) ( )

c s

sn t t sn

dl J dst

H w H w J w H H J

i e n H H J

⎜ ⎟∂⎝ ⎠

∴ Δ + −Δ = Δ − =

× − =

∫ ∫i i ,

2 1 2

2

. ) ( ) s

s

i e n H H J

n J

×

cf) & are perpendicular to each other


Boundary Condition (3)Boundary Condition (3)

t )

H

note)

The tangential component of the field is discontinuous across an interface where a free surface current exists

if both media have finite conductivity, currents are defined by volumecurrent density

1 2t tH H→→ =

current density surface currents do not exists

i e) discontinuous only for interface with an ideal perfect conductor or super i.e) discontinuous only for interface with an ideal perfect conductor or super conductor.

1 20 n nB B B∇ = ∴ =i


Interface Between Two Lossless Linear MediaInterface Between Two Lossless Linear Media

ε μ→Linear media permitivity : permeability :ε μσ→

→Linear media permitivity : , permeability : Lossless =0

SJρ→→SAssume, at interface, no free charge =0

no surface currents =0

1 111 2 1 2,t tt t t t

D BE E H Hε μ= ⇒ = = ⇒ = 11 2 1 22 2 2

,t t t tt t

E E H HD Bε

⇒ ⇒ 2

1 2 1 1 2 2 1 2 1 1 2 2,n n n n n n n nD D E E B B H Hμ

ε ε μ μ= ⇒ = = ⇒ =


Interface between a Dielectric and Perfect Conductor (1)Interface between a Dielectric and Perfect Conductor (1)

→Good conductor perfect conductor

( ) ( )

E

E D B H

→

⇒

Good conductor perfect conductor

Interior of perfect conductor (surface charge only) :

are zero in the interior of a conductor( , ) ( , )

,

E D B H

E D

⇒ are zero in the interior of a conductor

cf) In static case, may be zero, but ,H B may not be zero.

2 22 20, 0, 0, 0E H D B= = = =


Interface between a Dielectric and Perfect Conductor (2)Interface between a Dielectric and Perfect Conductor (2)

0 0E E1 2

2 1 2 2

1 1

0, 0

( ) , 0, 0 0

t t

s t

t sn sn t

E E

n H H J HH J J H

= =

× − = == = → =

if

2 1 2 2 1

1 2

( ) , 0,0, 0

s n n s

n n

n D D D DB B

n

ρ ρ− = = == =i

note) : outward normal from medium22n note)

E

: outward normal from medium2

At an interface between a dielectric and a perfect conductor

i l t d i t f (i t ) th d t f1

1 11

sn

E

E E ρε

= =

: is normal to and points away from(into) the conductor surface

1H:

1 1

) ( )

t sH H J

f H H J

= =

is tangential to the interface with a magnitude of

di ection


2 1 2) ( ) scf n H H J× − = direction

Wave Equation and Their Solutions (1)Wave Equation and Their Solutions (1)2 A∂2

2

22

2

AA JtVVt

με μ

ρμεε

∂∇ − = −

∂∂

∇ − = −∂

Wave equation :

2

( )

t

t t

ε

ρ ν

∂

′ΔiSolution : Assume an elemental point charge at time , located at the

i i f th di t origin of the coordinates.

V R tii

Spherical coordinates. depends only on . and because of spherical symmetry.

21 V

φ

∂ ∂⎛ ⎞

i (No dependence on ) Except at the origin,

2V∂22

1 VRR R R

με∂ ∂⎛ ⎞ −⎜ ⎟∂ ∂⎝ ⎠ 2 0

Vt

∂=

∂


Wave Equation and Their Solutions (2)Wave Equation and Their Solutions (2)New variable

( )2 2

1( , ) ( , )

1 1 1

V R t U R tR

U U

=

∂ ∂ ∂⎛ ⎞ ⎡ ⎤

New variable

( )2 2 22

2 2 2

1 1 1,

1 1

U UR U R t R U U RR R R R R R

U U U UU R RR R R R R R R

∂ ∂ ∂⎛ ⎞ ⎡ ⎤= − + = − +⎜ ⎟ ⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎣ ⎦⎡ ⎤∂ ∂ ∂ ∂ ∂⎡ ⎤− + = − + +⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

1

R R R R R R R∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

∴ 2 2 2 2

2 2 2 2

1 ( , ) ( , )0, 0U U U R t U R tR R R t R tt R t R

με με

με με

∂ ∂ ∂ ∂− = − =

∂ ∂ ∂ ∂+

i.e)

Any function of ( ) or of ( ) will satisfy the differential

( )

t R t R

f t R

με με

με

− +

−

Any function of ( ) or of ( ) will satisfy the differential

equation

is a wave equation which travels away from the origin


Wave Equation and Their Solutions (3)Wave Equation and Their Solutions (3)

( )f R i ti hi h t l t th i i h i l( )

( , ) ( )

f t R

U R t f t R

με

με

+ →

∴ = −

is a wave equation which travels to the origin physical nonsense

( , ) ( )

( , ) ( )

fR R t t

U R R t t f t t R R

μ

με

+ Δ + Δ

⎡ ⎤+ Δ + Δ = + Δ − + Δ⎣ ⎦

the function at at a later .

( )f t R με= −

0

.1 1lim

t

if t RR R ut t

με

με μεΔ →

Δ = Δ

Δ Δ= ⇒ = =

Δ Δ

: velocity of propagation0

1( , )

tt t

RV R t f tR u

με μεΔ →Δ Δ

⎛ ⎞= −⎜ ⎟⎝ ⎠

Rf tu

⎛ ⎞−⎜ ⎟⎝ ⎠

Determine


Wave Equation and Their Solutions (4)Wave Equation and Their Solutions (4)( )tρ νΔRecall potential function induced by a static point charge at( )

'( ) '

t

Rt vt v R u

ρ ν

ρρ

Δ

⎛ ⎞− Δ⎜ ⎟Δ ⎛ ⎞ ⎝ ⎠

Recall potential function induced by a static point charge at the origin

( )( ) ,4 4

1

t v R uV R f tR u

Rtu

ρπε πε

ρ

Δ ⎛ ⎞ ⎝ ⎠Δ = Δ − =⎜ ⎟⎝ ⎠

⎛ ⎞−⎜ ⎟⎝ ⎠∫

'

1( , ) '4 v

uV R t dvRπε

⎝ ⎠∴ = ∫ :R t

R⎛ ⎞

Retarded scalar potential

Scalar potential at a distance from the source at time

Rtu

⎛ ⎞→ −⎜ ⎟⎝ ⎠

Depends on the value of charge distribution at an earlier time

Retarded vector potential

R⎛ ⎞

'

( , ) '4 v

RJ tuA R t dv

Rμπ

⎛ ⎞−⎜ ⎟⎝ ⎠= ∫


Source Free Wave EquationSource Free Wave Equation

If the wave is in a simple (linear isotropic and homogeneous)

,H EE Ht t

ε μ σ

μ ε∂ ∂∇× = − ∇× =∂ ∂

If the wave is in a simple (linear, isotropic and homogeneous) non conducting medium. i.e) , ( =0)

0 ,t t

E H∂ ∂

∇ = ∇i i

2

0= From vector identity

2

2( )

) ( ) ( ) ( ) ( )

EE Ht t

cf A B C B A C C A B B A C A B C

μ με∂ ∂∇×∇× = − ∇× = −∂ ∂

× × = − = −i i i i

2

22

( ) ( )E E E

EE με

∇ ∇ − ∇ ∇ = −∇

∂∴∇ −

i i

2 210 if

t uμε= =

∂

2 22 2

2 2 2 2

1 10, 0E HE Hu t u t

∂ ∂∇ − = ∇ − =

∂ ∂ : Homogeneous vector wave equation


Time Harmonic FieldsTime Harmonic FieldsMaxwell's equationsMaxwell s equations - linear differential equations - sinusoidal time variation of source functions at given frequency

,E H - are sinusoidal with the same frequency

Ti h i d i id lTime harm →

→

onic steady state sinusoidalPhasors : Amplitude and phase information

independent of timej te ω

→ independent of time

cf) : time dependent factor


Time Harmonic Electromagnetics (1)Time Harmonic Electromagnetics (1)Vector phasors of field vectors : depend on space coordinates

( , , ; ) Re ( , , ) ,

( ) :

j tE x y z t E x y z e

E x y z

ω⎡ ⎤= ⎣ ⎦

Vector phasors of field vectors : depend on space coordinates

where vector phasor : complex quantity( , , ) :

( , , ; ) Re

E x y z

E x y z t j Et

ω∂ =∂

where vector phasor : complex quantity

( , , ) j tx y z e ω⎡ ⎤⎣ ⎦

( , , ) :

( , , )( , , ; ) Re j t

j E x y z

E x y zE x y z t dt ej

ω

ω

ω⎡ ⎤

= ⎢ ⎥⎣ ⎦

∫

where vector phasor

( , , )

j

E x y zj

ω

ω

⎣ ⎦

where : vector phasor

( )2

22

1, ,j jt t j

ω ωω

∂ ∂⇒ ⇒ ⇒

∂ ∂ ∫ i.e)


Time Harmonic Electromagnetics (2)Time Harmonic Electromagnetics (2)

Maxwell's equations

,

, ,

E H

Jρ

Maxwell's equations

Vector field phasors ( )

Source phasors ( ) Simple (linear, isotropic and homogeneous) media

,

,

E j H H J j E

E H

ωμ ωερε

∇× = − ∇× = +

∇ = ∇i i

0j te ω

⎫⎪⎬

= ⎪⎭

Assuming

ε

2 2

V A

V k V ρ

⎭

⎫∇ + = − ⎪⎬

Time harmonic wave equation for and

Non-homogeneous helmholtz's equations 2 2

2 2 , :k kA k A J uε ωω με ω μεμ

⎪⎬

= = =⎪∇ + = − ⎭

where wave-number


Time Harmonic Electromagnetics (3)Time Harmonic Electromagnetics (3)2∂ ∂22

2

22

2

) 0 0A Vcf A J A A j Vt tVVt

με μ με ωμε

ρμεε

∂ ∂∇ − = − ∇ + = ⇒∇ + =

∂ ∂∂

∇ − = −∂

i i ( )

22

2

22

0

tEE

tH

ε

με

∂∂

∇ − =∂∂

2 HH με ∂∇ − 2

( )

0

1 1Rj t j Ru u

t

ωωρ ρ

− −

=∂

Phasor solution

' '

'

1 1( , ) ' ( ) '4 4

1( ) '4

u uj t j t

v v

jkR

v

e eV R t dv V R e dv eR R

eV R dvR

ω ωρ ρπε πε

ρπε

−

= ⇒ = ⋅

⎧=⎪⎪

⎨

∫ ∫

∫

[V] E p essions fo the eta ded scala and

'

4

( ) '4

jkR

v

RJeA R dv

R

πεμπ

−

⎪⎨⎪ =⎪⎩ ∫

[Wb/m]

Expressions for the retarded scalar and vector potentials due to time harmonic sources


Time Harmonic Electromagnetics (4)Time Harmonic Electromagnetics (4)2 2

kR k R1 .... :2

2 2 ,

jkR k Re jkR

fk u fu uω π π λ

λ

− = − − +

= = = =

cf) Taylor series expansion.

'

12 1 1, ( ) '4

jkR

v

u uRif kR e V R dv

R

λρπ

λ πε−= ⇒ = = ⇒∫ then static potential

Procedure for determining the electric and magnetic fields due toProcedure for determining the electric and magnetic field

( ) ( )V R A R

s due to time harmonic charge and current distributions 1. Find phasors and

( ) )

( )

AE R V j A cf E Vt

B R A

ω ∂= −∇ − = −∇ −∂

= ∇×

2. Find phasors

( , )E R t 3. Find instantaneous


Source free Fields in Simple Media (1)Source-free Fields in Simple Media (1)Source free fields in simple media

E j H

H j E

ωμ

ωε

⎧∇× = −⎪∇× =⎪⎨

Source free fields in simple media

0

0

E

H

⎨∇ =⎪⎪∇ =⎩

i

i

2 2

2 2 2 2

0

0

E k E

H k H k ω με

⎧∇ + =⎪⎨∇ + = =⎪⎩

and Homogeneous vector Helmholtz’s equation

μ⎩Principl

E

e of duality : Source free Maxwell's equations in a simple mediaare invariant under the linear transformation

' , ' ,EE H H μη ηη ε

= = − =


Source free Fields in Simple Media (2)Source-free Fields in Simple Media (2)

0

( )

J E

H j E j E j E

σ σ

σσ ωε ω ε ωε

≠ ⇒ =

⎛ ⎞∴∇× = + = + =⎜ ⎟

If simple medium is conducting i.e)

( ) c

c

H j E j E j Ej

j

σ ωε ω ε ωεω

σε εω

∴∇× + +⎜ ⎟⎝ ⎠

= −

and [F/m] : complex permitivity

⋅cf) out of phase polariza

⋅ →

tion : power loss to overcome a fractional damping mechanism caused by the inertia the charged particle finite conductivity ohmic losses

' ''jε ε ε

→

= −

te co duct ty o c osses

Complex permitivity[F/m] ''εwhere : out of phase polarization and finite conductivityc jε ε ε= [F/m] ,

''

ε

σ ωε⇒ = ←

where : out of phase polarization and finite conductivity

equivalent conductivity representing all losses


Source free Fields in Simple Media (3)Source-free Fields in Simple Media (3)Complex permeability : out of phase component of magnetization

' '', '''jμ μ μ μ μ

μ μ= −

∴ =

Complex permeability : out of phase component of magnetization where ' for ferromagnetic materials Complex wavenumber

( ' '')c ck jω με ω μ ε ε= = −

Complex wavenumber

: in a lossy dielectric

Loss tangent

''tan ,'

''

c cε σδ δε ωε

ε

⎧ = ≅⎪⎪⎨⎪

where : loss angle

loss tangent'ε

σ ωε

⎪⎪⎩loss tangent

Good conductor & Good insulator : Good conductor

σ ωε : Good insulator


Source free Fields in Simple Media (4)Source-free Fields in Simple Media (4)π

∂

Cf) Electric hertz vector,

,A Vt

A

πμε π∂= = −∇∂

⎧ ∂⎪

i

: combine the vector and scalar potential and satisfy the Lorentz condition

0

AE Vt

VAt

με

∂= −∇ −⎪⎪ ∂⎨

∂⎪∇ + =⎪ ∂⎩i

combine continuity equation

0, ,

J

PJ J Pt t

ρ

ρ ρ∂ ∂∇ + = = = −∇∂ ∂

i i

with and

2

t t

Pπ

∂ ∂

∂

Single vector equation

22

2

2( )

Pt

PEt

ππ μεεππ με π

ε

∂∇ − = −

∂∂

= ∇ ∇ − = ∇×∇× −∂

i


Htπε ∂= ∇×∂

The electromagnetic spectrumThe electromagnetic spectrum

Field and Wave Electromagnetic - Seoul National...

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