PDF Field and Wave Electromagnetic - Seoul National University

(1)

Field and Wave Electromagnetic

Chapter7 Chapter7

The time varying fields and Maxwell’s equation

(2)

Introduction (1) Introduction (1)

Time static fields

0, ,

E D ρ D εE

∇× = ∇i = =

1) Electrostatic

2) Magnetostatic

0, , 1

B H J H B

E D B H

∇i = ∇× = = μ not

2) Magnetostatic

) and are not related to and for time st

e E D B H atic cases

not

e) and are not related to and for time static 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

(3)

Introduction (2) Introduction (2)

Time varying fields Time 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 ∇ ×H equation to be consiste

) 0

nt with the equation of continuity cf J for static. but J for time varying

t

ρ

∇ = ∇ = −∂

i i ∂

3) ∇iD =

ρ

and ∇iB = 0 never changes.

(4)

Faraday’s Law Faraday s Law

Michael 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

E B

t

∇× = −∂ ⇒

∂

as the gradient of a scalar potential

C S

E dl B dS

t

= − ∂

∫

ⁱ

∫

∂ ⁱ

(5)

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

d ∂ d ∂

∫ ∫ ( , since stationary 0)

C S

d d

E dl B ds ds

dt t dt t

∂ ∂

= − → =

∂ ∂

∫ i ∫ i ∵

11 2

Right hand rule (counter clock wise)

2

, 1 d , d 0

emf v E dl

dt dt

Φ Φ

=

∫

ⁱ = − >

Assume

dt dt

(6)

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

: emf induced in circuit with contour C Define v ∫ E dl : emf induced in circuit with contour C Define v = ∫

C

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 12

1

( 0)

2

right hand

E dl = E dl E = = − E dl = − = V V V

⇒

∫ i ∫ i ∵ ∫ i

inside contour

ca n be replaced with voltage source. But polarity of v V =

₁₂

depends on the change of the flux linkage

the change of the flux linkage

(7)

A Stationary Circuit in a Time Varying Magnetic Field (3) A Stationary Circuit in a Time Varying Magnetic Field (3)

∂B

0 . 12 0

[ ]

B v i e V

t

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

v d

dt Φ =

= − Φ ⇒

∫

ⁱ

Define : 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

(8)

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

(πr) i

A i l l f N t ₀

2

cos( ) sin 2

i ) ( 2 ) ) 2

B zB r wt

b

B d B r t d f ^π d

π

π φ

=

Φ

∫ ∫

^b

∫

A circular loop of N turns,

Find the emf induced in the loop

sol) each turn ( ₀

0 2

0

cos sin ) ( 2 ) ) 2

2

8 1) sin

SB ds zB wt z rdr cf d

b

b B wt

π φ π

π π

Φ = =

= −

∫

ⁱ

∫

0 ⁱ

∫

sol) each turn = (

( 2

z

2

0

8 1) cos

N

d Nb

v N B wt

dt

π π

∴ ⇒ Φ

∴ = − Φ = − −

N-turns

( [V]

2

y b

x

(9)

Transformers (1) Transformers (1)

mmf

j j k k

j 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_{1 1} N_{2 2} ℜΦ

1 1 2 2 :

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

N i N i

l

∴ − = ℜΦ

ℜ = ^{ℜ =}

μ

(10)

Transformers (2) Transformers (2)

)

a Ideal transformer

1 2

1 1 2 2

2 1

)

) a

i N N i N i

i N cf

μ → ∞ = ⇒ =

Ideal transformer

,

Faraday's law

1 1

) cf

v N d dt d

= Φ Φ

Φ Faraday s law

( No negative sign, careful of sign of flux )

v N

2 2

v N d dt

= Φ

(But flux is in the reverse direction) ¹ ¹

2 2

v N

N

∴ =

⎛ ⎞

effective load seen by the source connected to primary winding

1 2 2

1 2 1

1

1 2 2

2

( )_eff _L

N v

v N N

R R

i N N

i

⎛ ⎞

⎜ ⎟ ⎛ ⎞

⎝ ⎠

= = ⎛ ⎞ = ⎜⎝ ⎟⎠

⎜ ⎟

2 1

2 1 1

2

( )_eff _L

N

Z N Z

N

⎜ ⎟

⎝ ⎠

⎛ ⎞

∴ = ⎜ ⎟

⎝ ⎠ Impedance transformation

⎝ 2 ⎠

(11)

Transformers (3) Transformers (3)

) Real transformer b

1 1 2 2

) Real transformer

b

N i N i l

s

s s

μ

μ μ

− = Φ

2 2

1 1 1 1 2 2 2 2 1 2 1 2 2

1 2 1 2

1 1 12 2 12 2

( ), ( )

,

1 s s

N N i N N i N N N i N i

l l

di di di di

v L L v L L

dt dt dt dt

μ μ

⇒ Λ = Φ = − Λ = Φ = −

= − = −

∵

2 2

1 1 , 2 2, 12 1 2)

(where

For an ideal tran

dt dt dt dt

s s s

L N L N L N N

l l l

μ 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

⇒ ∴ =

∴ = <

(12)

Transformers (4) Transformers (4)

Equivalent ci rcui t Equivalent ci rcui t

1, 2 : :

winding resistance

leakage inductive reactance R R

X X₁, ₂ : :

:

leakage inductive reactance

power loss due to hysteresis and eddy current

nonlinear inductive reactance due to the nonlinear magnetization behavior of the ferro

c c

X X R X

magnetic core of the ferromagnetic core

(13)

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

F_m =qu×B F qu B

F F

×

→

Charge Seperation

Coulomb force of an attraction

and will balance each other to be in equilibrium.

m e

F F

→

∴

∫

and will balance each other to be in equilibrium.

Magnetic force per unit charge

2

21 1

, ,

( )

m m

F F

u B V E dl E

q q

V u B dl

= × = − ⋅ = −

∴ = × ⋅

∫

The emf g

' ( )

V =

∫

C u×B dl⋅ →

enerated around the closed loop is flux cutting emf

(14)

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

B = zBˆ ₀ constant u

0 1 2

,

( )

C

B zB u

V V V u B dl

=

= − =

∫

× ⋅ constant

a)

1' 2' 0

0

ˆ ˆ ˆ

(xu zB ) (ydl) uB h

= × ⋅

= −

∫

2

0 2 ( 0 )

, _l

V uB h

I P I R

R R

= = =

b)

1'

ˆ 0

Fm = I

∫

dl × = −B xIB h c) mechanical power

₀

m 2'

I

∫

( : n

2 2 2

0

)

m m

dl u B h

P F u F u

∴ = ⋅ = − ⋅ = R

egative direction to

_m _m

R

(15)

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)

', '

Fm q E u B

E E 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

E dl B ds u B dl

t

⋅ = − ∂ + × ⋅ →

∫ ∫

∂

∫

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

(16)

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 2 1 1

0

lim 1 ( ) ( )

S

S S

t

d d

dt dt B ds

B t t ds B t ds

Δ → t

Φ = ⋅

⎡ + Δ ⋅ − ⋅ ⎤

⎢ ⎥

⎣ ⎦

Δ

∫

∫ ∫

=

( )

2 1

( ) ( ) . . .

t

B t t B t B t t H O T t

d B ds

⎣ ⎦

Δ

+ Δ = +∂ Δ +

∂

∴ ⋅ =

cf) Taylor's series

2 1

lim 1 . . .

B ds B ds B ds H O T

∂ ⋅ + ⎡⎢⎣ ⋅ − ⋅ + ⎤⎥⎦

∫

^{B ds}

∫ ∫ ∫

∴ dt

2 1

0

3

lim . . .

S S ds t S B ds S B ds H O T

t t

dS dl u t

+Δ → ⎢⎣ + ⎥⎦

∂ Δ

Δ

= × Δ

∫ ∫ ∫ ∫

assuming side surface S as the area swept out by the conductor in time t3

f om di e gence theo em

2 2 1 1

S 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 ²

∫

₁ ¹

∫

⁽ ⁾

( )

' '

S S C

B ds B ds t u B dl

d B

B ds ds u B dl

dt t

d d

V E dl B d

∴ Δ ×

∴ ⋅ = ∂ ⋅ − × ⋅

∂

Φ

∫ ∫ ∫

∫ ∫

' '

C S

d d

V E dl B ds

dt dt

∴ =

∫

ⁱ = −

∫

⋅ = − Φ

(17)

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

static Time varying

0

∇× = E B

E ∂

∇× = −

∂ ,

D ρ D ε E

∇ i = = t

D ρ

∂

∇ i =

0, H J

B B μ H

∇× =

∇ i = =

0 H J D

t B

∇× = + ∂

∂

∇ B 0

∇ i =

(18)

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

Note ① Continuity equation 0

J

J ρ

∇ =

∇ ∂ i

Note Continuity equation ①

: for steady state current : time varying current

J t

ρ

∇ = −

∂

∂ i

②

: time varying current Vector identity

( H ) 0 J J

t ρ

∇ ∇× = = ∇ ⇒ ∇ = − ∂

∂

∴

i i i

contradiction

( ) 0 D

H J ∂ ρ ⎛ J ∂ ⎞ ρ D

∇ ∇× i = = ∇ + i = ∇ i ⎜ + ⎟ where = ∇ i

Displacement current density. [A/m²]

Time varying electric field and induced magnetic field→coupling

∴

( H ) 0 J J D

t t

H J D

ρ

∇ ∇× = = ∇ + = ∇ ⎜ + ⎟ = ∇

∂ ⎝ ∂ ⎠

∇× = + ∂

∂

i i i , where i

Time varying electric field and induced magnetic field→coupling

( )

t

F q E u B

∂

= + ×

Cf) Lorentz force equation,

(19)

Integral Form of Maxwell’s Equation Integral Form of Maxwell s Equation

→

Cf) Differential form Point function

E B S C

t

→

∇× = − ∂ ⇒

∂

Cf) Differential form Point function

Apply stokes's theorem over open surface with contour

( )

S S

E ds B ds

t

∇× = − ∂

∫ i ∫ ∂ i

c s

B d

E dl ds

t dt

∂ Φ

= − = −

∫ i ∫ ∂ i

① : Faraday's law

∂ D

② H

c s

s

dl I D ds t

D ρ D ds Q

= + ∂

∂

∇ = ⇒ =

∫ ∫

∫

i i

③

: Ampere's circuital law : Gauss law

∫

s

(20)

Ex 7 5 Ex. 7-5

(a) Displacement current = conduction current

dv

→

(a) Displacement current conduction current 1 conduction current current on the wire.

Apply circuit theorem

1 ^c 1 0 cos

C

i C dv C V t

dt ω ω

= =

2 Displacement cur

∂D rent. Reminding

1

H J D t

C A

μ ε d

∇× = + ∂

∂

=

Assuming the area A, plate separation d, permitivity , then Assume is uniform in the dielectric (ignoriE

0 i

vc V

E D E t

ng fringing effects) then

0

0 1 0

, sin

cos cos

c

D C

A

E D E t

d d

D A

i ds V t C V t i

t d

ε ε ω

ε ω ω ω ω

= = =

= ∂ = = =

∫

∂ ⁱ

(21)

Ex 7 5 Ex. 7-5

(b) Magnetic field intensity reminding Ampere's law

, C S

D D

H J H dl I ds

t t

∂ ∂

∇× = + = +

∂

∫

ⁱ

∫

∂ ⁱ

(b) Magnetic field intensity reminding Ampere s law

0 2

D =

∫

H dlⁱ =

π

r

①

②

①

surface S1 with ring C surface S2 with ring C

H

0, 2

D =

∫

C H dlⁱ =

π

r

①

1 0 cos

C

H

I J ds i C V t

φ

ω ω

⇒

=

∫

ⁱ = =

( Symmetry around the wire along the contour C) constant

_{1 0}

1 0

cos

, cos

S C

d

J ds i C V t

I i H_φ C V t

ω ω

= ∴ =

∫

② no conduction current, but displacement current

, cos

I id H_φ

ω ω

t

∴

π

(22)

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

E E A E

∇ = ∇ ∇× =

⎛ ⎞

∂ ∂ ∂

∇× ∴∇× ∇× ⇒ ∇× ⎜ +

i i

Vector identity Recall Faraday's law

▷

⎟ 0

Curl free

( )

E E A E

t t t

∇× = − ∂ ∴∇× = − ∂ ∇× ⇒ ∇× ⎜ ⎝ + ∂ ⎠

0

( V ) 0

⎟ =

∇× ∇ = Vector identity

d d f l

▷

[ / ]

E V

A A

E V E V V m

t t

= −∇

∂ ∂

+ = −∇ = −∇ −

∂ ∂

and reminding for electromagnetics for time varying i.e)

t t

∂ ∂

(23)

∂ A A 0

E V

t

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

^v

4

^v

dv A J dv

R R

μ ρ

πε ∫ = π ∫ : From the static condition

0

2 2

V ρ A μ

0

J

∇ = − ∇ = −

These are solution of poisson equation

and

₀

ε

(24)

J

R ρ

Quasi-static fields

- and vary slowly with time

- the range of interest R is small compared to the wavelength 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 , D ( , )

B A E V H J B H D E

t t μ ε

∂ ∂

= ∇× = −∇ − ∇× = + = =

∂ ∂

From the equations

t t

A J V A

t t

μ με

∂ ∂

⎛ ⎞

∂ ∂

∇×∇× = + ⎜ −∇ − ⎟

∂ ⎝ ∂ ⎠

⎝ ⎠

Recalling vec

( )

2

A A A

∇×∇× = ∇ ∇ i − ∇ tor identity

(25)

V

2

A

∂ ∂

⎛ ⎞

2

2 2

2

( ) V A

A A J

t t

A V

A J A

μ με με

με μ με

∂ ∂

⎛ ⎞

∴∇ ∇ − ∇ = − ∇ ⎜ ⎝ ∂ ⎟ ⎠ − ∂

∂ ⎛ ∂ ⎞

∇ + ∇ ∇ + ⎜ ⎟

i

A

2

J A

t t

A B A

με μ με

∇ − ∂ = − + ∇ ∇ + ⎜ ⎝ ∂ ⎟ ⎠

∇× = ∇

i

- we only designated but we are free to choose

- vector will b A A A

V V

∇× ∇

∂ ∂

e specified by giving and i

0 , 0 0

V V

A A

t t

με ∂ ∂

∇ + = = ∴∇ =

∂ ∂

i i

- let for static

Lorentz gauge for potentials g g p

(26)

0, 0

0

V A

t t

A

∂ ∂

= =

∂ ∂

∴∇ i = cf) For static,

2

A μ

0

J

∇ = −

then vector poisson equation

2

A

∂

- Then nonhomogeneous wave equation for vector potential becomes

2

A A

με ∂ t

∇ − = −

∂ μ J : Vector potential wave equation

(27)

Potential Functions(6) Potential Functions(6)

A , A

E V D V

t ρ ε ⎛ t ⎞ ρ

∂ ∂

= −∇ − ∂ ∇ i = ⇒ −∇ i ⎜ ⎝ ∇ + ∂ ⎟ ⎠ = Scalar potential wave equation

∨

2

( ) ,

t t

V A A V

t t

ρ με

ε

∂ ⎝ ∂ ⎠

∂ ∂

∴∇ + ∇ = − ∇ = −

∂ i i ∂

2 2

2

V V

t με ρ

ε

∴ ∇ − ∂ = −

∂

(28)

Boundary Condition (1) Boundary Condition (1)

Electric field's boundary condition

⊙

c s s v

E dl B ds D ds dv

t ρ

= − ∂ =

∫

ⁱ

∫

∂ ⁱ

∫

ⁱ

∫

Electric field s boundary condition

... ...

⊙

0 0 0

B ds h S

∂ → Δ → →

∫

ⁱ

From equation

when since area

0 0, 0

s ds h S

t → Δ → →

∫

∂

when since area ∴ E₁_t = E₂_t (E₁_tΔ −w E₂_tΔ =w 0)

(29)

Boundary Condition (2) Boundary Condition (2)

From equation

1 2 2 1 2 1 2

( ) ( )

( )

sD ds D n D n S n D D S s S

n D D D D

ρ

ρ ρ

= + Δ = − Δ = Δ

∴ − = − =

∫

ⁱ ⁱ ⁱ ⁱ

i From equation

2 ( 1 2) _s , 2_n 1_n _s

n D D D D

H

ρ ρ

∴ i = =

Magnetic field's boundary conditions

D

dl ⎛J ∂ ⎞ ds

= ⎜ + ⎟

∫

ⁱ

∫

ⁱ

1 2 1 2

2 1 2

( )

. ) ( )

c s

sn t t sn

dl J ds

t

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

(30)

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 volume current density

1t 2t

H 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.

∇iB = 0 ∴ B₁_n = B₂_n

(31)

Interface Between Two Lossless Linear Media Interface Between Two Lossless Linear Media

ε μ

→

Linear media permitivity : ε permeability : μ σ

→

Linear media permitivity : , permeability : Lossless =0

J

S

ρ

→

→ Assume, at interface, no free charge

S

=0

no surface currents =0

1 1 1

1_t 2_t

D

^t

,

1_t 2_t

B

^t

E = E ⇒ = ε H = H ⇒ = μ

₁

1 2 1 2

2 2 2

t t

,

t t

E E H H

D ε B

⇒ ⇒

2

1_n 2_n 1 1_n 2 2_n

,

1_n 2_n 1 1_n 2 2_n

D D E E B B H H

μ

ε ε μ μ

= ⇒ = = ⇒ =

(32)

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 , may not be zero. B

2

0,

2

0,

2

0,

2

0

E = H = D = B =

(33)

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

0 0

E₁ E₂

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 H

H J J H

= =

× − = =

= = → =

if

2 1 2 2 1

1 2

( ) , 0,

0, 0

s n n s

n n

n D D D D

B B

n

ρ ρ

− = = =

= =

i

note) n₂ : outward normal from medium2 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 f

1

1 1

1 s n

E

E E ρ

= = ε

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

: is tangential to the interface with a magnitude of

(34)

Wave Equation and Their Solutions (1) Wave Equation and Their Solutions (1)

2 A

2 ∂

2 2 2

2

A A J

t V V

t

με μ

με ρ

ε

∇ − ∂ = −

∂

∇ − ∂ = −

∂ Wave equation :

₂

( ) t

t t

ε

ρ ν

∂

Δ ′ i

Solution :

Assume an elemental point charge at time , located at the

i i f th di t

origin of the coordinates.

V R t

i i

Spherical coordinates.

depends only on . and because of spherical symmetry.

1 2 V

φ

∂ ⎛ ∂ ⎞ i

(No dependence on ) Except at the origin,

2V

2 ∂

2

1 V

R ∂^∂R^⎛⎜⎝R ^∂∂R ⎞ −⎟⎠

με

V₂ 0 t

∂ =

∂

(35)

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 t R

U U

=

∂ ⎛ ⎞ ⎡ ∂ ⎤ ∂

New variable

( )

2 2

2

2 2 2

1 1 1

,

1 1

U U

R U R t R U U R

R R R R R R

U U U U

U R R

R R R R R R R

∂ ⎛⎜ ⎞⎟= ⎡⎢− + ∂ ⎤⎥= − + ∂

∂ ⎝ ⎠ ⎣ ∂ ⎦ ∂

⎡ ⎤

∂ ⎡⎢− + ∂ ⎤⎥ = ⎢−∂ + ∂ + ∂ ⎥

∂ ⎣ ∂ ⎦ ⎣ ∂ ∂ ∂ ⎦

1

R ∂R ⎣ ∂R ⎦ R ⎣ ∂R ∂R ∂R ⎦

∴

² ₂ 1 ²₂ ² ( , )₂ ² ( , )₂

0, 0

U U U R t U R t

R R R t R t

t R t R

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

(36)

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

+ →

∴ = −

is a wave equation which travels to the origin physical nonsense

( , ) ( )

f

R 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 1

limt

if t R

R R

t t u

με

^{Δ →}

με

Δ = Δ

Δ = ⇒ Δ = =

Δ Δ

: velocity of propagation

0

( , ) 1

t t t

V R t f t R

R u

με

^{Δ →}

με

Δ Δ

⎛ ⎞

= ⎜⎝ − ⎟⎠

f t R u

⎛ − ⎞

⎜ ⎟

⎝ ⎠

Determine

(37)

Wave Equation and Their Solutions (4) Wave Equation and Their Solutions (4)

( )t ρ Δν Recall potential function induced by a static point charge ( ) at

( ) ' '

t t R v

t v R u

ρ ν

ρ ρ

Δ

⎛ − ⎞Δ

⎜ ⎟

Δ ⎛ ⎞ ⎝ ⎠

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

( ) ( ) ,

4 4

1

t v R u

V R f t

R u

t R u ρ

πε πε

ρ

Δ ⎛ ⎞ ⎝ ⎠

Δ = Δ ⎜⎝ − ⎟⎠=

⎛ − ⎞

⎜ ⎟

⎝ ⎠

∫

'

( , ) 1 '

4 ^v

V R t u dv

πε ^⎝ R ^⎠

∴ =

∫

:

R t

⎛ R⎞ Retarded scalar potential Scalar potential at a distance from the source at time

t R u

⎛ ⎞

→Depends on the value of charge distribution at an earlier time ⎜⎝ − ⎟⎠ Retarded vector potential

⎛ RR⎞

⎛ ⎞

(38)

Source Free Wave Equation Source Free Wave Equation

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

H , E

E H

t 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

) ( ) ( ) ( ) ( )

E H E

t t

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

μ ^∂ με ^∂

∇×∇× = − ∇× = −

∂ ∂

× × = i − i = i − i

2 2

2

( E) ( )E E

E με E

∇ ∇ − ∇ ∇ = −∇

∴∇ − ∂

i i

₂ 1₂

0 if

t = με = u

∂

2 2

2 2 2 2

1 1

0, 0

E H

u t u t

∂ ∂

∇ − = ∇ − =

∂ ∂

: Homogeneous vector wave equation

(39)

Time Harmonic Fields Time Harmonic Fields

Maxwell's equations Maxwell s equations

- linear differential equations

- sinusoidal time variation of source functions at given frequency ,

- are sinusoidal with the same frequency E H

Ti h i d i id l

Time harm →

→

onic steady state sinusoidal Phasors : Amplitude and phase information

independent of time e

j t^ω

→ independent of time

cf) : time dependent factor

(40)

Time Harmonic Electromagnetics (1) Time Harmonic Electromagnetics (1)

Vector phasors of field vectors : depend on space coordinates ( , , ; ) Re ( , , ) ,

( ) :

E x y z t E x y z e

j t

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 E

t ω

∂ =

∂

where vector phasor : complex quantity ⎡ ⎣ ( , , ) x y z e

^{j t}^ω

⎤ ⎦

( , , ) :

( , , )

( , , ; ) Re

^{j t}

j E x y z

E x y z

E x y z t dt e

j

ω

⎡ ⎤

= ⎢ ⎥

⎣ ⎦

∫

where vector phasor

( , , )

j E x y z

j

ω ω

⎣ ⎦

where : vector phasor

2

( )

2 2

, , 1

j j

t ω t ω j

ω

∂ ⇒ ∂ ⇒ ⇒

∂ ∂ ∫

i.e)

(41)

Maxwell's equations

,

, ,

E H ρ J

Maxwell's equations

Vector field phasors ( )

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

,

E j H H J j E

E H

ωμ ωε

ρ ε

∇× = − ∇× = +

∇i = ∇i

0

ej t^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 , k :

A k A J u

εμ ω με ω με ω

⎪⎬ = = =

∇ + = − ⎪⎭

where wave-number

(42)

Time Harmonic Electromagnetics (3) Time Harmonic Electromagnetics (3)

∂2² ∂

2

2 2 2

2

) A V 0 0

cf A J A A j V

t t

V V

t

με μ με ωμε

με ρ

ε

∂ ∂

∇ − = − ∇ + = ⇒ ∇ + =

∂ ∂

∇ − ∂ = −

∂

i i

( )

2 2

2 2 2

0 t

E E

t H

ε με

∂

∇ − ∂ =

∂

2 H

H με ^∂

∇ −

2

( )

0

1 1

j t R j R

u u

t

ω ω

ρ ⁻ ρ ⁻

∂ = Phasor solution

' '

'

1 1

( , ) ' ( ) '

4 4

( ) 1 '

4

u u

j t j t

v v

jkR

v

e e

V R t dv V R e dv e

R R

V R e dv

R

ω ω

ρ ρ

πε πε

ρ πε

−

= ⇒ = ⋅

⎧ =

⎪⎪⎨

∫ ∫

∫

[V]

E p essions fo the eta ded scala and

'

4

( ) '

4

jkR

v

R

A R Je dv

R πε

μ π

−

⎪⎨

⎪ =

⎪⎩

∫

[Wb/m]

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

(43)

2 2

kR k R

1 .... :

2

2 2

,

jkR k R

e jkR

k f u f

u u

ω π π λ

λ

− = − − +

= = = =

cf) Taylor series expansion.

'

2 1 1, ( ) 1 '

4

jkR

v

u u

if kR R e V R dv

R

λ

π ρ

λ πε

= ⇒ − = =

∫

⇒

then static potential

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

( ) ( )

V R A R

s due to time harmonic charge and current distributions

1. Find phasors and

( ) )

( )

E R V j A cf E V A

t

B R A

ω

^∂

= −∇ − = −∇ −

∂

= ∇×

2. Find phasors

(44)

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 media are invariant under the linear transformation

' , ' E ,

E

η

H H

η μ

η ε

= = − =

(45)

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

σ ωε ω ε σ ωε

≠ ⇒ =

⎛ ⎞

∴∇× = + = ⎜ + ⎟ =

If simple medium is conducting i.e)

( ) _c

c

H j E j E j E

j j

σ ωε ω ε ωε

ω ε ε σ

ω

∴∇× + ⎜ + ⎟

⎝ ⎠

= −

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

' j '' ε ε ε

→

= −

te co duct ty o c osses Complex permitivity

[F/m] where ε '' : out of phase polarization and finite conductivity

c j

ε =ε ε

[F/m] , where : out of phase polarization and finite conductivityε

(46)

Source free Fields in Simple Media (3) Source-free Fields in Simple Media (3)

Complex permeability : out of phase component of magnetization

' '', ''

'

μ μ jμ μ μ

μ μ

= −

∴ =

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

Complex wavenumber

( ' '')

c c

k =ω με =ω μ ε − jε Complex wavenumber

: in a lossy dielectric

Loss tangent

tan '' ,

'

''

c c

ε σ

δ δ

ε ωε ε

⎧ = ≅

⎪⎪⎨

⎪

where : loss angle

loss tangent ε ' σ ωε

⎪⎪⎩loss tangent Good conductor & Good insulator

: Good conductor σ ωε

: Good insulator

(47)

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

π

∂

Cf) Electric hertz vector, ,

A V

t

A

με ^∂π π

= = −∇

∂

⎧ ∂

⎪

i

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

0 E V A

t A V

με t

= −∇ −∂

⎪⎪ ∂

⎨ ∂

⎪∇ + =

⎪ ∂

⎩ i

combine continuity equation

0, ,

J

J J P P

t t

ρ

ρ ρ

∂ ∂

∇ + = = = −∇

∂ ∂

i i

with and

2

t t

π P

∂ ∂

∂

Single vector equation

2π με ^∂ π P

∇ − = −

(48)

The electromagnetic spectrum