CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class...

18
Time-Varying Fields 6-1 Proprietary of Prof. Lee, Yeon Ho CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S EQUATIONS 9 Static electric charges Static E and D 0 × = E v D i Ú No mutual relationship Steady electric currents Static H and B × = H J 0 = B i 9 Accelerated charges Time-varying fields E and H (time-varying currents) Intercoupled From this point time-varying quantities are denoted by script or italic letters. E , H , v ρ and J , : time-varying electric field, magnetic field, volume charge density and current density, respectively. Gauss’s laws, v i D and 0 = i B , still hold. ∇×E and ∇× H are modified to include the electromagnetic induction and the displacement current. Faraday’ experiment Maxwell’s hypothesis 9 Maxwell combined two curl equations for E and H Electromagnetic waves propagating with the same velocity as light 6.1 FARADAY’S LAW Moving a magnet near a wire loop, or moving a wire loop near a magnet Electromotive force(emf), or a voltage Faraday’s law The induced emf in a closed wire loop is equal to the negative time rate of change of the magnetic flux linkage with the loop. d emf dt Φ =− [ ] V (6-1) If the loop has N turns of wire, d emf N dt Φ =− [ ] V (6-2) Φ is the magnetic flux enclosed by a single turn of wire. 9 Using the electric field C emf d = l i v E (6-3) E is a function of time in general Constant, if / d dt Φ is constant

Transcript of CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class...

Page 1: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-1 Proprietary of Prof. Lee, Yeon Ho

CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S EQUATIONS Static electric charges → Static E and D → 0∇ × =E

v∇ = ρDi No mutual relationship Steady electric currents → Static H and B → ∇ × =H J 0∇ =Bi

Accelerated charges → Time-varying fields E and H

(time-varying currents) ↑ Intercoupled From this point time-varying quantities are denoted by script or italic letters. E , H , vρ and J , : time-varying electric field, magnetic field, volume charge density and current density, respectively. Gauss’s laws, v∇ = ρiD and 0∇ =iB , still hold. ∇ ×E and ∇ ×H are modified to include the electromagnetic induction and the displacement current. ↑ ↑ Faraday’ experiment Maxwell’s hypothesis

Maxwell combined two curl equations forE and H → Electromagnetic waves propagating with the same velocity as light

6.1 FARADAY’S LAW

Moving a magnet near a wire loop, or moving a wire loop near a magnet → Electromotive force(emf), or a voltage Faraday’s law The induced emf in a closed wire loop is equal to the negative time rate of change of the magnetic flux linkage with the loop.

demfdtΦ

= − [ ]V (6-1)

If the loop has N turns of wire,

demf NdtΦ

= − [ ]V (6-2)

Φ is the magnetic flux enclosed by a single turn of wire.

Using the electric field

Cemf d= ∫ liE (6-3)

E is a function of time in general ↑ Constant, if /d dtΦ is constant

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Static E is conservative → Closed line integral of E is always zero d∫ E li corresponds to the potential difference E is a non-conservative field → Closed line integral of E is emf V≠ −∇E

Combine Eq. (6-2) with Eq. (6-3), and use surface integral of B for Φ . Faraday’s law is then

demf d ddt

= = −∫ ∫l sC S

i iE B (6-4)

1N = S : a surface bounded by C dl and ds : the right-hand rule Minus sign : Lenz’s law The induced emf produces a current and therefore a magnetic flux In such a direction as to oppose the change in the flux linkage with the loop. 12 0V > in Fig. 6-1(a) 12 0V < in Fig. 6-1(b)

12V is an induced voltage, not an applied voltage that would produce an electric field in the opposite direction to E in the loop.

6-1.1 Transformer emf Stationary loop → / t∂ ∂ inside the integral

emf d dt

∂= = −

∂∫ ∫l sC S

i iBE (6-5)

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Time-Varying Fields 6-3 Proprietary of Prof. Lee, Yeon Ho

It is called the transformer emf ↑ Voltage induced in a stationary loop in B Applying Stokes’s theorem to Eq. (6-5)

( )S S

d dt

∂∇ × = −

∂∫ ∫BE s si i

or

t

∂∇ × = −

∂BE (6-6)

Faraday’s law in point form, one of the four equations of Maxwell. Example 6-1

Find induced E in a stationary conducting loop of radius a , which is placed in a spatially uniform time-varying magnetic flux density, ( ) coso zt B t= ω aB .

Solution

Assuming dl to be clockwise, from Eq. (6-5)

d dt

∂= −

∂∫ ∫l sC S

i iBE

( )2

2

2 cos

sin

o

o

a a B tt

a B t

φ

∂π = −π ω

∂= π ω ω

E

Rearranging it

( )12 sinoa B tφ φ φ= = ω ωa aE E

Figure 6-3 plots zB and φE , normalized to unity, as functions of t . Shade areas indicate the time intervals in which zB is increasing and φE therefore is given parallel φ−a .

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6-1.2 Motional emf

A loop moves in static B → Motional emf ↑ Due to magnetic force on free electrons of the conductor.

demf d ddt

= = −∫ ∫l B sC

i iES

(6-7)

S is a part of the surface S that is bounded by C , having nonzero B . S depends on time in general.

Referring to Fig. 6-4

o zB=B a in the region oy y≥

Page 5: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-5 Proprietary of Prof. Lee, Yeon Ho

From Eq. (6-7)

( ) o o o o o o

demf d ddtd B x v t B x vdt

= = −

= − = −

∫ ∫l B sC

i iES (6-8)

where zds a and C is counterclockwise emf → Clockwise current → B in z−a direction, opposing to the increase in the flux

Calculation of motional emf from magnetic force on free electrons Magnetic force on an electron

m e= ×BF v (6-9) ↑ ↑ Time dependent Motional electric field intensity is defined by

mm e= = ×BEF v (6-10)

The induced emf ( )memf d d= = ×∫ ∫l B l

C Ci iE v (6-11)

↓ mE d⊥ l on the top and bottom sides ↓ 0=B on the left side during 0 /o ot y v< < 0

o

x

o o x x o o ox xemf v B dx B x v

=

== = −∫ a ai (6-12)

An example motional emf referring to Fig. 6-5

Sliding bar moves on perfectly conducting rails

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With the switch 1S opened. On the loop that consists of bar, rails and gap between terminals 1 and 2

( )

l

m u

x l

o o x x o o ox u

emf d d

v B dx v B x=

=

= = ×

= =

∫ ∫

l B l

a a

Ci i

i

E v (6-13)

Perfectly conducting rails are at equi-potentials → 12 o o oV v B x=

From the other viewpoint. Magnetic forces mF on charges → + charges at l , - charges at u → Infinite current in x−a direction? → No current due to mE Both mF and mE force electrons to move in x− -direction in the bar. mE appears as an emf across terminals 1 and 2. 12 o o oV v B x= (6-14)

Direction calculation of motional emf from Eq. (6-7)

( ) ( )

o z z o o o o

o o o

demf ddtd dB dxdy B x y v tdt dt

B x v

= −

= − − = +⎡ ⎤⎣ ⎦

=

B s

a a

i

i

S

S (6-15)

With the switch 1S closed

Induced emf → Current in clockwise direction → B in z− -direction. Opposing to the change in Φ .

Power calculation with switch 1S closed Current I in the bar moving with v in B . From Eq. (5-125b)

( )

x l

I o yx u

o o y

I d I B dx

IB x

=

== × = −

= −∫ ∫F l B a

aC

The mechanical power m I o o oP IB x v= − =F iv (6-16) Power dissipated in the resistor R

12

e

o o o

P IVIB x v

=

= (6-17)

m eP P= , the conservation of energy.

Page 7: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

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6-1.3 A Loop Moving in a Time-Varying Magnetic Field A conducting loop moves in a time-varying magnetic field Total emf = transformer emf + motional emf

( )emf d d dt

∂= = − + ×

∂∫ ∫ ∫l s lC S C

i i iBE Bv (6-18)

Applying Stokes’s theorem

( )t

∂∇ × = − + ∇ × ×

∂B

E Bv (6-19)

The instantaneous emf

d demf ddt dtΦ

= − = − ∫ sS

iB (6-20)

The time derivative may not be taken inside the integral, because the bounded surface S may move with time and contribute to the change in the magnetic flux linkage with the loop.

Example 6-2

A rectangular conducting loop with sides a and b is in a time-varying magnetic flux density ( )sino o yB t= ωB a . The loop rotates with an angular speed ω about x -axis. The rotation angle of the normal to the loop surface is tϕ = ω − α . Find at time t (a) transformer emf (b) motional emf (c) instantaneous emf

Solution (a) Total magnetic flux enclosed by the loop at t

( )( )

sin

sin coso o y s

o o

d B t ab

B t ab

Φ = = ω

= ω ϕ∫ s a ai iB

(6-21)

Fixing ϕ to a constant value, we obtain the transformer emf

( )cos coso o odemf B ab tdtΦ

= − = − ω ω ϕ (6-22a)

For the given unit vector sa , according to the right-hand rule, the line integral of E should be done along the loop passing the points in the order 1-2-3-4. The minus sign in Eq. (6-22a) indicates that the terminal II is at the higher potential, or the positive terminal, at

0t = .

Page 8: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

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(b) Motional emf , from Eq. (6-11) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2121

4123

sin

sin

S o o y x

S o o y x

emf d

b B t dx

b B t dx

= ×

⎡ ⎤= ω − × ω⎣ ⎦

⎡ ⎤+ ω × ω⎣ ⎦

∫∫

B l

a a a

a a a

Ci

i

i

v

( ) sin sino oB ab t= ω ω ϕ (6-22b) The left and right sides contribute no motional emf because ×Bv is perpendicular to the conductor.

(c) By substituting ( )tω − α for ϕ in Eq. (6-21), the instantaneous magnetic flux enclosed by the loop

( ) ( )sin coso oB t ab tΦ = ω ω − α Instantaneous emf

( ) ( ) ( ) ( )cos cos sin sino o o o odemf B ab t t B ab t tdtΦ

= − = − ω ω ω − α + ω ω ω − α

It is the same as the sum of the results in Eqs. (6-22a) and (6-22b).

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Time-Varying Fields 6-9 Proprietary of Prof. Lee, Yeon Ho

6-2 DISPLACEMENT CURRENT DENSITY Ampere’s circuital law, ∇ × =H J (6-23) Divergence of both sides ∇ ∇ × = ∇H Ji i ↓ ↓ =0. =0, the continuity condition

Vector identity

The continuity condition for time-varying fields, /v t∇ = −∂ ∂iJ ρ → Eq. (6-23) is not valid for time-varying fields

A parallel-plate capacitor A closed loop C around the lead → Bounded surface 1S or 2S Ampere’s law to Fig. 6-7(a)

d I=∫ lC

iH : I penetrates 1S

Ampere’s law to Fig. 6-7(b)

0d =∫ lC

iH : No current penetrates 2S

An obvious contradiction!

Page 10: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-10 Proprietary of Prof. Lee, Yeon Ho

A current is not the only source of a time-varying magnetic field

An increased or decreased I → Magnetic field between electrodes Electric field between electrodes

zoA

aEQ (6-24)

Q : total charge on a conducting plate A : area of the plate oε : permittivity of free space. Time derivatives of both sides, with /I d dt= Q

( )oz

It A

∂ ε=

∂a

E : a current density (6-25)

The displacement current density is defined by

D t∂

≡∂DJ (6-26)

The generalized Ampere’s law

t

∂∇ × = +

∂D

H J (6-27)

The generalized Ampere’s law in integral form

d dt

∂⎡ ⎤= +⎢ ⎥∂⎣ ⎦∫ ∫l sC S

i iDH J (6-28)

The current density J includes both the conduction and convection currents

Check the generalized Ampere’s law Divergence of both sides of Eq. (6-27)

( )t∂

∇ ∇ × = ∇ + ∇∂

i i iH DJ

Using Gauss’s law

( )vt∂

∇ ∇ × = ∇ +∂

i iH J ρ : always true

↓ ↓ =0. =0, the continuity condition A vector identity Example 6-3

An ac voltage sinoV V t= ω is applied across an air-gap parallel-plate capacitor as shown in Fig. 6-8. Ignoring edge effects, (a) show that the displacement current is equal to the conduction current. (b) find the magnetic fields around the conductor lead and in the gap.

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Time-Varying Fields 6-11 Proprietary of Prof. Lee, Yeon Ho

Solution (a) For electrodes separated by d

sinoo o

V td

= ε = ε ωD E

Displacement current density

cosoD o

Vt

t d∂

= = ωε ω∂DJ

Displacement current in the capacitor of /oC A d= ε

cos

cos

oD D oS

o

VI d A td

CV t

= = ωε ω

= ω ω

∫ siJ

Conduction current in the lead

cosC odVI C CV tdt

= = ω ω

We have D CI I=

(b) We first apply the generalized Ampere’s law to a closed loop around the capacitor lead.

The displacement current is zero in the capacitor lead, because of no electric field in a perfect conductor.

1 Cd I=∫ lC

iH 12 CIφπρ =H or

1 2CI

φ=πρaH (6-29)

The conduction current is zero in the gap of the capacitor 2 Dd d=∫ ∫l s

C Si iH J (6-30)

For the capacitor plate of radius a , using D CI I=

2 2CD

DII

a a≡ =π π

J (6-31)

Inserting Eq. (6-31) in Eq. (6-30)

2

2 22 CI aφ

πρπρ =

πH

Thus

22CI a φ

ρ=

πaH ( )aρ ≤ (6-32a)

2CI

φ=πρaH ( )aρ ≥ (6-32b)

At a far radial distance aρ > , the magnetic field is continuous along the axial direction; the magnetic fields in Eqs. (6-29) and (6-32b) are identical.

Page 12: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-12 Proprietary of Prof. Lee, Yeon Ho

6-3 MAXWELL’S EQUATIONS Unification of two theories

Electricity ↔ Magnetism ↑ Displacement current density

→ Differential wave equation → Existence of electromagnetic waves (Same velocity as light)

Maxwell’s equations

t∂

∇ × = −∂B

E (6-33a)

t∂

∇ × = +∂D

H J (6-33b)

v∇ =iD ρ (6-33c)

0∇ =iB (6-33d)

Any electromagnetic field must satisfy the four equations

Constitutive equations = εD E (6-34a) = μB H (6-34b) Conduction and convection current densities = σEJ (6-35a) v= EJ ρ (6-35b)

Page 13: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-13 Proprietary of Prof. Lee, Yeon Ho

Lorentz force equation ( )q= + ×E BF v (6-36) Continuity equation

v

t∂

∇ = −∂

iJ ρ (6-37)

Example 6-4

Show that the time-varying fields ( )coso yE kx t= − ωaE and ( )cosoo z

o

E kx tε= − ω

μaH ,

where o ok = ω μ ε , satisfy Maxwell’s equations in free space. Solution

Substituting E and H in Eq. (6-33a)

( ) ( )cos cosoo y o o z

o

E kx t E kx tt⎡ ⎤ε∂⎡ ⎤∇ × − ω = − μ − ω⎢ ⎥⎣ ⎦ ∂ μ⎢ ⎥⎣ ⎦

a a

or ( ) ( )sin sino z o o o zkE kx t E kx t− − ω = −ω μ ε − ωa a The equation is true for o ok = ω μ ε .

Substituting the fields in Eq. (6-33b), with 0=J in free space

( ) ( )cos cosoo z o o y

o

E kx t E kx tt

⎡ ⎤ε ∂ ⎡ ⎤∇ × − ω = ε − ω⎢ ⎥ ⎣ ⎦μ ∂⎢ ⎥⎣ ⎦a a

or

( ) ( )sin sinoo y o o y

o

kE kx t E kx tε

− ω = ωε − ωμa a

The equation is true for o ok = ω μ ε .

Substituting the fields in Eq. (6-33c) ( )cos 0o o yE kx t⎡ ⎤∇ = ∇ ε − ω =⎣ ⎦ai iD Gauss’s law for D is satisfied in free space of 0v =ρ .

Substituting the fields in Eq. (6-33d)

( )cos 0oo o z

o

E kx t⎡ ⎤ε

∇ μ − ω =⎢ ⎥μ⎢ ⎥⎣ ⎦ai

Gauss’s law for B is satisfied.

In the same way, it can be shown that the time-varying fields ( )coso yE kx t= + ωaE and

( )cosoo z

o

E kx tε= − + ω

μaH also satisfy Maxwell’s equations.

Page 14: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-14 Proprietary of Prof. Lee, Yeon Ho

6-3.1 Maxwell’s Equations in Integral Form The point form is useful in describing the local effect ↑

A local consequence observed at a point in space is directly related to a local source at the point.

The integral form is useful in describing the nonlocal effect ↑

A local consequence at a point is related to a source distributed in the neighborhood the point.

The integral form is easier to understand, because the steps of integration, involving increments, dot product, summation, and so on, may be more realistic than the differentiation. The point form can be converted to the integral form by the use of divergence and Stokes’s theorems, or vice versa.

Maxwell’s equations in integral form

d dt

∂= −

∂∫ ∫l sC S

i iBE (6-38a)

d dt

∂⎡ ⎤= +⎢ ⎥∂⎣ ⎦∫ ∫l sC S

i iDH J (6-38b)

vd dv=∫ ∫s

S ViD ρ (6-38c)

0d =∫ s

SiB (6-38d)

The positive direction of C is related to the direction of ds by the right-hand rule. 6-3.2 Electromagnetic Boundary Conditions

Boundary conditions for E and H

1 2t t=E E (6-39a)

1 2t t snJ− =H H (6-39b)

snJ is the surface current density normal to 1tH and 2tH .

The boundary condition for D and B 1 2n n s− =D D ρ (6-39c)

1 2n n=B B (6-39d)

sρ is the free surface charge density.

Page 15: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-15 Proprietary of Prof. Lee, Yeon Ho

At the interface between two lossless dielectrics → 0s sJ= =ρ 1 2t t=E E (6-40a) 1 2t t=H H (6-40b) 1 2n n=D D (6-40c) 1 2n n=B B (6-40d)

At the interface between a lossless dielectric(medium 1) and a perfect conductor(medium 2)

→ 0= = = =E H D B in the conductor 1 0t =E (6-41a) 1t sJ=H (6-41b) 1n s=D ρ (6-41c) 1 0n =B (6-41d) and 2 2 0t t= =E H (6-42a) 2 2 0n n= =D B (6-42b)

sρ and sJ are the direct sources of static fields D and H . sρ and sJ may not be the direct sources of time-varying fields.

An electromagnetic wave impinging on a perfect conductor → The incident and reflected waves induce time-varying sρ and sJ on the conductor surface.

Page 16: CHAPTER 6. TIME-VARYING FIELDS AND MAXWELL’S …icc.skku.ac.kr/~yeonlee/Electromagnetics/Class 6_Time... · 2012. 10. 23. · Time-Varying Fields 6-2 Proprietary of Prof. Lee, Yeon

Time-Varying Fields 6-16 Proprietary of Prof. Lee, Yeon Ho

6-4 RETARDED POTENTIALS Scalar electric potential V from a static volume charge density vρ .

Vector magnetic potential A from a steady current density J .

'

'4vdvV ρ

=πε∫V R

(6-43a)

'

'4dvμ

=π∫JA

V R (6-43b)

For time-varying vρ and J , they becomes the retarded scalar potential or the retarded vector potential.

0∇ =iB still holds for the time-varying field Time-varying vector magnetic potential is defined by

= ∇ ×B A (6-44) Inserting Eq. (6-44) in Faraday’s law

( )

t

t

∂∇ × = −

∂∂

= − ∇ ×∂

BE

A (6-45)

Rewriting it

0t

∂⎛ ⎞∇ × + =⎜ ⎟∂⎝ ⎠E

A (6-46)

We define the time-varying scalar potential V

Vt

∂+ ≡ −∇∂

EA (6-47)

or

Vt

∂= −∇ −

∂E

A (6-48)

It can reduce to the previous relation, V= −∇E , in the steady system

By substituting Eq. (6-44) in the generalized Ampere’s law

t

∂∇ ×∇ × = μ + μ

∂DA J (6-49)

Inserting Eq. (6-48) in Eq. (6-49) and using ( ) 2∇ ×∇ × = ∇ ∇ − ∇A A Ai

2

22

Vtt

∂ ∂⎛ ⎞∇ − με = −μ + ∇ ∇ + με⎜ ⎟∂∂ ⎝ ⎠iAA J A (6-50)

Lorentz condition for potentials

Vt

∂∇ = −με

∂iA (6-51)

It reduces to 0∇ =Ai in the steady system.

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Time-Varying Fields 6-17 Proprietary of Prof. Lee, Yeon Ho

From Eqs. (6-50) and (6-51), the inhomogeneous wave equation for vector potential

2

22t

∂∇ − με = −μ

∂AA J (6-52)

It reduces to vector Poisson’s equation as Eq. (5-53) in the steady system. The solution is given by a traveling wave of velocity 1/ με .

Rewriting Gauss’s law using Eq. (6-48)

( )2

v

V Vt t

∇ =ε

∂ ∂⎛ ⎞= ∇ −∇ − = −∇ − ∇⎜ ⎟∂ ∂⎝ ⎠

i

i i

E

A A

ρ

(6-53)

Using the Lorentz condition, we obtain the inhomogeneous wave equation for scalar potential

2

22

vVVt

∂∇ − με = −

ε∂ρ (6-54)

The solution of the inhomogeneous differential wave equation

( ) ( )'

/, '

4V

t vt dv

μ −=

π∫rR

R

JA (6-55)

( ) ( )'

/ ',

4v

V

t v dvV t

−=

πε∫rR

R

ρ (6-56)

which are called the retarded vector potential and the retarded scalar potential. At time t , at a distance R from the source. A and V are determined by the values of J or vρ given at an early time ( )/t v− R . ↑ Time delay /vR is the time taken by the wave during traveling the distance R .

Example 6-5 Find retarded vector potential at a point p due to an infinitesimal current element located at the origin, carrying a current cosoI I t= ω .

Solution Rewriting Eq. (6-55)

( ) ( ) ( )' '

/ /, ' '

4 4 zV L

t v I t vt dv dl

μ − μ −= =

π π∫ ∫r aR R

R R

JA (6-57)

For an infinitesimally small current element of height h , we assume R to be constant. Then, Eq. (6-57) reduces to

( )cos /4 o zh I t R vR

μ= ω −⎡ ⎤⎣ ⎦π

A a

The vector potential at a distance R is retarded by /R v with respect to I at the origin. The retardation is due to the traveling time of the electromagnetic wave from the origin to p .

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Time-Varying Fields 6-18 Proprietary of Prof. Lee, Yeon Ho