(1831-1879)

April 2, 2013

(with Time-Varying Fields)

t

DJH

t

BE

B

D

0

Differential Form Integral Form

Gauss’s Law

Faraday’s Law

Gauss’s Law for

Magnetism

Ampere’s Law

vD

t

BE

S

QSdD

Sdt

BldE

SC

0 B

0S SdB

t

DJH

Sdt

DJldH

SC

Page 256

Ampere’s Law JH

Current Produces a magnetic field

I H

H

t

DJH

Sdt

DJldH

SC

PED

0

This term is present in material media and

in free space. It doesn't necessarily involve

any actual movement of charge, but it does

have an associated magnetic field, just as

does a current due to charge motion.

This term is associated with the

polarization of the individual

molecules of the dielectric

material. Polarization results

when the charges in molecules

move a little under the influence

of an applied electric field.

t

P

t

E

t

DJD

0Displacement

Current

Displacement Field

No conduction current enters cylinder surface R, while

current I leaves through surface L. Consistency of

Ampère's law requires a displacement current ID = I to

flow across surface R

Capacitor Imaginary

cylindrical surface

Example to illustrate the physical meaning of the displacement current Id

tVtVs cos)( 0

- Assume wires are perfect conductors and capacitor is filled with perfect dielectric

tCVtVdt

dC

dt

dVCI c

C sincos 001

dC II 21

tCVtVd

A

dsytd

Vy

tsd

t

DI

Sd

sinsin

ˆcosˆ

00

02

td

Vy

d

VyED c cosˆˆ 0

Page 300

Even though the displacement current

does not carry real charge, it nonetheless

behave like a real current.

Displacement current

?

)10sin(2

1

/102

9

7

d

c

r

I

mAtI

mS

Solution:

t

EAAJI dd

tAA

t

A

IE c

sin

10

102

sin102 10

7

3

ttId cos10885.0cos10 1210

Wire Ic

EAJAIc

Page 300

Field

Components

General Form Medium 1: Dielectric

Medium 2: Dielectric

Medium1: Dielectric

Medium 2: Conductor

Tangential E

Normal D

Tangential H

Normal B

0)(ˆ212 EEn

sDDn )(ˆ212

sJHHn

)(ˆ212

0)(ˆ212 BBn

tt EE 21

snn DD 21

tt HH 21

nn BB 21

021 tt EE

snD 102 nD

st JH 1 02 tH

021 nn BB

The boundary conditions derived previously for electrostatics

and magnetostatics remain valid for time-varying fields as well

Pages 301-303

s The surface charge density at the boundary;

Normal components of all fields are along , the outward unit vector of medium 2. 2n̂

Direction of Js is orthogonal to (H1-H2).

Surface current density at the boundary sJ

Implies that the tangential components are equal in magnitude and

Parallel in direction tt EE 21

Medium 1

Medium 2 2n̂

1E

tE1

nE1

2E

tE2

nE2

ED

Under static conditions, the charge density ρV and the current density J

at a given point in a material are totally independent of one another.

dvdt

ddv

dt

d

dt

dQI

v

v

vv

Net positive charge within v

Net current flowing across S out of v

vv

dvJsdJI

tJ v

Page 301

tJ v

0 J

0

t

Kirchhoff’s Current Law

0i

iI

Page 302

Charge-Current Continuity Relation

tJ v

tE v

0

v

v

dt

EJ

/vE

Relaxation time constant

rt

v

t

vv eet /)/(

00)(

Solution

t= 0 r r3

decay

t)(

ovof

%100

Copper

Mica

mS

mF

/108.5

/10854.8

7

12

0

mS /10

6

15

0

s

r

19105.1

s

r

4103.5

(Excess charges in a point within the material and how fast it decays)

Page 303

r

ovof

%37

ovof

%5

0)(&0)( AV

Dynamic Case Static Case 0/ t

ABB

VEE

0

0

0

B

t

BE

)( At

E

Vt

AE

)(

Relation between E & B and V & A ?

Page 303

t

A

Create an additional E field

'E

AB

t

AVE

0)(

t

AE

vdR

RRV

v

iv

)(

4

1)(

Static field

Retarded scalar potential

vdR

tRtRV

v

iv

),(

4

1),(

Dynamic field with no retardation

vdR

uRtRtRV

v

piv

)/,(

4

1),(

vdR

uRtRJtRA

v

pi

)/,(

4),(

Similarly:

Retarded vector potential

Page 304

vdR

uRtRtRV

v

piv

)/,(

4

1),(

vdR

uRtRJtRA

v

pi

)/,(

4),(

V and A are linearly dependent

on ρv and J, respectively.

AB

t

AVE

E and B are linearly dependent

on V and A.

tJ v

ρ and J have the same functional

dependence on time

V, A, E, D, B and H have the same functional dependence on time and the relationships

interconnecting all these quantities obey the rules of linear systems. We can use

sinusoidal function to determine the response of the system due to the source with any

type of time dependence Page 305

(Steady-State Sinusoidal Time Dependence)

Time harmonic responses of the retarded scalar and vector potentials

Suppose: tRtR iviv cos)(),(

tj

iviv eRetR )(~),(

In phasor notation

)/()(~)/,( puRtj

ivpiv eReuRtR

For retarded charge density:

tjRjk

ivpiv eeReuRtR )(~)/,(

Page 306

puk

Wavenumber

vdeR

eReeRVetRV tj

v

Rjk

ivtj

)(~

4

1)(

~),(

vdR

eRRV

v

Rjk

iv

)(~

4

1)(

~

vdR

eRJRA

v

Rjk

i

)(

~

4

1)(

~

Similarly:

AH~1~

Page 306

tjRjk

ivpiv eeReuRtR )(~)/,(

Ej

HorHjE

Hj

EorEjH

~1~~~

~1~~~

Page 307

t

DJH

t

BE

B

D

0

In a nonconducting medium (J=0)

tj

tj

eREetRE

eRHetRH

)(~

),(

)(~

),(

vdR

eRRV

v

Rjk

iv

)(~

4

1)(

~

vdR

eRJRA

v

Rjk

i

)(

~

4

1)(

~

AH~1~

Ej

HorHjE

Hj

EorEjH

~1~~~

~1~~~

Pages 305-307

VE~~

tjeRVetRV )(~

),(

tjeRAetRA )(~

),(

)10sin(10ˆ),( 1 0 k ztxtzE

?& kH jkzejxzE 10ˆ)(

~Phasor form

jkz

jkz

ek

jy

ej

zyx

zyx

j

Ej

zH

10ˆ

0010

///

ˆˆˆ1

~1)(

~

jexjexeextzE

eeeexkztxtzE

jkzjkzj

jkz

tjj

jkz

10ˆ)2

sin()2

cos(10ˆ10ˆ),(~

10ˆ)2

10cos(10ˆ),(

2

10210 10

)&16( 00

jkzek

jxHj

zE 2

210ˆ

~1~

jkzek

jyzH

10ˆ)(

~

jkzejxzE 10ˆ)(~

k=?

22 k

)/(133103

10444

8

10

00 mradc

k

ztyeek

jyeezHetzH tjjkztj 13310sin11.0ˆ10

ˆ)(~

),( 10

2

21010

k