CH 7. Time-Varying Fields and Maxwell’s Equations

Chung-Ang University Field & Wave Electromagnetics

CH 7. Time-Varying Fields and Maxwell’s Equations


7.1 Introduction

Fundamentl RelationsElectrostatic

ModelMagnetostatic

Model

Governing equations

Constitutive relations

(linear and isotropic media)

D

E 0

JH

B

0

In the static case, electric field vectors and

and magnetic field vectors and form separate and independent pairs.

ED BH1

E D

B H

Fundamental Relations for Electrostatic and Magnetostatic Models

In a conducting medium, static electric and magnetic fields may both exist and

form an electromagnetostatic field.


7.1 Introduction

A static electric field in a conducting medium causes a steady current to flow that,

In turn, gives rise to a static magnetic field.

The electric field can be completely determined from the static electric charges or

potential distributions.

The magnetic field is a consequence; it does not enter into the calculation of the

electric field.

In this chapter we will see that a changing magnetic field gives rise to an electric field,

and vice versa.


7.2 Faraday’s Law of Electromagnetic Induction

Michael Faraday, in 1831, discovered experimentally that a current was induced in a

conducting loop when the magnetic flux linking the loop changed.

The quantitative relationship between the induced emf and the rate of change of flux

linkage, based on experimental observation, is known as Faraday’s law.

Fundamental Postulate for Electromagnetic Induction

t

BE (7 - 1)

Equation 7-1 expresses a point-function relationship; that is, it applies to every point in

space, whether it be in free space or in a material medium.

The electric field intensity in a region of time-varying magnetic flux density is therefore

nonconservative and cannot be expressed as the gradient of a scalar potential.



Taking the surface integral of both sides of Eq.(7-1) over an open surface and applying

Stokes’s theorem, we obtain

.sdt

BldEsd

t

BE

sc

(7 - 2)

7–2.1 A STATIONARY CIRCUIT IN A TIME-VARYING MAGNETIC FIELD

Equation (7-2) is valid for any surface S with a bounding contour C, whether or not a

physical circuit exists around C.

For a stationary circuit with a contour C and surface S, Eq(7 - 2) can be written as

.sdBdt

dldE

sc (7 - 3)



If we define c

ldE

s

sdB

= emf induced in circuit with contour C (V) (7 – 4)

= magnetic flux crossing surface S (Wb), (7 – 5)

then Eq.(7 - 3) becomes dt

d (V). (7 – 6)

Equation (7 – 6) states that the electromotive force induced in a stationary closed circuit

is equal to the negative rate of increase of the magnetic flux linking the circuit.

This is a statement of Faraday’s law of electromagnetic induction.



7-2.2 TRANSFORMERS

A transformer is an alternating-current (a-c) device that transforms voltage, currents,

and impendances.

(a) Schematic diagram of a transformer.

FIGURE 7-1

)(1tv )(

2tv

)(1ti )(

2ti

N 1 N 2RL

+

_

+

_

For the closed path in the magnetic circuit

In Fig.7-1(a) traced by magnetic flux ,

we have, from Eq.(6-101),

( Eq.(6-101), )

,2211

iNiN

. k

kkj

jj IN

(7 – 7)

.S

l

(7 – 8)

( core of length l, cross-sectional area S,

permeability )Substituting Eq.(7-8) in Eq.(7-7)

.2211

S

liNiN (7 – 9)

back



X 1 X 2

(b) An equivalent circuit.

FIGURE 7-1

Ideal transformer

R1 R2N 1 N 2i1 i2

X c Rc RLv1 v2

+

_

+

_

a) Ideal transformer.

For an ideal transformer we assume that , and

.2211

S

liNiN Eq.(7 - 9) becomes .

1

2

2

1

NN

ii

(7 – 10)



Eq.(7 - 10) states that the ratio of the currents in the primary and secondary

windings of an ideal transformer is equal to the inverse ratio of the numbers

of turns.

.1

2

2

1

NN

ii

dt

dNv

11,

22 dt

dNv

.2

1

2

1

NN

vv

Faraday’s law tells us that

(7 - 11) and (7 – 12)

From Eqs. (7 – 11) and (7 – 12) we have (7 – 13)

Thus, the ratio of the voltages across the primary and secondary windings of an ideal

transformer is equal to the turns ratio.



,

2

11 )()(

2

RNN

R Leff ,

)/(

)/(1

212

221

1

1)(iNNvNN

ivR eff

RLWhen the secondary winding is terminated in a load resistance , as shown in Fig.

7-1(a) click, the effective load seen by the source connected to primary winding is

or (7 – 14a)

For a sinusoidal source and a load impedance , it is obvious that the effective

load seen by the source is , an impedance transformation.

)(1tv Z L

ZNN L)/( 212

.

2

11 )()(

2

ZNNZ Leff

We have (7 – 14b)

b) Real transformer.

Referring to Eq. (7 – 9), we can write the magnetic flux linkages

of the primary and secondary windings as

.2211

S

liNiN



),(2211

2

111 iNNiNN l

S

).(2

2

212122 iNiNNN l

S

,212

111 dt

diLdtdiLv ,2

21

122 dtdiLdt

diLv

,2

11 NL l

S

,2

22 NL l

S

.2112 NNL l

S

(7 – 15)

(7 – 16)

Using Eqs. (7 - 15) and (7 - 16) in Eqs. (7 - 11) and (7 - 12), we obtain

(7 – 17) (7 – 18)

where (7 – 19) the self-inductance of the primary winding.

(7 – 20) the self-inductance of the secondary winding.

(7 – 21) the mutual inductance between the primary and

secondary windings.



For an ideal transformer there is no leakage flux, and .2112 LLL

,2112 LLL kFor real transformers, k < 1 , (7 -22)

Where k is called the coefficient of coupling.

For real transformers we have the following real-life conditions.

the existence of leakage flux ( k < 1 ),

noninfinite inductances, nonzero winding resistances,

the presence of hysteresis and eddy-current losses.

The nonlinear nature of the ferromagnetic core further compounds the difficulty of

an exact analysis of real transformers.



Eddy currents.

When time-varying magnetic flux flows in the ferromagnetic core, an induced emf will

result in accordance with Faraday’s law.

This induced emf will produce local currents in the conducting core normal to the

magnetic flux.

These currents are called eddy currents.

Eddy currents produce ohmic power loss and cause local heating.

This is the principle of induction heating.

In transformers, eddy-current power loss is undesirable and can be reduced by using

core materials that have high permeability but low conductivity (high and low ).

For low-frequency, high-power applications an economical way for eddy-current

power loss is to use laminated cores.



7-2.3 A MOVING CONDUCTOR IN A STATIC MAGNETIC FIELD.

☉ ☉

☉

☉

☉

☉

☉

☉

☉

☉

☉

☉

2

1B

Figure 7-2 A conducting bar moving in a magnetic field.

u

u

dl

BuqF mA force will cause the

freely movable electrons in the conductor to

drift toward one end of the conductor and

leave the other end positively charged.

This separation of the positive and negative

charges creates a Coulombian force of attr-

action.

The charge-separation process continues

until the electric and magnetic forces

balance each other and a state of equilibri-

um is reached.



BuqF m/

ldBuc

)(

.)(2

121ldBuV

'

To an observer moving with the conductor there is no apparent motion, and the magnetic force per unit charge can be interpreted as an induced electric field acting along the conductor and producing a voltage

(7 – 23)

If the moving conductor is a part of a closed circuit C, then the emf generated around the circuit is

(V). (7 – 24)

This is referred to as a flux cutting emf or a motional emf. Obviously, only the part of the circuit that moves in a direction not parallel to the magnetic flux will contribute in Eq. (7 – 24).



).( BuEqF

BuEE .BuEE

7-2.4 A MOVING CIRCUIT IN A TIME-VARYING MAGNETIC FIELD.

ldBusdt

BldE

csc

)(

Lorentz’s force equation (7 – 31)

The force on q can be interpreted as caused by an electric field , whereE

(7 – 32) (7 – 33)

Hence, when a conducting circuit with contour C and surface S moves with a velocity

in a field , we use Eq. (7 – 33) in Eq. (7 – 2) to obtain

u),( BE

.sdt

BldEsd

t

BE

sc

(V). (7 – 34)

( Eq. 7-2 )

Eq. (7 – 34) is the general form of Faraday’s law for a moving circuit in a time-varying magnetic field.

or



C1 C2

tt Let us consider a circuit with contour that moves from at time t to at time

in a changing magnetic field .B

C

B C2

C1

S 2

S1tu

ld

3Sd

FIGURE 7-5A moving circuit in a time-varying magnetic field.

The time-rate of change of magnetic flux

through the contour is

sdBdt

d

dt

ds

.)()(1

1212

0lim

sdtB

sdttB

t sst

(7 – 35)



)( ttB

.,..)(

)()( TOHtt

tBtBttB

in Eq. (7 - 35) can be expanded as a Taylor’s series:

,...1

lim2 1

120

s sTOHdBdB

tsd

t

BsdB

dt

dsstss

(7 – 36)

Substitution of Eq. (7 – 36) in Eq. (7 – 35) yields

(7 – 37)

An element of the side surface is .3

tuldds (7 – 38)

Apply the divergence theorem for at time t to the region sketched in Fig. 7 – 5 :

,312

312sss d

sBd

sBd

sBdvB

V (7 – 39)

B



0 B

.)(12

12

ldButdsBd

sB

css Using Eq. (7 – 38) in Eq. (7 – 39) and noting that , we have

,)( ldBusdt

BsdB

dt

dcss

dt

d

sdBdt

ds

ldEc

Combining Eqs. (7 – 37) and (7 – 40), we obtain

(7 – 40)

which can be identified as the negative of the right side of Eq. (7 – 34).

If we designate

emf induced in circuit C measured in the moving frame, (7 – 42)

(7 – 41)

Eq. (7 – 34) can be written simply as

(V),

(7 – 43)



dt

d

Eq. (7 – 43) is of the same form as Eq. (7 – 6).

If a circuit is not in motion, reduces to , and Eqs. (7 – 43) and (7 – 6) are exactly the same.

Faraday’s law that the emf induced in a closed circuit equals the negative time-rate of increase of the magnetic flux linking a circuit applies to a stationary circuit as well as a moving one.


7.3 Maxwell’s Equations

The fundamental postulate for electromagnetic induction assures us that a time-varying magnetic field gives rise to an electric field.

Time-varying case: 0 E .t

BE

,JH

, D .0 B

The revised set of two curl and two divergence equations from Table 7 – 1:

,t

BE

(7 – 47a) (7 – 47b)

(7 – 47c) (7 – 47d)

The mathematical expression of charge conservation is the equation of continuity :

.t

J

(7 – 48)

Divergence of Eq. (7 – 47b) : ,0)( JH (7 – 49) (null identity)

since Eq. (7 – 48) asserts that does not vanish in a time-varying situation,

Eq. (7 – 49) is, in general, not true.

J



How should Eqs. (7 – 47a, b, c, d) be modified so that they are consistent with

Eq. (7 – 48)?

First of all, a term must be added to the right side of Eq. (7 – 49) :

Using Eq. (7 – 47c) in Eq. (7 – 50), we have

which implies that

Eq. (7 – 52) indicates that a time-varying electric field will give rise to a magnetic field, even in the absence of a current flow.

The additional term is necessary to make Eq. (7 – 52) consistent with the principle of conservation of charge.

t /

.0)(t

JH

),()(t

DJH

.t

DJH

tD /

(7 – 50)

(7 – 51)

(7 – 52)

The term is called displacement current density.tD /



In order to be consistent with the equation of continuity in a time varying situation, both of the curl equations in Table 7 – 1 must be generalized.

The set of four consistent equations to replace the inconsistent equations,

Eqs. (7 – 47a, b, c, d), are

,t

DJH

, D

.0 B

,t

BE

(7 – 53a)

(7 – 53b)

(7 – 53c)

(7 – 53d)

They are known as Maxwell’s equations.



7-3.1 INTEGRAL FORM OF MAXWELL’S EQUATIONS.

The four Maxwell’s equations in (7 – 53a, b, c, d) are differential equations that are valid at every point in space.

In explaining electromagnetic phenomena in a physical environment we must deal with finite objects of specified shapes and boundaries.

It is convenient to convert the differential forms into their integral-form equivalents.

We take the surface integral of both sides of the curl equations in Eqs. (7 – 53a, b) over an open surface S with contour C and apply Stokes’s theorem to obtain

sdt

BldE

sc

.)( sdt

DJldH

sc

(7 – 54a)

(7 – 54b)



Taking the volume integral of both sides of the divergence equations in Eqs. (7 – 53c, d) over a volume V with a closed surface S and using divergence theorem, we have

dvsdDvs

.0 sdBs

(7 – 54c)

(7 – 54d)

The set of four equations in (7 – 54a, b, c, d) are the integral form of Maxwell’s

equations.



t

DJH

D

0 B

t

BE

dt

dldE

c

sdt

DIldH

sc

QsdDs

0 sdBs

Maxwell’s Equations

Differential Form Integral Form Significance

Faraday’s law

Ampere’s circuital law

Gauss’s law

No isolated magnetic charge

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CH 7. Time-Varying Fields and Maxwell’s Equations

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