EE 307 Chapter 9 - Maxwells Eqns

8/14/2019 EE 307 Chapter 9 - Maxwells Eqns

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Chapter 9: Maxwells Equations

Topics Covered Faradays Law

Transformer and Motional

Electromotive Forces

Displacement Current

Magnetization in Materials Maxwells Equations in Final

Form

Time Varying Potentials

(Optional)

Time Harmonic Fields (Optional)

Homework: 3, 7, 9, 12, 13, 16,

18, 21, 22, 30, 33

All figures taken from primary textbook unless otherwise cited.


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Faradays Law (1)

We have introduced several methods of examining magnetic fields in terms of forces,energy, and inductances.

Magnetic fields appear to be a direct result of charge moving through a system and

demonstrate extremely similar field solutions for multipoles, and boundary condition

problems.

So is it not logical to attempt to model a magnetic field in terms of an electric one? This is

the question asked by Michael Faraday and Joseph Henry in 1831. The result is Faradays

Law for induced emf

Induced electromotive force (emf) (in volts) in any closed circuit is equal to the time rate of

change of magnetic flux by the circuit

where, as before, is the flux linkage, is the magnetic flux, N is the number of turns in the

inductor, and t represents a time interval. The negative sign shows that the induced voltageacts to oppose the flux producing it.

The statement in blue above is known as Lenzs Law: the induced voltage acts to oppose the

flux producing it.

Examples of emf generated electric fields: electric generators, batteries, thermocouples, fuel

cells, photovoltaic cells, transformers.

dt

dN

dt

dVemf


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Faradays Law (2)

To elaborate on emf, lets consider a battery circuit. The electrochemical action within the battery results and in emf produced electric field, Ef

Acuminated charges at the terminals provide an electrostatic field Eethat also exist that

counteracts the emf generated potential

The total emf generated in the between the two open terminals in the battery is therefore

Note the following important facts An electrostatic field cannot maintain a steady current in a close circuit since

An emf-produced field is nonconservative

Except in electrostatics, voltage and potential differences are usually not equivalent

IRldEldEV

P

N

e

P

N

femf

L

e IRldE 0

P

Nf

Lf

L

ef

ldEldEldE

EEE

0


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Electromotive Forces (1)

For a single circuit of 1 turn

The variation of flux with time may be caused by three ways

1. Having a stationary loop in a time-varyingB field

2. Having a time-varying loop in a static Bfield

3. Having a time-varying loop in a time-varying Bfield

A stationary loop in a time-varyingB field

SL

emf

emf

SdBdt

dldEV

dt

d

dt

dV

dt

BdE

SdBdt

dSdEldEV

SSL

emf

One of Maxwells for time varying fields


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A time-varying loop in a static Bfield

BuE

ldBuSdE

TheoremsStokesby

uBlV

IlBF

BIlF

ldBuldEV

BuQ

FE

fieldEmotionalain

BuQF

m

LL

m

emf

m

m

LLemf

m

m

_'_

____

Some care must be used when applying this equation

1. The integral of presented is zero in the portion of the

loop where u=0. Thus dl is taken along the portion

of the lop that is cutting the field where u is not

equal to zero

2. The direction of the induced field is the same as that

of Em. The limits of the integral are selected in the

direction opposite of the induced current, thereby

satisfying Lenzs Law




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A time-varying loop in a time-varying Bfield

Budt

BdE

ldBuSddt

BdldEV

m

LSL

emf

One of Maxwells for time varying fields




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Conducting element is stationary and themagnetic field varies with time

Assume the bar is held stationary at y =0.08 m

and B= 4cos(106t)azmWb/m2

Assume the length between the two conducting

rails the bar slides along is 0.06 m


Electromotive Forces: Example1

dt

BdE

Sddt

BdV

m

S

emf

Vt

t

txy

dxdyt

SdatdtdSd

dtBdV

S

S

z

S

emf

)10sin(2.19)10sin()10)(10)(4(06.008.0

)10sin()10)(10)(4(

)10sin()10)(10)(4(

))10cos(004.0(

6

663

663

663

6


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Conductor moves at a velocity u= 20aym/s inconstant magnetic field B=4azmWb/m2

Assume the length between the two

conducting rails the bar slides along is 0.06 m

mV

xdx

adxaaV

BuE

ldBuldEV

x

L

zyemf

m

LL

emf

8.406.008.0

08.008.0

004.020


Electromotive Forces: Example 2


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Conductor moves at a velocity u= 20aym/s in timevarying magnetic field B=4cos(106t-y)azmWb/m2

Assume the length between the two conducting

rails the bar slides along is 0.06 m


Electromotive Forces: Example 3

L

xzy

z

S

zemf

LSL

emf

adxayta

adxdyaytdt

dV

ldBuSddt

Bd

ldEV

)10cos()4)(10(20

)10cos()4)(10(

63

63

)10cos(240)10cos(240

)10cos(4000)10cos(4000

)10cos()4(10)10cos()10(8)4)(10(

)10cos()4)(10(20

)10cos()4(10)10cos()4)(10(

)10cos()4)(10(20

)10cos()4(10)10cos()4)(10(

)10cos()4)(10(20

)10sin(10)4)(10(

66

66

63623

63

6363

63

6363

63

663

tytV

txytxV

xtxytV

xyt

xtxytV

dxyt

xtxytV

dxyt

adxdyaytV

emf

emf

emf

emf

emf

z

S

zemf


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Lets now examine time dependent fields from the perspective on Amperes Law.

t

DJH

t

DJ

t

DD

ttJJ

JJH

JJH

tJ

JH

JH

d

v

d

d

d

v

0

0

0

Another of Maxwells for time varying fields

This one relates Magnetic Field Intensity to conduction

and displacement current densities

Displacement Current (1)

This vector identity for the cross product is mathematically

valid. However, it requires that the continuity eqn. equals

zero, which is not valid from an electrostatics standpoint!

Thus, lets add an additional current density termto balance the electrostatic field requirement

We can now define the displacement current density as

the time derivative of the displacement vector


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Using our understanding of conduction and displacement current density. Lets test thistheory on the simple case of a capacitive element in a simple electronic circuit.

Idt

dQSdD

tSdJldH

ISdJldH

IISdJldH

Sdt

DSdJI

t

DJH

SS

d

L

enc

SL

enc

SL

d

22

2

1

0

If J =0 on the second surface then Jd must be

generated on the second surface to create a time

displaced current equal to current on surface 1


Amperes circuit law to a closed path provides the following eqn.

for current on the first side of the capacitive element

However surface 2 is the opposite side of the capacitor and has no

conduction current allowing for no enclosed current at surface 2

Based on the equation for displacement current density, we can

define the displacement current in a circuit as shown


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Show that Ienc on surface 1 and dQ/dt on surface 2of the capacitor are both equal to C(dV/dt)

dt

dVC

dt

dV

d

S

dt

dES

dt

dDS

dt

dS

dt

dQI

surfacefrom

dt

dVCdt

dV

d

SSJI

dt

dV

dt

DJ

d

VED

sc

dd

d

1__



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It was James Clark Maxwell that put all of this together and reduced electromagnetic field

theory to 4 simple equations. It was only through this clarification that the discovery of

electromagnetic waves were discovered and the theory of light was developed.

The equations Maxwell is credited with to completely describe any electromagnetic field

(either statically or dynamically) are written as:

Maxwells Time Dependent Equations

Differential Form Integral Form Remarks

Gausss Law

Nonexistence of the

Magnetic Monopole

Faradays Law

Amperes Circuit Law

t

DJH

t

BE

B

D v

0

0 SdBS

Sdt

DJldH

SL

SL

SdBt

ldE

dvSdDv

S


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A few other key equations that are routinely used are listed over the next couple of slides

Maxwells Time Dependent Equations (2)

0

0

12

21

21

21

n

sn

n

n

aBB

aDD

KaHH

aEE

Continuity Equation

Compatibility Equations

Boundary Conditions for Perfect Conductor

Boundary Conditions

Equilibrium Equations

0E

m

m

Jt

BE

B

t

DJH

Dv

m= free magnetic density

0J

0H

0nB

0tE

tJ

v

Lorentz Force Law

BuEQF


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Time Varying Potentials

Jt

A

A

t

VV

ApotentialsforConditionLorentzApply

t

VA

gchoobyconditionsfieldvectortheLimit

t

A

t

VJAA

yields

AAA

identityvectortheApplying

t

A

t

VJA

t

AV

tJ

t

EJA

dtDdJABH

LawCircuitsAmpereApplying

v

2

22

2

22

2

22

2

2

2

0:____

:sin______

:

:___

11

:__'_

At

VE

t

AVE

Vt

AE

t

AE

Att

BE

LawsFaradayApplying

AB

AfromBofDefinition

R

dvJA

RdvV

potentialsField

v

v

v

v

2

0

:_'_

:____

4

4

:_


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Wave Equation

00

00

2

22

2

22

2

22

2

22

1

1

___

u

cn

c

u

t

BB

t

E

E

yieldsspacefreeIn

Jt

AA

t

V

V v

Refractive index

Speed of the wave in a medium

Speed of light in a vacuum

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EE 307 Chapter 9 - Maxwells Eqns

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