maxwell equations inel 4151 ch 9 sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

Maxwell EquationsMaxwell EquationsINEL 4151 ch 9INEL 4151 ch 9

Sandra Cruz-Pol, Ph. D.Sandra Cruz-Pol, Ph. D.ECE UPRMECE UPRM

MayagMayagüüez, PRez, PR

Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM

Electricity => MagnetismElectricity => MagnetismElectricity => MagnetismElectricity => Magnetism

In 1820 Oersted discovered that a steady In 1820 Oersted discovered that a steady current produces a magnetic field while current produces a magnetic field while teaching a physics class. teaching a physics class.


Would magnetism would Would magnetism would produce electricity?produce electricity?

Eleven years later, Eleven years later, and at the same time, and at the same time, Mike Faraday in Mike Faraday in London and Joe London and Joe Henry in New York Henry in New York discovered that a discovered that a time-varying time-varying magnetic magnetic field would produce field would produce an electric current! an electric current!

dt

dNVemf

Faraday’s Law:

Len’s Law = (-)Len’s Law = (-)

If N=1 (If N=1 (1 loop1 loop)) The time changeThe time change

can refer to can refer to BB or or SS


Two cases of Two cases of

BB changes changes SS (area) changes (area) changes


BuQFE

SdBt

dlE

mm

sL

/

sL

dSt

BdlE

sS

dSt

BdSE

t

BE

Stoke’s theo.

dlBudSELS

BuE

3 cases:3 cases:

Stationary loop in Stationary loop in time-varying time-varying B B fieldfield

Moving loop Moving loop in static in static B B fieldfield

Moving loop Moving loop in in time-varying time-varying B B field field

Cruz-Pol, Electromagnetics UPRMCruz-Pol, Electromagnetics UPRM

t

BE

BuE

PE 9.1 If B=PE 9.1 If B=


ExampleExample 9.3 Circuit with 10 9.3 Circuit with 10-3-3 m m22 cross-section, cross-section, radius =10cm, radius =10cm, ii11(t)(t)=3 sin =3 sin t [A], 50 Hz, Nt [A], 50 Hz, N11=200 =200 turns, Nturns, N22= 100 , = 100 , =500=50000, find V, find Vemfemf on both coils on both coils

Find reluctance and use Faraday’s LawFind reluctance and use Faraday’s Law


Example PE 9.3 A magnetic core of uniform cross-Example PE 9.3 A magnetic core of uniform cross-section 4 cmsection 4 cm22 is connected to a 120V, 60Hz is connected to a 120V, 60Hz

generator. generator. Calculate the induced emf VCalculate the induced emf V22 in the secondary coil. in the secondary coil.

NN11= 500, N= 500, N22=300=300 Use Faraday’s LawUse Faraday’s Law

Answer; 72 cos(120Answer; 72 cos(120t) Vt) V


Some termsSome terms

EE = electric field intensity [V/m] = electric field intensity [V/m] DD = electric field = electric field densitydensity [VF/m2] [VF/m2] HH = magnetic field intensity, [A/m] = magnetic field intensity, [A/m] BB = magnetic field = magnetic field densitydensity, [Teslas], [Teslas]

Take a look at a capacitorTake a look at a capacitor

dielectricdielectric is an electrical is an electrical insulatorinsulator

d.c., C = open circuitsd.c., C = open circuits What’s the current in an open circuit?What’s the current in an open circuit?


Maxwell’s Eqs.Maxwell’s Eqs.

the equation of continuity the equation of continuity Maxwell addedMaxwell added the term to Ampere’s the term to Ampere’s

Law so that it not only works for Law so that it not only works for staticstatic conditions but also for conditions but also for time-varyingtime-varying situations. situations.

This added term is called the This added term is called the displacement displacement current densitycurrent density, , JJdd,, while while J=JJ=Jcc is the is the

conduction current. conduction current.

tJ v

t

D

For For staticstatic fields we had: fields we had:But But the the divergence of the curl of ANY divergence of the curl of ANY vector vector 00

The The continuity of the current continuity of the current requires:requires:

We need to define:We need to define:

Now when we take the divergence of curl Now when we take the divergence of curl

Therefore:Therefore:

Cruz-Pol, Electromagnetics, UPRMCruz-Pol, Electromagnetics, UPRM

JH

JJ d

0

t

J v

JH

0

dJJH

dJJH

0

This doesn’t work for time-varying fields!


Maxwell put them togetherMaxwell put them togetherIf we didn’t have the displacement If we didn’t have the displacement current,current, J Jdd, then Ampere’s Law:, then Ampere’s Law:

IIdSJdlH enc

SL

1

02

SL

dSJdlHI

S2

S1L

Idt

dQdSD

dt

ddSJdlH

SS

d

L

22

At low frequencies J>>Jd, but at radio frequencies both terms are comparable in magnitude.

For For dynamic dynamic fields we had:fields we had:

Therefore:Therefore:

Substituting in curl of HSubstituting in curl of H

This is Maxwell’s equation for Ampere’s LawThis is Maxwell’s equation for Ampere’s Law

Cruz-Pol, Electromagnetics, UPRMCruz-Pol, Electromagnetics, UPRM

t

D

t

D

tJJ v

d

t

DJ d

t

DJH


Maxwell Equations Maxwell Equations in General Form in General Form

Differential formDifferential form Integral FormIntegral FormGauss’sGauss’s Law for Law for EE field.field.

Gauss’sGauss’s Law for Law for HH field. Nonexistence field. Nonexistence of monopole of monopole

Faraday’sFaraday’s Law Law

Ampere’sAmpere’s Circuit Circuit LawLaw

vD

0 B

t

BE

t

DJH

v

v

s

dvdSD

0s

dSB

sL

dSBt

dlE

sL

dSt

DJdlH


ElectromagneticsElectromagnetics

This is the principle of This is the principle of motors, hydro-electric motors, hydro-electric generators and generators and transformers operation.transformers operation.

sL

dSt

DJdlH

*Mention some examples of em waves

This is what Oersted discovered This is what Oersted discovered accidentally:accidentally:


Phasors & complex #’sPhasors & complex #’s

Working with Working with harmonic fieldsharmonic fields is easier, but is easier, but requires knowledge of requires knowledge of phasorphasor, let’s review , let’s review

complex numberscomplex numbers and and phasorsphasors


t

DJH

t

BE

0 B

vD

Maxwell Equations Maxwell Equations for for Harmonic fieldsHarmonic fields

Differential form* Differential form*

Gauss’sGauss’s Law for E field. Law for E field.

Gauss’sGauss’s Law for H field. Law for H field. No monopoleNo monopole

Faraday’sFaraday’s Law Law

Ampere’sAmpere’s Circuit Law Circuit Law

vE

0 H

HjE

EjJH

* (substituting and )ED BH

Earth Magnetic Field Declination Earth Magnetic Field Declination from 1590 to 1990from 1590 to 1990


NewsNewsJan 2011Jan 2011The Earth's magnetic north The Earth's magnetic north pole is slowly heading toward pole is slowly heading toward Russia, according to scientists, Russia, according to scientists, but one of the places being but one of the places being affected by this is Tampa affected by this is Tampa International Airport.International Airport.

Airport officials closed its main Airport officials closed its main runway this week until Jan. 13 runway this week until Jan. 13 to adjust the taxiway signs to adjust the taxiway signs accounting for the magnetic accounting for the magnetic pole shift, pole shift, Tampa Bay Online reports.

http://www2.tbo.com/content/2011/jan/05/060831/shift-of-earths-magnetic-north-pole-impacts-tampa-/news-money/

maxwell equations inel 4151 ch 9 sandra cruz-pol, ph. d. ece uprm mayag ü ez, pr

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