Maxwell Equations INEL 4151 ch 9

ElectromagnetismElectromagnetismINEL 4151INEL 4151

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

MayagMayagüüez, PRez, PR

In summaryIn summary Stationary ChargesStationary Charges

QQ Steady currentsSteady currents

II Time-varying Time-varying

currentscurrents I(t)I(t)

Electrostatic fields\Electrostatic fields\EE

Magnetostatic fieldsMagnetostatic fields HH

Electromagnetic Electromagnetic (waves!)(waves!)E(t)E(t) & & H(t)H(t)

Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM

OutlineOutline Faraday’s Law & Origin of ElectromagneticsFaraday’s Law & Origin of Electromagnetics Transformer and Motional EMFTransformer and Motional EMF Displacement Current & Maxwell EquationsDisplacement Current & Maxwell Equations Review: Phasors Review: Phasors and Time Harmonic fieldsand Time Harmonic fields

Faraday’s LawFaraday’s Law9.29.2


Electricity => 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.

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


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 & (Joe) Henry London & (Joe) Henry in New York 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!

dtdNVemf

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



Electromagnetics was born!Electromagnetics was born! This is Faraday’s Law -This is Faraday’s Law -

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

*Mention some examples of em waves

Faraday’s LawFaraday’s Law For For NN=1 and =1 and BB=0=0


dtdNVemf

Example PE 9.3 Example PE 9.3 A magnetic core of uniform cross-section 4 A magnetic core of uniform cross-section 4 cmcm22 is connected to a 120V, 60Hz generator. is connected to a 120V, 60Hz generator. Calculate the induced emf Calculate the induced emf VV22 in the secondary coil. in the secondary coil.NN11= 500, = 500, NN22=300=300

Use Faraday’s LawUse Faraday’s Law

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

Transformer & Motional Transformer & Motional EMFEMF

9.39.3

Two cases of Two cases of BB changes changes SS (area) changes (area) changes


Stoke’s theorem

dlBudSELS

BuE

Three (3) cases:Three (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


BuE

ExampleExample

+V 1

__

+V 2

_

x

y

S= 0.5 m2

R1=300

R2=200

The resistors are in parallel, but V2≠V1

PE 9.1PE 9.1

VVemfemf variation with S variation with S https://https://

www.youtube.com/www.youtube.com/watch?v=i-j-watch?v=i-j-1j2gD28&feature=relat1j2gD28&feature=relateded


Transformer ExampleTransformer Example


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

Displacement Current, JDisplacement Current, Jdd

9.49.4


Maxwell noticed something Maxwell noticed something was missing…was missing…

And added And added JJdd, the , the displacement currentdisplacement current

IIdSJdlH encSL

1

02

SL

dSJdlHI

S2

S1

L

IdtdQdSD

dtddSJdlH

SSd

L

22

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

Maxwell’s Equation Maxwell’s Equation in Final Formin Final Form

9.49.4


Summary of TermsSummary of Terms E E = electric field intensity [V/m]= electric field intensity [V/m] DD = electric field density = electric field density [C/m[C/m22]] HH = magnetic field intensity, [A/m] = magnetic field intensity, [A/m] B B = magnetic field density, [Teslas]= magnetic field density, [Teslas] J J = current density [A/m= current density [A/m22]]

Maxwell Equations Maxwell Equations in General Form in General Form

Differential formDifferential form Integral FormIntegral FormGaussGauss’’ss Law Law for for EE field.field.

GaussGauss’’ss Law Law for for HH field. Nonexistence field. Nonexistence of monopole of monopole FaradayFaraday’’ss LawLaw

AmpereAmpere’’ss Circuit Circuit LawLaw

vD

0 B

tBE

tDJH

v

vs

dvdSD

0s

dSB

sL

dSBt

dlE

sL

dStDJdlH


MaxwellMaxwell’’s Eqs.s Eqs. Also the equation of continuityAlso the equation of continuity

Maxwell addedMaxwell added the term to Ampere the term to Ampere’’s 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, while , while JJ is the conduction is the conduction current.current.

tJ v

tD

Relations & B.C.Relations & B.C.


Time Varying Time Varying PotentialsPotentials

9.69.6

We had definedWe had defined Electric & Magnetic potentials:Electric & Magnetic potentials:

Related to B as:Related to B as:Substituting into Faraday’s law:Substituting into Faraday’s law:

Identity: the curl of the gradient of Identity: the curl of the gradient of a scalar = zero.. Choose Va scalar = zero.. Choose V

Electric & Magnetic potentials:Electric & Magnetic potentials: If we take the divergence of If we take the divergence of EE::

Or Or

Taking the curl of: & add Ampere’sTaking the curl of: & add Ampere’swe getwe get


Electric & Magnetic potentials:Electric & Magnetic potentials: If we apply this If we apply this vector identityvector identity

We end up with We end up with


Electric & Magnetic potentials:Electric & Magnetic potentials: We now use the We now use the Lorentz condition:Lorentz condition:

To get:To get:

and: and:


Which are both wave equations.

Who was NikolaTesla?Who was NikolaTesla? Find out what inventions he madeFind out what inventions he made His relation to Thomas EdisonHis relation to Thomas Edison Why is he not well know?Why is he not well know?

Time Harmonic Time Harmonic FieldsFields

Phasors ReviewPhasors Review

9.79.7

Time Harmonic FieldsTime Harmonic Fields DefinitionDefinition: is a field that varies periodically : is a field that varies periodically

with time.with time.Example: A sinusoidExample: A sinusoid

Let’s review Phasors!Let’s review Phasors!



Phasors & complex #Phasors & complex #’’ssWorking with Working with harmonic fieldsharmonic fields is easier, but is easier, but

requires knowledge of requires knowledge of phasorphasor, let, let’’s review s review complex numberscomplex numbers and and phasorsphasors


COMPLEX NUMBERS:COMPLEX NUMBERS: Given a complex number Given a complex number zz

wherewhere

sincos jrrrrejyxz j

magnitude theis || 22 yxzr

angle theis tan 1

xy


Review:Review: Addition, Addition, Subtraction, Subtraction, Multiplication, Multiplication, Division, Division, Square Root, Square Root, Complex ConjugateComplex Conjugate


For a Time-varying phaseFor a Time-varying phase

Real and imaginary parts are:Real and imaginary parts are:


PHASORSPHASORS For a sinusoidal current For a sinusoidal current equals the real part of equals the real part of tjj

o eeI

joeI

tje

sI

The complex term which results from The complex term which results from dropping the time factor is called the dropping the time factor is called the phasor current, denoted by (phasor current, denoted by (s comes from sinusoidal)


To To changechange back to back to time time domaindomain

The phasor is The phasor is 1.1.multiplied by the time factor, multiplied by the time factor, e e jjtt, , 2.2.and taken the real part.and taken the real part.

}Re{ tjseAA


Advantages of Advantages of phasorsphasors TimeTime derivativederivative in time is equivalent to in time is equivalent to

multiplying its phasor by multiplying its phasor by jj

TimeTime integralintegral is equivalent to dividing by is equivalent to dividing by the same term.the same term.

sAjtA

jA

tA s

Time Harmonic Time Harmonic FieldsFields

9.79.7


Time-Harmonic fields Time-Harmonic fields (sines and cosines)(sines and cosines)

The wave equation can be derived from The wave equation can be derived from Maxwell equations, indicating that the Maxwell equations, indicating that the changes in the fields behave as a wave, changes in the fields behave as a wave, called an called an electromagneticelectromagnetic wave or field. wave or field.

Since any periodic wave can be Since any periodic wave can be represented represented as a sumas a sum of sines and cosines of sines and cosines (using Fourier), then we can deal only with (using Fourier), then we can deal only with harmonic fields to simplify the equations.harmonic fields to simplify the equations.


tDJH

tBE

0 B

vD

Maxwell Equations Maxwell Equations for Harmonic fields for Harmonic fields

(phasors)(phasors)Differential form* Differential form*

GaussGauss’’ss Law for E field. Law for E field.

GaussGauss’’ss Law for H field. Law for H field. No monopoleNo monopole

FaradayFaraday’’ss Law Law

AmpereAmpere’’ss Circuit Law Circuit Law

vE

0 H

HjE

* (substituting and )ED BH

ExampleExampleUse Maxwell equations:Use Maxwell equations:In Phasor formIn Phasor form

In time-domainIn time-domain


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


Maxwell Equations INEL 4151 ch 9

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