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Outline Faraday’s Law & Origin of emag Maxwell Equations explain waves Phasors and Time Harmonic fields Maxwell eqs for time-harmonic fields Maxwell eqs for time-harmonic fields
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ElectromagnetismElectromagnetismINEL 4152 CH 9INEL 4152 CH 9
Sandra Cruz-Pol, Ph. D.Sandra Cruz-Pol, Ph. D.ECE UPRMECE UPRM
MayagMayagüüez, PRez, PR
In summaryIn summary Stationary ChargesStationary Charges
Steady currentsSteady currents
Time-varying Time-varying currentscurrents
Electrostatic fieldsElectrostatic fields
Magnetostatic fieldsMagnetostatic fields
Electromagnetic Electromagnetic (waves!)(waves!)
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
OutlineOutline Faraday’s Law & Origin of emagFaraday’s Law & Origin of emag Maxwell Equations explain Maxwell Equations explain waveswaves Phasors Phasors and Time Harmonic fieldsand Time Harmonic fields
Maxwell eqs for time-harmonic fieldsMaxwell eqs for time-harmonic fields
Faraday’s LawFaraday’s Law9.29.2
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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:
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
dtdNVemf
Transformer & Motional Transformer & Motional EMFEMF
9.39.3
Three ways B can vary by Three ways B can vary by having…having…
1.1. A stationary loop in a t-varying B fieldA stationary loop in a t-varying B field2.2. A t-varying loop area in a static B fieldA t-varying loop area in a static B field3.3. A t-varying loop area in a t-varying B fieldA t-varying loop area in a t-varying B field
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
1. Stationary loop in 1. Stationary loop in a time-varying B fielda time-varying B field
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
2. Time-varying loop area 2. Time-varying loop area in a static B fieldin a static B field
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
3. A t-varying loop area in 3. A t-varying loop area in a t-varying B fielda t-varying B field
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Transformer ExampleTransformer Example
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Displacement Current, Displacement Current, JJdd9.49.4
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Summary of TermsSummary of Terms E E = electric field intensity [V/m]= electric field intensity [V/m] DD = electric field density = electric field density 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]]
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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.
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Time Varying Time Varying PotentialsPotentials
9.69.6
We had definedWe had defined Electric Scalar & Magnetic Vector potentials:Electric Scalar & Magnetic Vector potentials:
Related to B as:Related to B as: To find out what happens for time-varying fieldsTo find out what happens for time-varying fieldsSubstitute into Faraday’s law:Substitute into Faraday’s law:
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Electric & Magnetic potentials:Electric & Magnetic potentials: If we take the divergence of If we take the divergence of EE::
We have:We have:
Taking the curl of: & add Ampere’sTaking the curl of: & add Ampere’swe getwe get
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Electric & Magnetic potentials:Electric & Magnetic potentials: We use the We use the Lorentz condition:Lorentz condition:
To get:To get:
and: and:
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Which are both wave equations.
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. Ex. SinusoidEx. Sinusoid
Let’s review Phasors!Let’s review Phasors!
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Review:Review: Addition, Addition, Subtraction, Subtraction, Multiplication, Multiplication, Division, Division, Square Root, Square Root, Complex ConjugateComplex Conjugate
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
For a Time-varying phaseFor a Time-varying phase
Real and imaginary parts are:Real and imaginary parts are:
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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 dropping the time factor is called the is called the phasor current, denoted by (phasor current, denoted by (s comes from sinusoidal)
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
How to How to changechange back from back from Phasor to tPhasor to time domainime domain
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
Time Harmonic Time Harmonic FieldsFields
9.79.7
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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.
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
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
Ex. Given E, find HEx. Given E, find H EE = E = Eoo cos( cos(t-t-z) z) aaxx
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
Ex. 9.23Ex. 9.23 In free space, In free space,
Find Find k, Jk, Jdd and and H using phasors and H using phasors and Maxwells eqs. Recall:Maxwells eqs. Recall:
Cruz-Pol, Electromagnetics Cruz-Pol, Electromagnetics UPRMUPRM
mVkztE /)10cos(50 8