10. DECONTAMINATION PROCEDURES - INEL Environmental Restoration
Maxwell Equations INEL 4151 ch 9
description
Transcript of 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)
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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
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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:
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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
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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
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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
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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
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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
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Transformer ExampleTransformer Example
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Find reluctance and use Faraday’s LawFind reluctance and use Faraday’s Law
Displacement Current, JDisplacement Current, Jdd
9.49.4
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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
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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
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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.
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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
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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
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Electric & Magnetic potentials:Electric & Magnetic potentials: We now use the We now use the Lorentz condition:Lorentz condition:
To get:To get:
and: and:
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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!
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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
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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
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Review:Review: Addition, Addition, Subtraction, Subtraction, Multiplication, Multiplication, Division, Division, Square Root, Square Root, Complex ConjugateComplex Conjugate
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For a Time-varying phaseFor a Time-varying phase
Real and imaginary parts are:Real and imaginary parts are:
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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)
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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
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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
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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.
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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
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Earth Magnetic Field Declination Earth Magnetic Field Declination from 1590 to 1990from 1590 to 1990
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