Lecture: Maxwell’s Equations
Maxwell’s Equations A dynamical theory of the electromagnetic field James Clerk Maxwell, F R S Philosophical Transactions of the Royal Society of London, 1865 155, 459-512,
Maxwell’s Equations - Rutgers ECE
4 1 Maxwell’s Equations The next simplest form of the constitutive relations is for simple homogeneous isotropic dielectric and for magnetic materials: D =E B =μH (1 3 4) These are typically valid at low frequencies The permittivity and permeability μ are related to the electric and magnetic susceptibilities of the material as follows
Maxwell’s Equations - University of Delaware
Maxwell’s Equations Vector Theorems Constitutive Relationships 2 Basic Theory Generalized Current Derivation of Poynting’s Theorem 3 The Frequency Domain Phasors and Maxwell’s Equations Complex Power Boundary Conditions D S Weile Maxwell’s Equations
Chapter 13 Maxwell’s Equations and Electromagnetic Waves
Collectively they are known as Maxwell’s equations The above equations may also be written in differential forms as 0 000 0 t t ρ ε µµε ∇⋅ = ∂ ∇× =− ∂ ∇⋅ = ∂ ∇× = + ∂ E B E B E BJ G G G G G GG (13 3 1) where ρand are the free charge and the conduction current densities, respectively In
The Mathematical Theory of Maxwell’s Equations
1 1 Maxwell’s Equations Electromagnetic wave propagation is described by particular equations relating five vector fields E, D, H, B, J and the scalar field ρ, where E and D denote the electric field (in V/m) and electric displacement (in As/m2) respectively, while H and B denote the
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equations or partial di erential equations where a whole sleuth of mathematical methods and numerical methods can be deployed Therefore, it is prudent to derive the di erential operator form of Maxwell’s equations 2 1 Gauss’s Divergence Theorem The divergence theorem is one of the most important theorems in vector calculus [31{34]
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The Uniqueness of Maxwells Equations - WTAMU
The short answer is that Maxwell's equations are neither redundant nor over-specified because only six of Maxwell's equations are dynamical The other two can be thought of as initial conditions Note that although not typically written down explicitly as part of Maxwell's equations, boundary conditions are also considered part of the system
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Lecture:
at Jefferson Laboratory, January 15-26th2018 FMarhauser
day, January 1 , 2018This Lecture
ˉThis lecture provides theoretical basics useful for follow-up lectures on resonators and waveguidesSources of electromagnetic fields
Some clarifications on all four equations
Time-varying fields AEwave equation
Example: Plane wave
ˉPhase and Group Velocity
ˉWave impedance
2A dynamical theory of the electromagnetic field
James Clerk Maxwell, F. R. S.
Philosophical Transactions of the Royal Society of London, 1865 155, 459-512, published 1 January 1865 -Originally there were 20 equationsSources of Electromagnetic Fields
5ˉElectromagnetic fields arise from 2 sources:
Electrical charge (Q)
Electrical current (ܫ
to quantify the effects of fields: ௌelectric current density -total electric current per unit area S (or ܫൌௌԦܬȉ݀ԦܵStationary charge creates electric field
Moving charge creates magnetic field
ˉIf either the magnetic or electrical fields vary in time, both fields are 6DifferentialForm
D= electric flux density/displacement field (Unit: As/m2)E= electric field intensity (Unit: V/m)
ʌ= electric charge density (As/m3)
H= magnetic field intensity (Unit: A/m)
B= magnetic flux density (Unit: Tesla=Vs/m2)
J= electric current density (A/m2)
Ɋ=permeability of free space
or orGauss's law
Gauss's law for magnetism
Ampğre's law
Faraday's law of induction
(1) (2) (3) (4) form the basic of the classic electromagnetismLorentz ForceDiǀergence (Gauss') Theorem
7 outwardfluxofvectorfield(Ԧܨ {divCurl (Stokes') Theorem
8Green's Theorem
{curlIntegralofcurlofvectorfield(Ԧܨ
lineintegralofvectorfield(ԦܨEdžample͗ Curl (Stokes') Theorem
9Integralofcurlofvectorfield(Ԧܨ
lineintegralofvectorfield(ԦܨExample: Curl (Stokes) Theorem
10 Example: Closed line integrals of various vector fields {curlIntegralofcurlofvectorfield(Ԧܨ
lineintegralofvectorfield(ԦܨNo curlSome curlStronger curl
No net curl
11DifferentialFormIntegralForm
D= electric flux density/displacement field (Unit: As/m2)E= electric field intensity (Unit: V/m)
H= magnetic field intensity (Unit: A/m)
B= magnetic flux density (Unit: Tesla=Vs/m2)
J= electric current density (A/m2)
Gauss' theorem
Stokes' theorem
Ɋ=permeability of free space
Gauss's law
Gauss's law for magnetism
Ampğre's law
Faraday's law of induction
ʌ= electric charge density (C/m3=As/m3)
121. Uniform field
Electric Flux & 1stMaxwell Equation
-angle between field and normal vector to surface mattersGauss: Integration over closedsurface
2. Non-Uniform field
Example: Metallic plate,
assume only surface charges on one sideDefinition of Electric Flux
13Gauss: Integration over closedsurface
Example: Capacitor
Electric Flux & 1stMaxwell Equation
1. Uniform field
-angle between field and normal vector to surface matters2. Non-Uniform field
Definition of Electric Flux
14Integration of over closed spherical surface S
Examples of non-uniform fields
Point charge Q
Principle of Superposition holds:
Electric Flux & 1stMaxwell Equation
pointing out radiallyAdd charges
15Uniform field
Magnetic Flux & 2ndMaxwell Equation
Gauss: Integration over closedsurface
Non-Uniform field
Definition of Magnetic Flux
-There are no magnetic monopoles -All magnetic field lines form loopsClosed surface:
Flux lines out = flux lines in
What about this case?
Flux lines out > flux lines in ?
-No. In violation of 2ndMadžwell's law, i.e. integration over closed surface, no holes allowed -Also: One cannot split magnets into separate poles, i.e. there always will be aNorth and South pole
16Magnetic Flux & 3rdMaxwell Equation
Faraday's law of induction
If integration path is not changing in time
-Change of magnetic flux induces an electric field along a closed loop -Note: Integral of electrical field over closed loop may be non-zero, when induced by a time-varying magnetic field -Electromotive force (EMF) ם charge traveling once around loop 17 -Change of magnetic flux induces an electric field along a closed loopMagnetic Flux & 3rdMaxwell Equation
-Electromotive force (EMF) ם -Note: Integral of electrical field over closed loop may be non-zero, when induced by a time-varying magnetic fieldIf integration path is not changing in time
charge traveling once around loop -or voltage measured at end of open loopFaraday's law of induction
18 Ampère's (circuital) Law or 4thMaxwell Equation -Note that ௌԦܬȉ݀Ԧܵ haǀe arbitrary shape as long as эS is its closed boundary -What if there is a capacitor? -While current is still be flowing (charging capacitor): tangential to a circle at any radius r of integration {conduction current IRight hand side of equation:
Left hand side of equation:
19 Ampère's (circuital) Law or 4thMaxwell Equation {displacement current I -But one may also place integration surface Sbetween plates AEcurrent does not flow through surface here -This is when the displacement field is required as a corrective 2ndsource term for the magnetic fields tangential to a circle at any radius r of integration ; Gauss's law {conduction current ILeft hand side of equation:
20 conduction current displacement current -In resistive materials the current density Jis proportional to the electric field =1/the electric resistivity (ё·m) -Generally (ʘ, T) is a function of frequency and temperaturePresence of Resistive Material
21-We can derive a wave equation:
Time-Varying E-Field in Free Space
;Faraday's law of induction ; || curl ; Ampğre's law ; Gauss's lawǢԦܬൌܧ ; we presumed no charge -Assume charge-free, homogeneous, linear, and isotropic mediumHomogeneous wave equation
22Time-Varying B-Field in Free Space
; Ampğre's law ; Faraday's law ;no moving charge (Ԧܬ -We can derive a wave equation: -Assume charge-free, homogeneous, linear, and isotropic medium ; || curl 23Time-Harmonic Fields
Example: Plane Wave in Free Space
-kis a wave vector pointing in direction of wave propagation -Wave is unconstrained in plane orthogonal to wave direction, i.e. has surfaces of constant phase (wavefronts), wave vector kis perpendicular to the wavefront -One may align propagation of wave (k) with z-direction, which simplifies the equation -Magnitude of field (whether it is Eor B) is constant everywhere on plane, but varies with time and in direction of propagation -We know speed of light in linear medium:Example: Plane Wave in Free Space
Wikipedia CC BY-SA 2.0
26Example: Plane Wave in Free Space
kis the wavenumber [1/m]Phase velocity
Group velocity
-Acknowledging that kis generally a vector: ੦మ, i.e. a dependency with the angular frequency, we can denote the relation of kwith the wavelength = Phase velocity = speed of light 27Wave Impedance
-Similarly for the magnetic field considering -All field components are orthogonal to propagation direction AEthis means that the plane wave is a Transverse-Electric-Magnetic (TEM) wave -We then can find for the electrical field components considering -Considering the absence of charges in free space and 4thMaxwell equation, we find: -Furthermore for plane wave, due to 3rdMaxwell equation we know that magnetic field is orthogonal to electrical field and can derive for time-harmonic field: 28Wave Impedance
-We obtained two sets of independent equations, that lead to two linearly independent solutions -The wave equation for the electric field components yields:-Utilizing the Ansatz:ܧ௫ൌܧ௫ǡ݁ି௭ܧ௫ǡ݁ା௭ܧ௬ൌܧ௬ǡ݁ି௭ܧ
1a)2a)2b)1b)
;2a);1a) we can derive the corresponding magnetic field components: -Using the substitution ܼ vacuum impedanceZis the wave impedance in OhmsAppendix
Presence of Dielectric Material
30-For linearmaterials ris relative permittivity ris relative permeability -Particularly, the displacement current was conceived by Maxwell as the separation (movement) of the (bound) charges due to the polarization of the medium (bound charges slightly separate inducing electric dipole moment) -For homogeneous, linear isotropic dielectric material -For anisotropic dielectric materialԦܲ ߳cǡܧ -Material may be non-linear, i.e. Pis not proportional to E(AEhysteresis in ferroelectric materials) -Generally P(ʘ) is a function of frequency, since the bound charges cannot act immediately to the applied field (c(ʘ) AEthis gives rise to losses permanent and induced electric dipole moments
Similar Expressions for Magnetization
-For magnetic fields the presence of magnetic material can give rise to a magnetization by microscopic electric currents or the spin of electrons -The magnetization vector describes the density of the permanent or induced magnetic dipole moments in a magnetic material-Herein ࣲ௩is the magnetic susceptibility, which described whether is material if appealed or
retracted by the presence of a magnetic field -The relative permeability of the material can then be denoted as: -Magnetization may occur in directions other than that of the applied magnetic field -Example: If a ferromagnet (e.g. iron) is exposed to a magnetic field, the microscopic dipoles align with the field and remain aligned to some extent when the magnetic field vanishes (magnetization vector M) AEa non-linear dependency between Hand Moccursquotesdbs_dbs47.pdfusesText_47