Lecture: Maxwell’s Equations

Maxwell’s Equations (integral form) • Note the symmetry now of Maxwell’s Equations in free space, meaning when no charges or currents are present 22 22 2 hh1

Lecture: Maxwell’s Equations - USPAS

－Introduction to Maxwell’s Equations • Sources of electromagnetic fields • Differential form of Maxwell’s equation • Stokes’ and Gauss’ law to derive integral form of Maxwell’s equation • Some clarifications on all four equations • Time-varying fields wave equation • Example: Plane wave － Phase and Group Velocity

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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

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Lecture:

at Jefferson Laboratory, January 15-26th2018 F

Marhauser

day, January 1 , 2018

This Lecture

ˉThis lecture provides theoretical basics useful for follow-up lectures on resonators and waveguides

Sources of electromagnetic fields

Some clarifications on all four equations

Time-varying fields AEwave equation

Example: Plane wave

ˉPhase and Group Velocity

ˉWave impedance

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, published 1 January 1865 -Originally there were 20 equations

Sources of Electromagnetic Fields

ˉElectromagnetic 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 ܫൌ׭ௌԦܬȉ݀Ԧܵ

Stationary charge creates electric field

Moving charge creates magnetic field

ˉIf either the magnetic or electrical fields vary in time, both fields are 6

DifferentialForm

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 or

Gauss'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 Force

Diǀergence (Gauss') Theorem

7 outwardfluxofvectorfield(Ԧܨ {div

Curl (Stokes') Theorem

Green's Theorem

{curl

Integralofcurlofvectorfield(Ԧܨ

lineintegralofvectorfield(Ԧܨ

Eǆample͗ Curl (Stokes') Theorem

Integralofcurlofvectorfield(Ԧܨ

lineintegralofvectorfield(Ԧܨ

Example: Curl (Stokes) Theorem

10 Example: Closed line integrals of various vector fields {curl

Integralofcurlofvectorfield(Ԧܨ

lineintegralofvectorfield(Ԧܨ

No curlSome curlStronger curl

No net curl

DifferentialFormIntegralForm

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)

1. Uniform field

Electric Flux & 1stMaxwell Equation

-angle between field and normal vector to surface matters

Gauss: Integration over closedsurface

2. Non-Uniform field

Example: Metallic plate,

assume only surface charges on one side

Definition of Electric Flux

Gauss: Integration over closedsurface

Example: Capacitor

Electric Flux & 1stMaxwell Equation

1. Uniform field

-angle between field and normal vector to surface matters

2. Non-Uniform field

Definition of Electric Flux

Integration of over closed spherical surface S

Examples of non-uniform fields

Point charge Q

Principle of Superposition holds:

Electric Flux & 1stMaxwell Equation

pointing out radially

Add charges

Uniform 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 loops

Closed 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 a

North and South pole

Magnetic 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 loop

Magnetic 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 field

If integration path is not changing in time

charge traveling once around loop -or voltage measured at end of open loop

Faraday'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 I

Right 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 I

Left 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 temperature

Presence 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 medium

Homogeneous wave equation

Time-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 23

Time-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

Example: 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 27

Wave 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: 28

Wave 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:ܧ௫ൌܧ௫ǡ௣݁ି௜௞௭൅ܧ௫ǡ௥݁ା௜௞௭ܧ௬ൌܧ௬ǡ௣݁ି௜௞௭൅ܧ

1a)2a)2b)1b)

;2a);1a) we can derive the corresponding magnetic field components: -Using the substitution ܼ vacuum impedanceZis the wave impedance in Ohms

Appendix

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 ߳଴c௜ǡ௝ܧ -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

[PDF] Lecture: Maxwell’s Equations - USPAS

Maxwell’s Equations (integral form) - University of Hawaiʻi