[PDF] Lecture: Maxwell’s Equations - USPAS

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

Lecture: Maxwell’s Equations - USPAS

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, published 1 January 1865

Chapter 1 Maxwell’s Equations

To solve Maxwell’s equations (1 15)–(1 18) we need to invoke specific material properties, i e P = f(E) and M = f(B), which are denoted constitutive relations 1 4 Maxwell’s Equations in Differential Form For most of this course it will be more convenient to express Maxwell’s equations in differential form

Chapter 13 Maxwell’s Equations and Electromagnetic Waves

Ampere−Maxwell law 000 dId E dt µµε Φ ∫Bs⋅=+ GG v Electric current and changing electric flux produces a magnetic field 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

MAXWELL’S EQUATIONS, ELECTROMAGNETIC WAVES, AND STOKES PARAMETERS

MAXWELL EQUATIONS, EM WAVES, & STOKES PARAMETERS 7 nˆ ×(H 2 − H1) = 0 (finite conductivity) (3 10) The boundary conditions (3 1), (3 2), (3 4), (3 9), and (3 10) are useful in solving the differential Maxwell equations in different adjacent regions with continuous physical properties and then linking the partial solutions to

3 Maxwells Equations and Light Waves

In Maxwell’s original notation, the equations were not nearly so compact and easy to understand original form of Maxwell’s equations But, he was able to derive a value for the speed of light in empty space, which was within 5 of the correct answer The modern vector notation was introduced by Oliver Heaviside and Willard Gibbs in 1884

On the Notation of Maxwells Field Equations

equations, for example, contains the vector potential A , which today usually is eliminated Three Maxwell equations can be found quickly in the original set, together with O HM ’s law (1 6) , the F ARADAY-force (1 4) and the continuity equation (1 8) for a region containing char ges The Original Quaternion Form of Maxwell‘s Equations

32-1 Chapter 32

MAXWELL’S CORRECTION TO AMPERE’S LAW As we mentioned in the introduction, Maxwell de-tected a logical flaw in Ampere’s law which, when corrected, gave him the complete set of equations for the electric and magnetic fields With the complete set of equations, Maxwell was able to obtain a theory of light

Maxwell’s Equations & The Electromagnetic Wave Equation

Maxwell’s Equations in vacuum t E B t B E B E o o w w u w w u x x PH 0 0 • The vacuum is a linear, homogeneous, isotropic and dispersion less medium • Since there is no current or electric charge is present in the vacuum, hence Maxwell’s equations reads as • These equations have a simple solution interms of traveling sinusoidal waves,

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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

2

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

5

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

8

Green's Theorem

{curl

Integralofcurlofvectorfield(Ԧܨ

lineintegralofvectorfield(Ԧܨ

Edžample͗ Curl (Stokes') Theorem

9

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

11

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)

12

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

13

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

14

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

15

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

16

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
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