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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1
Maxwell"s Equations
1.1 Maxwell"s EquationsMaxwell"s equations describe all (classical) electromagnetic phenomena:
E B ∂t H J D ∂t D B 0 (Maxwell"s equations) (1.1.1) The rst isFaraday"s law of induction, the second isAmp`ere"s lawas amended byMaxwell to include the displacement current∂
D /∂t , the third and fourth areGauss" laws for the electric and magnetic fields.The displacement current term
D /∂t in Amp`ere"s law is essential in predicting the existence of propagating electromagnetic waves. Its role in establishing charge conser- vation is discussed in Sec. 1.7. Eqs. (1.1.1) are in SI units. The quantitiesEandHare the electric and magnetic field intensitiesand are measured in units of [volt/m] and [ampere/m], respectively. The quantitiesDandBare the electric and magneticflux densitiesand are in units of [coulomb/m 2 ] and [weber/m 2 ], or [tesla].Dis also called theelectric displacement, andB, themagnetic induction.
The quantities
ρandJare thevolume charge densityandelectric current density (charge flux) of anyexternalcharges (that is, not including any induced polarization charges and currents.) They are measured in units of [coulomb/m 3 ] and [ampere/m 2 The right-hand side of the fourth equation is zero because there are no magnetic mono- pole charges. Eqs. (1.3.17)-(1.3.19) display the induced polarization terms explicitly.The charge and current densities
Jmay be thought of as thesourcesof the electro-
magnetic fields. For wave propagation problems, these densities are localized in space; for example, they are restricted to flow on an antenna. The generated electric and mag-netic fields areradiatedaway from these sources and can propagate to large distances to21. Maxwell"s Equations
the receiving antennas. Away from the sources, that is, in source-free regions of space,Maxwell"s equations take the simpler form:
E B ∂t H D ∂t D 0 B 0 (source-free Maxwell"s equations) (1.1.2) The qualitative mechanism by which Maxwell"s equations give rise to propagatingelectromagnetic elds is shown in the gure below.For example, a time-varying currentJon a linear antenna generates a circulating
and time-varying magnetic fieldH, which through Faraday"s law generates a circulating electric fieldE, which through Amp`ere"s law generates a magnetic field, and so on. The cross-linked electric and magnetic fields propagate away from the current source. A more precise discussion of the fields radiated by a localized current distribution is given in Chap. 15.1.2 Lorentz ForceThe force on a charge
q moving with velocityvin the presence of an electric and mag- netic fieldE,Bis called the Lorentz force and is given by:
F =q( E v B (Lorentz force) (1.2.1) Newton"s equation of motion is (for non-relativistic speeds): md v dt= F =q( E v B (1.2.2) where m is the mass of the charge. The forceFwill increase the kinetic energy of the charge at a rate that is equal to the rate of work done by the Lorentz force on the charge, that is,v F. Indeed, the time-derivative of the kinetic energy is: W kin 12 m v v ?dW kin dt=m v ·d v dt= v F =q vE(1.2.3)
We note that only the electric force contributes to the increase of the kinetic energy- the magnetic force remains perpendicular tov, that is,v v B 0.1.3. Constitutive Relations3
Volume charge and current distributions
Jare also subjected to forces in the
presence of fields. The Lorentz forceper unit volumeacting onJis given by:
f E JB(Lorentz force per unit volume) (1.2.4)
wherefis measured in units of [newton/m 3 ]. IfJarises from the motion of charges within the distribution , thenJ v(as explained in Sec. 1.6.) In this case, f E v B (1.2.5)By analogy with Eq. (1.2.3), the quantityv
f v E JErepresents thepower
per unit volumeof the forces acting on the moving charges, that is, the power expended by (or lost from) the fields and converted into kinetic energy of the charges, or heat. It has units of [watts/m 3 ]. We will denote it by: dP loss dV= J E (ohmic power losses per unit volume) (1.2.6) In Sec. 1.8, we discuss its role in the conservation of energy. We will find that elec- tromagnetic energy flowing into a region will partially increase the stored energy in that region and partially dissipate into heat according to Eq. (1.2.6).1.3 Constitutive RelationsThe electric and magnetic flux densitiesD
Bare related to the field intensitiesE
Hvia the so-calledconstitutive relations, whose precise form depends on the material in which the fields exist. Invacuum, they take their simplest form: D 0 E B 0 H (1.3.1) where 0 0 are thepermittivityandpermeabilityof vacuum, with numerical values: 0 8 85410 12 farad/m 0 4 10 7 henry/m (1.3.2)