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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 by

Maxwell 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, and

B, 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 propagating

electromagnetic “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 v

E(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 on

Jis given by:

f E J

B(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 J

Erepresents 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 8 854
10 12 farad/m 0 4 10 7 henry/m (1.3.2)

The units for

0 and 0 are the units of the ratios D/E and B/H , that is, coulomb/m 2 volt/m coulomb volt m farad m weber/m 2 ampere/m weber ampere m henry m

From the two quantities

0 0 , we can define two other physical constants, namely, thespeed of lightand thecharacteristic impedanceof vacuum: c 0 1 0 0 3 10 8 m/sec 0 0 0

377 ohm

(1.3.3)41. Maxwell"s Equations The next simplest form of the constitutive relations is for simple homogeneous isotropic dielectric and for magnetic materials: D E B H (1.3.4) These are typically valid at low frequencies. The permittivity and permeability are related to theelectric and magnetic susceptibilitiesof the material as follows: 0 1 0 1 m (1.3.5)

The susceptibilities

m are measures of the electric and magnetic polarization properties of the material. For example, we have for the electric flux density: D E 0 1 E 0 E 0 E 0 E

P(1.3.6)

where the quantityP 0 Erepresents the dielectric polarization of the material, that is, the average electric dipole moment per unit volume. In a magnetic material, we have B 0 H M 0 H m H 0 1 m H

H(1.3.7)

whereM m His themagnetization, that is, the average magnetic moment per unit volume. The speed of light in the material and the characteristic impedance are: c= 1 (1.3.8) Therelative permittivity, permeabilityandrefractive indexof a material are defined by: rel 0 1 rel 0 1 m ,n=⎷ rel rel (1.3.9) so that n 2 rel rel . Using the definition of Eq. (1.3.8), we may relate the speed of light and impedance of the material to the corresponding vacuum values: c= 1 1 0 0 rel rel =c 0 rel rel =c 0 n 0 0 rel rel 0 rel rel 0 rel n=η 0 n rel (1.3.10)

For a non-magnetic material, we have

0 , or, rel

1, and the impedance

becomes simply 0 /n , a relationship that we will use extensively in this book. More generally, constitutive relations may be inhomogeneous, anisotropic, nonlin-quotesdbs_dbs18.pdfusesText_24