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PHYS3003 Light and Matter 07/11/2006

2 Classical electromagnetism

2.1 Introduction ethods of analysing physical motions fall into two general categories: the Lagrangian approach, such as classical mechanics, whereby the motion of a single element or particle is followed; and

the Eulerian technique, in which the physical system is described by position-dependent bulk properties or fields. The two approaches are in principle equivalent and, for example, the behaviour of a gas may be described either by following the positions and momenta of its constituent atoms or by analysing the variations in pressure, temperature, density and flow. For a given system, we adopt whichever approach proves the simpler: the motions of a few, otherwise isolated charged particles are therefore well described by a Lagrangian approach, featuring the Coulomb interaction or the formulae for interactions between dipoles.

Eulerian analysis is more appropriate for more complex systems, and in particular for electromagnetic fields, whose propagation is interesting even in the absence of individual charged particles. Classical electrodynamics allows wave theory to be applied to the propagation of radio waves and light. In this section, then, we shall be concerned with equations relating the temporal and spatial variations of electric and magnetic fields, from which wave equations may be determined. We shall see that the electric and magnetic fields are intimately related and, indeed, we shall describe in passing the fundamental but perhaps surprising origin of this relation.

This section therefore concentrates on four differential equations which, in combination with the formula for charged particle motion and some further defining equations, between them embody everything that is known about electromagnetism. These equations imply Coulomb’s law and the Biot-Savart formula, and they govern the behaviour of electrical components and the propagation of light. At high speeds, these four equations cease to be consistent with Newtonian mechanics, and require instead the relativistic Lorentz transformation; yet they preceded Einstein’s theory of special relativity by more than half a century. They were identified and rationalised by the Scottish physicist James Clerk Maxwell, whose name they now bear.

J. Clerk Maxwell, ‘A dynamical theory of the electromagnetic field,’ Phil.

65)

07/11/2006 PHYS3003 Light and Matter


2.1.1 Vector differentials We shall make frequent reference in this chapter to three vector derivatives: the gradient, divergence and curl. The gradient (grad) of a scalar field φ is a vector given by

=∇

zyx ∂∂φ

∂∂φ

∂∂φφ ,, (2.1)

and points in the direction of steepest change of the field φ. The divergence (div) refers to a vector field A = [Ax,Ay,Az] ≡ Ax i+ Ay j+ Az k, where i, j and k are unit vectors along the x, y and z axes, and is a scalar which indicates the net flux of the vector quantity into the region,

z

A

y

A

x

A zyx

∂∂

∂∂

∂∂

++=⋅∇ A (2.2)

The curl is also a vector derived from a vector field, and is a measure of the moment of the field, or vorticity, at the measurement point. It is given by

zyx

xyzxyz

AAAzyx

kji

y

A

x

A

x

A

z

A

z

A

y

A

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

−−−=×∇ ,,A

(2.3)

None of these vector differentials is dependent upon the choice of coordinate axes with respect to which they are calculated.

2.2 Maxwell’s equations That the entire Eulerian nature of electromagnetic dynamics can be described by four equations is one of the most satisfying and beautiful aspects of modern physics. To understand these equations properly, however, we must be familiar with the algebra of vector calculus – a luxury which Maxwell himself was denied. We shall therefore concentrate on Maxwell’s equations in their vector form.

We should first define the framework in which we operate. Electromagnetism is concerned with the variation of electric and magnetic fields with time and position; our coordinates will therefore be x, y, z and t, and we shall be interested in four vector quantities: the electric field strength E, the magnetic field strength H, the electric displacement D and the magnetic flux density B. Free charges are characterized by their density ρ and current density J, while bound charges allow an electric polarization P and magnetization M .


Classical electromagnetism 25

Maxwell’s equations are commonly written in both an integral form – relating volume, surface and line integrals of field components – and in a more compact differential form (obtained by applying the integral form to an elemental volume or surface), in terms of the nabla operator, ∇. The first, third and fourth equations are effectively Gauss’s law, Faraday’s law and Ampère’s law, respectively, while the second defines the continuity of magnetic flux lines and implies the absence of magnetic monopoles.

ρ=⋅∇ D Gauss vdd ∫∫∫∫∫ =⋅ ρSD (2.4)

0=⋅∇ B 0d =⋅∫∫ SB (2.5)

t∂

∂ BE −=×∇ Faraday ∫∫∫ ⋅−=⋅ S

BsE dd

t∂∂

(2.6)

t∂

∂ DJH +=×∇ Ampère ∫∫∫ ⋅

+=⋅ S

DJsH dd

t∂∂

(2.7)

Although these four equations are those now linked with Maxwell’s name, he found it necessary to write several additional formulae relating the fields to material properties. Three of these equations, which are sometimes called the constitutive formulae, may also be written in alternative forms, depending upon whether we are concerned with macroscopic or microscopic properties.

ED rεε 0= or PED += 0ε (2.8)

HB rµµ0= or ( )MHB += 0µ (2.9)

EJ σ= or ( )

×+==

BvEv

vJ

qm&

ρ (2.10)

0=+⋅∇t∂ρ∂

J (2.11)

Finally, the electric field strength E and magnetic flux density B may alternatively be written in terms of the vector and scalar potentials, A and ϕ:

t∂

∂−−∇= AE ϕ (2.12)

AB ×∇= (2.13)

The variables that appear in Maxwell’s equations and the constitutive formulae are summarized below, along with their SI units.

the del or nabla operator

zyx

zyx

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

kji ++≡

=∇ ,,

Maxwell’s equations

the permittivity of free space, ε0 = 8.854×10-12 F.m-1 the permeability of free space, µ0 = 4π×10-7 H.m-1= 1.257×10-6 H.m-1 constitutive equations

scalar and vector potentials



electric field strength E V.m-1

electric displacement D C.m-2

magnetic flux density B Tesla

magnetic field strength H A.m-1

current density J A.m-2

free charge density ρ C.m-3

vector potential A V.s.m-1

scalar potential ϕ V

permittivity of free space ε0 F.m-1

relative permittivity εr

permeability of free space µ0 H.m-1

relative permeability µr

conductivity σ m.Ω-1

2.2.1 Converting between SI and c.g.s. (Gaussian) units Many of the less recent texts and articles use the Gaussian or c.g.s. (centimetre-gramme-second) system of units, in which charge is measured in electrostatic units, e.s.u., and the magnetic flux density etc. are given in electromagnetic units, e.m.u.. The conversions are given below.

1 V.m-1 ≈ 1 / (3×10-4) esu

1 C ≈ 3×109 esu

1 = 0.1 emu ≈ 3×109 esu

1 T = 104 emu = 104 Gauss

1 A.m-1 = 4π×10-3 emu = 4π×10-3 Oersted

Equations in SI can be converted to Gaussian units by the rules

c BSI → BGaussian

ε0 → 1/4π

µ0 ε0 → 1/c2

J. D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York (1998) pp 783-4 E. M. Purcell, Electricity and Magnetism, 2nd edition, McGraw-Hill, New York (1985) backmatter



2.2.2 Alternative forms of Maxwell’s equations Maxwell’s reconciliation of the various laws of electromagnetism is all the more remarkable for being achieved without the benefit of the vector calculus that we use today. In place of vector quantities, such as the electric field, Maxwell therefore wrote each component separately, which increases both the number of equations and their complexity. By his own reckoning, Maxwell’s set of formulae, including the constitutive equations, numbered 20, in 20 unknowns. The 1865 article is nonetheless remarkable in its clarity.

Casting Maxwell’s equations into the form presented here was the work of a singularly persistent and incisive mathematician called Oliver Heaviside, who developed the methods of vector analysis for that purpose.

Further reduction is possible, however, by adopting the approach of special relativity (with which, of course, electromagnetism is so intimately related) and writing vectors and fields in four dimensions rather than three. We therefore construct position and momentum vectors,

[ ][ ]c

Eppp

ctzyx

zyx ,,,

,,,

=

=

p

r (2.14)

a combined current density and charge density vector,

[ ]ρcJJJ zyx ,,,=J (2.15)

and vector potential with the scalar potential as its fourth term,

= cAAA zyx

ϕ,,,A . (2.16)

The nabla operator ∇ is then extended to embrace the fourth, time dimension,

∂∂

∂∂

∂∂

∂∂=

ctzyx,,, (2.17)

in which case the four Maxwell’s equations and the condition for conservation of charge (2.11) may be written

0

2

=⋅=

J

JA (2.18)

2.2.3 Maxwell’s 1865 article Maxwell’s original article, ‘A dynamical theory of the electromagnetic field,’ (Phil. Trans. Royal Soc. London 155 459-512) is surprisingly accessible; it is an instructive exercise to work out the correspondence between today’s nomenclature and the terminology adopted by Maxwell.



2.3 Coulomb’s law We illustrate the application of Maxwell’s equations by using them to derive two familiar equations of basic electromagnetism, beginning with Coulomb’s law. We shall assume a charge distribution which is spherically symmetrical about the origin, and which may thus be written as a function of radius, r, alone as R(r). The spherical symmetry requires that the E and D fields are similarly functions of radius alone, with radial components only.

We shall now apply the first of Maxwell’s equations to a sphere of radius a about the origin.

( )

( )∫ ∫ ∫

∫∫∫∫∫

=

=⋅π π

φθφ2

0 0 0

2 dddsinr

drd

a

rrR

vRSD

(2.19)

The right-hand side is straightforward: it is the volume integral of the charge distribution within the sphere, and thus is simply the total charge enclosed,

( ) ( )∫=a

rrRrrQ0

2 d4π (2.20)

The left-hand side involves a surface integral of the scalar product D.dS, where dS is a vector equal in magnitude to an element of the surface da, and in a direction normal to the surface. At all points on the sphere, dS will thus point in the same direction as the position vector r , as will the electric displacement D which, as we have mentioned, does not have components except in the radial direction. Since the vectors D and dS are parallel, their scalar product will simply be the product of their magnitudes, and the left-hand side of our expression can thus be written

( )

( )aDa

aaD

24

dd.

π=

= ∫∫∫∫ SD (2.21)

Combining our results, we thus find that the electric displacement for the spherically symmetric charge distribution is given by

( ) ( )24 a

aQaD

π= (2.22)

and thus, using equation (2.8), that the electric field will be given by

( ) ( )2

04 a

aQaE

πε= (2.23)

which is, of course, simply Coulomb’s equation for the electric field around a point charge. The above result is slightly more general, in that it applies to any spherically-symmetric charge distribution.



2.4 Ampere’s law: the magnetic field from a current-carrying wire Whereas the first two of Maxwell’s equations relate the integral over a closed surface to an integral over the volume enclosed, the second pair relate surface integrals to line integrals around the edge of the open surfaces. We shall illustrate this by applying the fourth of Maxwell’s equations to find the tangential and axial components of the magnetic field around a straight current-carrying wire; we shall then apply the second equation in order to show that the radial component of the field is zero.

2.4.1 The tangential component of the magnetic field The geometry is shown in figure 2.1: we take as our first surface a disc, of

radius a, centred on and normal to the current-carrying wire. There is thus cylindrical symmetry about the wire, and the tangential component Ht of the magnetic field may be assumed to be a function of the radial coordinate alone. Applying the fourth of Maxwell’s equations to the surface, we find

( )aaHtπ2

d.d.

=

= ∫∫∫ sHSJ (2.24)

The left-hand side of this expression - the integral of the current density over the surface - is simply the total current passing through the disc, I; applying equation (2.9) for the magnetic flux density in a vacuum, we thus obtain

( )a

IaBt π

µ2

0= (2.25)

which is the expression we would obtain by integrating the Biot-Savart equation, and is a formal statement of the ‘right-hand grip rule’.

2.4.2 The axial component of the magnetic field The axial and radial field components remain to be determined, and for the first of these we again apply Maxwell’s fourth equation, this time taking as our surface the rectangular loop shown in figure 2.2. The line integral can be broken into four sections, of which the two radial parts are equal and opposite (because of the axial symmetry) and therefore cancel. No current flows through the surface, so the right-hand side of the expression will be zero,

( ) ( ) 0dd0

0

=+ ∫∫b

a

b

a zcHzaH (2.26)

This expression remains true whatever value of c we take, and must therefore hold as c becomes infinite. Making the reasonable assumption that the magnetic field induced by the current falls to zero at large distances from the wire, we allow the second integral of equation (2.26) to fall to zero, from which - again converting to the magnetic flux density B - we obtain

( ) 0=aBa (2.27)

Figure 2.1 Surface for determining the tangential component of the magnetic field around a current-carrying wire.

Figure 2.2 Surface for determining the axial component of the magnetic field around a current-carrying wire.

I

z

b

0

c a r

I



2.4.3 The radial component of the magnetic field To find the radial component of the magnetic field, we apply the second of Maxwell’s equations to a cylinder with the wire at its centre, shown in figure 2.3. The surface integral may be broken into three parts - the two ends of the cylinder, and the tubular section in between. Translational symmetry means that the contributions from the two end sections would cancel, even had we not already shown in equation (2.27) that the axial field is in any case zero.

The remaining part of the surface integral must thus satisfy

( ) 0d =∫∫tube

r SaB (2.28)

Since our geometry has cylindrical symmetry, the radial component is the same at all points on the surface, and we readily deduce that

( ) 0=aBr (2.29)

2.5 Poisson’s equation In general, such fortunate symmetries as we have exploited in section 2.4 may not occur. It will then be necessary to solve Maxwell’s equations from the beginning, either algebraically if the geometry is simple or, in most practical cases, numerically. For problems of pure electrostatics - in which the electric fields are constant and there are no magnetic fields - Maxwell’s equations may be simplified by writing the electric field in terms of an electric potential ϕ(x, y, z),

( )zyx ,,ϕ−∇=E (2.30)

Applying the gradient operator ⋅∇ to both sides of this equation yields

( )zyx ,,ϕ∇⋅−∇=⋅∇ E (2.31)

Substituting this into the first of Maxwell’s equations, using equation (2.8) to convert the electric field into a displacement, and defining the Laplacian or

‘del-squared’ operator 2∇ by

( ) ( ) ϕ∂∂

∂∂

∂∂ϕϕ

=∇⋅∇=∇

2

2

2

2

2

22 ,,,,,,

zyxzyxzyx (2.32)

we arrive at Poisson’s equation,

( ) ( )r

zyxzyx

εερϕ

0

2 ,,,, −=∇ (2.33)

To determine the spatial variation of a static electric field, we thus have to solve Poisson’s equation for the appropriate charge distribution, subject to such boundary conditions as may pertain.

Figure 2.3 Surface for determining the radial component of the magnetic field around a current-carrying wire.

If a magnetic field is present, or if the electric field varies with time, then the scalar potential is insufficient without the addition of a vector potential A. The electric and magnetic fields are related to the scalar and vector potentials by

AB

AE

×∇=

−−∇=t∂

∂φ

I



2.5.1 Electrostatic boundary conditions at conducting surfaces The presence of electric fields within a conductor will cause currents to flow until the redistribution of charge causes the electric fields to be cancelled. The electric potential ϕ(x, y, z) will then be the same at all points within the conductor. For problems in electrostatics, in which the steady-state may be assumed to have been achieved, conducting surfaces may be taken to be at constant potentials; the difference in potential between conducting surfaces corresponds to the standing voltage measured between them. The boundary conditions in electrostatic analysis are thus that the electric potential assume an appropriate constant value at the surface of each conductor present. We shall illustrate the application of boundary conditions by analysing the electric field present in a coaxial particle detector comprising a long tube with a rod at its centre, as shown in figure 2.4.

The detector is formed from a metal tube of inner diameter b along whose axis runs a conducting rod of diameter a. The remaining volume within the tube is filled with a low-pressure gas which may be taken to be charge-free and have a relative permittivity εr of unity. The electric potential will thus be found by solving Poisson’s equation, with ρ(x, y, z )= 0, subject to the boundary conditions that the rod and tube be equipotentials with values of V0 and zero respectively. We shall neglect end effects, and may thus assume a cylindrical symmetry.

In cylindrical polar coordinates, the 2∇ operator may be written as

2

2

2

2

22 11

∂φ∂

∂θ∂

∂∂

∂∂ ++

=∇rr

rrr

(2.34)

Substituting this form into equation (2.33), setting ρ(r)=0 and noting that the cylindrical symmetry renders all derivatives with respect to θ and φ equal to zero, we arrive at

( )

01 =

r

rr

rr ∂∂ϕ

∂∂

(2.35)

the general solution of which is a standard mathematical exercise, yielding

( ) BrAr += lnϕ (2.36)

By applying the boundary conditions ϕ(a) = V0, ϕ(b) = 0, we find that A = ϕ0/ln(a/b) and B = ϕ0lnb/ln(b/a), hence

( ) ( ) ( )

( ) ( ) rabrE

rbab

r

r

1

ln

lnlnln

0

0

ϕ

ϕϕ

=

−= (2.37)

Figure 2.4 Coaxial particle detector

The special case of Poisson’s equation with no free charge,

( ) 0,,2 =∇ zyxϕ

is known as Laplace’s equation.

to derive the electric field from the potential in this case, we need to know that in cylindrical polar coordinates,

zrr ∂∂

∂θ∂

∂∂ ++=∇ 1

a b

V



2.6 Continuity conditions A further application of Maxwell’s equations, which completes the basic knowledge required for electromagnetic analysis, is to the conditions applying at the boundary between two different media. For this, we shall return to the integral forms of the equations.

We shall analyse the boundary conditions in terms of components of the electric and magnetic fields parallel and perpendicular to the interface between the two media, as shown in figure 2.5. Let us first consider the parallel component of the electric field E, for which we apply the third of Maxwell’s equations, relating the integral around the edge of a rectangular loop to an integral over the area that it encloses. We shall assume that the interface is essentially flat (and thus has translational symmetry) over the scale of our analysis, and indeed we shall later ensure this by taking the limit as lengths a and b tend to zero. Breaking the loop integral into four parts, we can write Maxwell’s third equation for this case as

∫∫∫∫ ⋅−=+ABCD

D

C

B

A txExE S

Bddd 2//,1//, ∂

∂ (2.38)

where we have omitted the integrals along the sections B-C and D-A which are equal and opposite and therefore cancel. Taking the limit a→0, the right-hand side falls to zero, and we thus find that the parallel component of the electric field strength is continuous across the boundary

2//,1//, EE = (2.39)

Applying the fourth of Maxwell’s equations using an identical analysis, we find that the parallel component of the magnetic field strength is similarly continuous,

2//,1//, HH = (2.40)

In order to determine the conditions relating field components perpendicular to the interface, we follow a similar analysis using the first pair of Maxwell’s equations, this time referring to a closed surface containing a volume of integration but once again taking the limit as its dimensions fall to zero. In the absence of surface charge, we obtain

2,1, ⊥⊥ = DD (2.41)

and

2,1, ⊥⊥ = BB (2.42)

These relations are equally valid for time-dependent fields and, as we shall see later, prove to hold the key to the proper calculation of the reflection and refraction of light at boundaries in optical media.

Figure 2.5 Geometries for determining boundary conditions of components (a) parallel and (b) perpendicular to the interface between two media. Thus, using equation (6.1), the parallel components of D are related by

2

2//,

1

1//,

εεDD

=

The parallel components of B are hence related by

2

2//,

1

1//,

µµBB

=

Hence

2,21,1 ⊥⊥ = EE εε

Hence

2,21,1 ⊥⊥ = HH µµ

1 2

A

B C

D

b

a

(a)

(b)

a

b



2.7 The electromagnetic wave equation One of the great successes of Maxwell’s equations is their ability to describe completely the nature of electromagnetic radiation: radio waves, thermal infra-red, and light. Two of Maxwell’s equations express a spatial or temporal derivative of the electrical or magnetic field with the opposite derivative of the other; with a little vector algebra, one of the fields can be eliminated, leaving a wave equation defining the other.

Starting with Maxwell’s third and fourth equations, we wish to eliminate the magnetic field terms to yield an equation that describes the electric field alone. Since the third equation involves a temporal derivative of B, and the fourth the spatial derivative (curl), we shall proceed by differentiating each equation with respect to the opposite variable. The third and fourth equations thus become

( )2

2

ttt

t

∂∂

∂∂

∂∂

∂∂

DJH

BE

+=×∇

×−∇=×∇×∇ (2.43)

We may now use equation (2.9) to relate the magnetic flux density to the field strength, which may then be eliminated between the two equations, yielding

+−=×∇×∇

2

2

0 ttr ∂∂

∂∂µµ DJ

E (2.44)

Applying equation (2.8) to write this completely in terms of the electric field strength E now gives

+−=×∇×∇

2

2

00 tt rr ∂∂εε

∂∂µµ EJ

E (2.45)

Finally, we use vector algebra to re-write the left-hand side:

( )

+−=∇−⋅∇∇

2

2

002

tt rr ∂∂εε

∂∂µµ EJ

EE (2.46)

If there are no free charges (ρ = 0) and no currents flow (J = 0), then we arrive at the wave equation governing electromagnetic radiation

2

2

002

trr ∂

∂εεµµ EE =∇ (2.47)

Comparing this with equation (1.14), we see that the speed of electromagnetic

waves is given by rrv µεµε 001= . In vacuum, this correctly gives the

defined speed of light, 1800 s.m1099792.21 −×== µεc

Our general strategy is to reduce two of Maxwell’s equations, and two or more of the constitutive equations, to a single differential wave equation defining a just one field term:

In a conductor with conductivity σ, the current density is not zero but is related to the electric field strength by

EJ σ= .

If the mass of the charge carriers causes significant inertia in the response to the electric field, the conductivity becomes complex and frequency dependent. We shall see later that it may be readily determined by Newtonian mechanics.

differentiate equations to

allow electric or magnetic field

to be eliminated

apply vector relations to

produce wave equation

use constitutive equations to

reduce electric & magnetic

fields to single functions



2.7.1 Example: plane wave propagating along the z-axis To clarify the vector manipulations leading to the wave equation (2.47), we shall repeat the derivation for the specific case of plane waves propagating along the z-axis, f(r , t) = f(z, t). We shall constrain our search to waves in which the electric field E lies along the x-axis, so that it may be written as

( ) ( )[ ]0,0,,, tzEt x=rE (2.48)

From Faraday’s law, eqn (2.6), we therefore obtain

( ) ( ) ( )

∂∂

=

∂∂

−∂

∂=

∂∂− 0,

,,0

,,

,,0

z

tzE

y

tzE

z

tzE

txxxB

(2.49)

Integrating this with respect to time gives

( ) ( )

∂∂

−= ∫ 0,,

,0, 0 dtz

tzEt xBrB (2.50)

so that, dropping the constant field B0 (which is not part of the wave), the magnetic field component is directed along the y-axis,

( ) ( )[ ]0,,,0, tzBt y=rB (2.51)

where our application of Faraday’s law gave

z

E

t

Bxy

∂∂

−=∂

∂ (2.52)

The constitutive equations (2.8, 2.9) allow us to write the D and H fields,

( ) ( )[ ]( ) ( )[ ]0,,,0,

0,0,,,

0

0

ry

xr

tzBt

tzEt

µµεε

==

rH

rD (2.53)

and application of Ampère’s law hence gives us another relation,

∂∂

+=

∂∂−

0,0,0,0,1

00 t

E

z

Bx

ry

r

εεµµ

J (2.54)

so that, in the absence of currents J,

t

E

z

Bx

ry

r ∂∂

=∂

∂− εεµµ 0

0

1 (2.55)

Differentiating eqn (2.52) with respect to z and eqn (2.55) with respect to t gives in both cases a term ∂

2By / ∂z ∂t, which may be eliminated to give

2

2

002

2 1

z

E

t

E x

rr

x

∂∂

=∂

∂µµεε

(2.56)

We shall show shortly that plane electromagnetic waves can have no field component along their direction of propagation, hence the component Ez must here be zero. Our choice Ey = 0 corresponds to selecting the polarization state of the wave, which we shall examine later in detail.



2.7.2 The magnetic field of an electromagnetic wave In our derivation of the electromagnetic wave equation, we eliminated the magnetic field – linked to the electric field through Maxwell’s equations – to give an equation defining the electric field alone. We could equally have eliminated the electric field, to give a wave equation for B or H. It turns out, though, that at optical frequencies, interactions that are described by the electric field are nearly always the more important. Determining the magnetic field is nonetheless quite straightforward, for we simply substitute our solution for the electric field back into Maxwell’s equations. Faraday’s and Ampère’s laws give alternative forms of the relationships (written above, using the constitutive equations, as (2.52, 2.55),

z

D

t

H

z

E

t

Bxyxy

∂∂

−=∂

∂∂

∂−=

∂∂

; (2.57)

If the fields are sinusoidal functions of (ω t – k z), then it follows that

( ) ( ) xyxy DkHEkB ωω == ; (2.58)

The magnetic field component is therefore proportional to (and in phase with) the electric field component; the fields are perpendicular to both each other and to the direction of propagation. In a given medium, an electromagnetic wave is therefore completely characterized by knowledge of a single field.

2.7.3 Magnetism, relativity and the origin of Maxwell’s equations Magnetism appears a curious phenomenon: it is intrinsically associated with currents and varying electric fields and, in an apparent collapse of symmetry, exerts a force upon moving charges at right-angles to both itself and their direction of motion. Furthermore, it can only be measured by its effect upon moving charges, be they free particles or current loops. The magnetic field in fact proves to be an artificial, intermediate quantity that describes the relativistic correction to the electrostatic force when charges move and fields change with time. The net force on a charged particle moving near a current-carrying but electrically neutral wire proves not to be exactly zero because of the different speeds of the positive and negative charges in the wire relative to the moving particle. Since magnetism is itself a manifestation of special relativity, it should be no surprise that the equations that describe it are consistent with the relativistic Lorentz transform.

It is for this reason that a single field is generally sufficient to define an electromagnetic wave in a given medium, and also why the electric field usually accounts for the stronger interaction with matter, for magnetic interactions imply relativistic phenomena. It is also rare for the magnetic response of a material to be fast enough to respond at optical frequencies. We therefore tend to characterize electromagnetic radiation through the electric field component; the magnetic field can always be calculated by application of Faraday’s or Ampère’s law, as shown in section 2.7.2.

Figure 2.6 The magnetic force F between two current-carrying but electrically neutral conductors acts in the plane of the conductors and is due to the imperfect cancellation of the forces of attraction and repulsion when the positive and negative charges move relative to each other.

I F

v

B



2.8 Electromagnetic energy flow Life on earth would not exist if electromagnetic waves did not carry energy, which is associated with both the electric and magnetic fields and therefore moves with the wave magnitude. Maxwell’s equations allow us to derive some precise and fairly general expressions for the energy flow associated with an electromagnetic wave.

We begin with results from electro- and magneto-statics, that the energy densities We and Wm associated with electric and magnetic fields are given by

HB

ED

⋅=

⋅=

2

12

1

m

e

W

W (2.59)

For the specific but common case of linear, isotropic materials, we may apply the constitutive equations (2.8) and (2.9) to give a total energy density

20

202

1HEW rr µµεε += (2.60)

Using the relationship (2.58) between electric and magnetic fields, this becomes

20

20

202

1EEEW rrr εεεεεε =+= (2.61)

The energy associated with an electromagnetic wave in isotropic media is therefore equally divided between the electric and magnetic fields. The energy flow S per unit area per unit time is simply the product of the energy density with the speed of propagation of the wave:

2

0

0

20

E

vES

r

r

r

µµεε

εε

=

= (2.62)

If the energy flow is represented in magnitude and direction by a vector S, then the conservation of energy allows us to write an expression similar to that for the conservation of charge, eqn (2.11):

0=∂

∂+⋅∇t

WS (2.63)

The temporal derivative may be obtained from eqn (2.59):

∂∂⋅+

∂∂⋅=

∂∂

ttt

We DE

ED

2

1 (2.64)

The constant here is, in vacuo, the reciprocal of the impedance of free space,

Ω==

7.376000 εµZ



A similar expression describes the magnetic contribution. For linear media, D.∂E/∂t and E.∂D/∂t are equal and, inserting the magnetic term, we may write

∂∂⋅+

∂∂⋅=

∂∂

ttt

W BH

DE (2.65)

Faraday’s and Ampère’s laws allow us to re-write this as the Poynting theorem:

( ) ( )

( ) JEHE

EHJHE

⋅−×⋅−∇=

×∇⋅−−×∇⋅=∂

∂t

W

(2.66)

The final term, E.J, corresponds to resistive power dissipation. In its absence, comparison with eqn (2.63) suggests that the energy flow associated with electromagnetic radiation is given by

HES ×= (2.67)

The vector S is known as the Poynting vector, and this expression is valid even for anisotropic media. Being perpendicular to both the electric and magnetic fields, the Poynting vector in isotropic media is collinear with the wavevector k; in anisotropic media, however, S and k may diverge.

Application of the constitutive equations shows the equivalence of eqns (2.62) and (2.67). Note that the energy flow is obtained by multiplying the energy density by the phase velocity rather than the group velocity, for within a wave packet the energy moves with the maxima of the electric field (rather than the intensity). To calculate the average energy flux for wave packets in dispersive media, we need to add together the individual contributions of the constituent frequency components of the wave packet.

We have used here the vector identity

( ) ( )( )BA

ABBA

×∇⋅−×∇⋅≡×⋅∇

The Poynting vector S has a magnitude equal to the intensity of the wave, and its orientation indicates the direction of energy flow.

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