THE CERN ACCELERATOR SCHOOL

Theory of Electromagnetic Fields

Part I:Maxwell's Equations

Andy Wolski

The Cockcroft Institute,and the University of Liverpool,UK

CAS Specialised Course on RF for Accelerators

Ebeltoft,Denmark,June 2010

Theory of Electromagnetic Fields

In these lectures,we shall discuss the theory of electromagnetic

elds,with an emphasis on aspects relevant to RF systems in

accelerators:

1.Maxwell's equations

Maxwell's equations and their physical signicance

Electromagnetic potentials

Electromagnetic waves and their generation

Electromagnetic energy

2.Standing Waves

Boundary conditions on electromagnetic elds

Modes in rectangular and cylindrical cavities

Energy stored in a cavity

3.Travelling Waves

Rectangular waveguides

Transmission lines

Theory of EM Fields 1 Part I:Maxwell's Equations

Theory of Electromagnetic Fields

I shall assume some familiarity with the following topics:

vector calculus in Cartesian and polar coordinate systems;

Stokes'and Gauss'theorems;

Maxwell's equations and their physical signicance;

types of cavities and waveguides commonly used in

accelerators.

The fundamental physics and mathematics is presented in

many textbooks;for example:

I.S.Grant and W.R.Phillips,\Electromagnetism,"

2nd Edition (1990),Wiley.

Theory of EM Fields 2 Part I:Maxwell's Equations

Summary of relations in vector calculus

In cartesian coordinates:

gradf rf

@f

@x

;

@f

@y

;

@f

@z

!

(1)

div

~

A r

~

A

@A

x

@x

+

@A

y

@y

+

@A

z

@z

(2)

curl

~

A r

~

A

^x ^y ^z

@

@x

@

@y

@

@z

A

x

A

y

A

z

(3)

r

2

f

@

2

f

@x

2

+

@

2

f

@y

2

+

@

2

f

@z

2

(4)

Note that ^x,^y and ^z are unit vectors parallel to the x,y and z

axes,respectively.

Theory of EM Fields 3 Part I:Maxwell's Equations

Summary of relations in vector calculus

Gauss'theorem:

Z

V

r

~

AdV =

I

S

~

A d

~

S;(5)

for any smooth vector eld

~

A,where the closed surface S

bounds the volume V.

Stokes'theorem:

Z

S

r

~

A d

~

S =

I

C

~

A d

~

`;(6)

for any smooth vector eld

~

A,where the closed loop C bounds

the surface S.

A useful identity:

rr

~

A r(r

~

A) r

2

~

A:(7)

Theory of EM Fields 4 Part I:Maxwell's Equations

Maxwell's equations

r

~

D= r

~

B=0

r

~

E =

@

~

B

@t

r

~

H=

~

J +

@

~

D

@t

James Clerk Maxwell

1831 { 1879

Note that is the electric charge density;and

~

J is the current

density.

The constitutive relations are:

~

D ="

~

E;

~

B =

~

H;(8)

where"is the permittivity,and is the permeability of the

material in which the elds exist.

Theory of EM Fields 5 Part I:Maxwell's Equations

Physical interpretation of r

~

B =0

Gauss'theorem tells us that for any smooth vector eld

~

B:

Z

V

r

~

BdV =

I

S

~

B d

~

S;(9)

where the closed surface S bounds the region V.

Applied to Maxwell's equation r

~

B =0,Gauss'theorem tells

us that the total ux entering a bounded region equals the

total ux leaving the same region.

Theory of EM Fields 6 Part I:Maxwell's Equations

Physical interpretation of r

~

D =

Applying Gauss'theorem to Maxwell's equation r

~

D =,we

nd that:

Z

V

r

~

DdV =

I

S

~

D d

~

S =Q;(10)

where Q =

R

V

dV is the total charge within the region V,

bounded by the closed surface S.

The total ux of electric displacement crossing a closed surface

equals the total electric charge enclosed by that surface.

In particular,at a distance r from the

centre of any spherically symmetric

charge distribution,the electric

displacement is:

~

D =

Q

4r

2

^r;(11)

where Q is the total charge within

radius r,and ^r is a unit vector in the

radial direction.

Theory of EM Fields 7 Part I:Maxwell's Equations

Physical interpretation of r

~

H =

~

J +

@

~

D

@t

Stokes'theorem tells us that for any smooth vector eld

~

H:

Z

S

r

~

H d

~

S =

I

C

~

H d

~

`;(12)

where the closed loop C bounds the surface S.

Applied to Maxwell's equation

r

~

H =

~

J +

@

~

D

@t

,Stokes'

theorem tells us that the

magnetic eld

~

H integrated

around a closed loop equals the

total current passing through

that loop.For the static case

(constant currents and elds):

I

C

~

H d

~

`=

Z

S

~

J d

~

S =I:(13)

Theory of EM Fields 8 Part I:Maxwell's Equations

The displacement current and charge conservation

The term

@

~

D

@t

in Maxwell's equation r

~

H =

~

J +

@

~

D

@t

is known

as the displacement current density,and has an important

physical consequence.

Since,for any smooth vector eld

~

H:

r r

~

H 0;(14)

it follows that:

r

~

J +r

@

~

D

@t

=r

~

J +

@

@t

=0:(15)

This is the continuity equation,that expresses the local

conservation of electric charge.The signicance is perhaps

clearer if we use Gauss'theorem to express the equation in

integral form:

I

S

~

J d

~

S =

dQ

dt

;(16)

where Q is the total charge enclosed by the surface S.

Theory of EM Fields 9 Part I:Maxwell's Equations

Physical interpretation of r

~

E =

@

~

B

@t

Applied to Maxwell's equation r

~

E =

@

~

B

@t

,Stokes'theorem

tells us that a time-dependent magnetic eld generates an

electric eld.

In particular,the total electric eld around a closed loop equals

the rate of change of the total magnetic ux through that loop:

Z

S

r

~

E d

~

S =

I

C

~

E d

~

`=

@

@t

Z

S

~

B d

~

S:(17)

This is Faraday's law of electromagnetic

induction:

E =

@

@t

;(18)

where E is the electromotive force (the

integral of the electric eld) around a

closed loop,and is the total magnetic

ux through that loop.

Theory of EM Fields 10 Part I:Maxwell's Equations

Solving Maxwell's equations

Maxwell's equations are of fundamental importance in

electromagnetism,because they tell us the elds that exist in

the presence of various charges and materials.

In accelerator physics (and many other branches of applied

physics),there are two basic problems:

Find the electric and magnetic elds in a system of charges

and materials of specied size,shape and electromagnetic

characteristics.

Find a system of charges and materials to generate electric

and magnetic elds with specied properties.

Theory of EM Fields 11 Part I:Maxwell's Equations

Example:elds induced by a bunch in an accelerator

Theory of EM Fields 12 Part I:Maxwell's Equations

Linearity and superposition

Neither problem is particularly easy to solve in general;but

fortunately,there are ways to decompose complex problems

into simpler ones...

Maxwell's equations are linear:

r

~

B

1

+

~

B

2

=r

~

B

1

+r

~

B

2

;(19)

and:

r

~

H

1

+

~

H

2

=r

~

H

1

+r

~

H

2

:(20)

This means that if two elds

~

B

1

and

~

B

2

satisfy Maxwell's

equations,so does their sum

~

B

1

+

~

B

2

.

As a result,we can apply the principle of superposition to

construct complicated electric and magnetic elds just by

adding together sets of simpler elds.

Theory of EM Fields 13 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

Perhaps the simplest system is one in which there are no

charges or materials at all:a perfect,unbounded vacuum.

Then,the constitutive relations are:

~

D ="

0

~

E;and

~

B =

0

~

H;(21)

and Maxwell's equations take the form:

r

~

E =0 r

~

B=0

r

~

E =

@

~

B

@t

r

~

B=

1

c

2

@

~

E

@t

where 1=c

2

=

0

"

0

.

There is a trivial solution,in which all the elds are zero.But

there are also interesting non-trivial solutions,where the elds

are not zero.To nd such a solution,we rst\separate"the

electric and magnetic elds.

Theory of EM Fields 14 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

If we take the curl of the equation for r

~

E we obtain:

rr

~

E r(r

~

E) r

2

~

E =

@

@t

r

~

B:(22)

Then,using r

~

E =0,and r

~

B =

1

c

2

@

~

E

@t

,we nd:

r

2

~

E

1

c

2

@

2

~

E

@t

2

=0:(23)

This is the equation for a plane wave,which is solved by:

~

E =

~

E

0

e

i(

~

k~r!t)

;(24)

where

~

E

0

is a constant vector,and the phase velocity c of the

wave is given by the dispersion relation:

c =

!

jkj

=

1

p

0

"

0

:(25)

Theory of EM Fields 15 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

Similarly,if we take the curl of the equation for r

~

B we

obtain:

r

2

~

B

1

c

2

@

2

~

B

@t

2

=0:(26)

This is again the equation for a plane wave,which is solved by:

~

B =

~

B

0

e

i(

~

k~r!t)

;(27)

where

~

B

0

is a constant vector,and the phase velocity c of the

wave is again given by the dispersion relation (25):

c =

!

jkj

=

1

p

0

"

0

:

Theory of EM Fields 16 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

Although it appears that we obtained independent equations

for the electric and magnetic elds,we did so by taking

derivatives.Therefore,the original Maxwell's equations impose

constraints on the solutions.

For example,substituting the solutions (24) and (27) into

Maxwell's equation:

r

~

E =

@

~

B

@t

;(28)

we nd:

~

k

~

E

0

=!

~

B

0

:(29)

This imposes a constraint on both the directions and the

relative magnitudes of the electric and magnetic elds.

Theory of EM Fields 17 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

Similarly we nd:

r

~

B =

1

c

2

@

~

E

@t

)

~

k

~

B

0

=

!

c

2

~

E

0

(30)

r

~

E =0 )

~

k

~

E

0

=0 (31)

r

~

B =0 )

~

k

~

B

0

=0 (32)

These equations impose constraints on the relative amplitudes

and directions of the electric and magnetic elds in the waves:

~

E

0

,

~

B

0

,and

~

k are mutually perpendicular;

The eld amplitudes are related by

E

0

B

0

=c.

Theory of EM Fields 18 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

Theory of EM Fields 19 Part I:Maxwell's Equations

Example:plane electromagnetic waves in free space

Note that the wave vector

~

k can be chosen freely.We can refer

to a wave specied by a particular value of

~

k as a\mode"of

the electromagnetic elds in free space.

The frequency of each mode is determined by the dispersion

relation (25):

c =

!

jkj

=

1

p

0

"

0

:

A single mode represents a plane wave of a single frequency,

with innite extent in space and time.More realistic waves can

be obtained by summing together (superposing) dierent

modes.

Theory of EM Fields 20 Part I:Maxwell's Equations

Electromagnetic potentials

Sometimes,problems can be simplied by working with the

electromagnetic potentials,rather than the elds.

The potentials and

~

A are dened as functions of space and

time,whose derivatives give the elds:

~

B = r

~

A;(33)

~

E = r

@

~

A

@t

:(34)

Note that because the elds are obtained by taking derivatives

of the potentials,there is more than one set of potential

functions that will produce the same elds.This feature is

known as gauge invariance.

Theory of EM Fields 21 Part I:Maxwell's Equations

Electromagnetic potentials:the Lorenz gauge

To dene the potentials uniquely,we need to specify not just

the elds,but also an additional condition { known as a gauge

condition { on the potentials.

For time-dependent elds (and potentials),the conventional

choice of gauge is the Lorenz gauge:

r

~

A+

1

c

2

@

@t

=0;(35)

where c is the speed of light.

The Lorenz gauge is convenient because it allows us to write

wave equations for the potentials in the presence of sources,in

a convenient form.

Theory of EM Fields 22 Part I:Maxwell's Equations

Wave equations for the potentials in the Lorenz gauge

If we take Maxwell's equation r

~

D =,and substitute for the

electric eld in terms of the potentials (34),we nd:

r

~

E =r

2

@

@t

r

~

A =

"

0

:(36)

Then,using the Lorenz gauge (35),we nd:

r

2

1

c

2

@

2

@t

2

=

"

0

:(37)

The Lorenz gauge allows us to write a wave equation for the

scalar potential ,with a source term given by the charge

density ;and without the appearance of either the vector

potential

~

A or the current density

~

J.

Theory of EM Fields 23 Part I:Maxwell's Equations

Wave equations for the potentials in the Lorenz gauge

We can nd a similar wave equation for the vector potential,

~

A.

Substituting

~

B =r

~

A into Maxwell's equations,we obtain:

r

~

B =rr

~

A r(r

~

A) r

2

~

A =

0

~

J +

1

c

2

@

~

E

@t

:(38)

Then,using the Lorenz gauge (35),and substituting for the

electric eld

~

E =r

@

~

A

@t

,we nd:

r

2

~

A

1

c

2

@

2

~

A

@t

2

=

0

~

J:(39)

Using the Lorenz gauge allows us to write a wave equation for

the vector potential,with a source term given by the current

density

~

J,and without the appearance of either the charge

density ,or the scalar potential .

Theory of EM Fields 24 Part I:Maxwell's Equations

Wave equations for the potentials in the Lorenz gauge

The wave equations for the potentials are useful,because they

allow us to calculate the elds around time-dependent charge

and current distributions.

The general solutions to the wave

equations can be written:

(~r;t) =

1

4"

0

Z

(~r

0

;t

0

)

j~r ~r

0

j

dV

0

;(40)

~

A(~r;t) =

0

4

Z

~

J(~r

0

;t

0

)

j~r ~r

0

j

dV

0

;(41)

where:

t

0

=t

j~r ~r

0

j

c

:(42)

Theory of EM Fields 25 Part I:Maxwell's Equations

Generating electromagnetic waves:the Hertzian dipole

It is easy to show that,in the static case,the expression for the

scalar potential gives the result expected from Coulomb's law.

A more interesting exercise is to calculate the elds around an

innitesimal oscillating dipole (a Hertzian dipole).We can

model the current associated with a Hertzian dipole oriented

parallel to the z axis as:

~

I =I

0

e

i!t

^z:(43)

We can think of the current as being associated with a charge

oscillating between two points either side of the origin,along

the z axis.

Theory of EM Fields 26 Part I:Maxwell's Equations

Generating electromagnetic waves:the Hertzian dipole

Since the current is located only at

the origin,it is straightforward to

perform the integral (41) to nd

the vector potential:

~

A(~r;t) =

0

4

(I

0

`)

e

i(kr!t)

r

^z;(44)

where:

k =

!

c

:(45)

Note that`is the length of the

dipole:strictly speaking,we take

the limit`!0,but with the

amplitude I

0

`remaining constant.

Theory of EM Fields 27 Part I:Maxwell's Equations

Generating electromagnetic waves:the Hertzian dipole

Having obtained the vector potential,we can nd the magnetic

eld from

~

B =r

~

A.For the curl in spherical polar

coordinates,see Appendix A.The result is:

B

r

= 0;(46)

B

= 0;(47)

B

=

0

4

(I

0

`)k sin

1

kr

i

e

i(kr!t)

r

:(48)

The electric eld can be obtained from r

~

B =

1

c

2

@

~

E

@t

.

The result is:

E

r

=

1

4"

0

2

c

(I

0

`)

1 +

i

kr

e

i(kr!t)

r

2

;(49)

E

=

1

4"

0

(I

0

`)

k

c

sin

i

k

2

r

2

+

1

kr

i

e

i(kr!t)

r

;(50)

E

= 0:(51)

Theory of EM Fields 28 Part I:Maxwell's Equations

Generating electromagnetic waves:the Hertzian dipole

At distances from the dipole large compared with the

wavelength,kr 1,and we can nd approximate expressions

for the dominant eld components:

E

i

1

4"

0

(I

0

`)

k

c

sin

e

i(kr!t)

r

;(52)

B

i

0

4

(I

0

`)k sin

e

i(kr!t)

r

:(53)

This is known as the\far eld regime".Note that the elds

take the form of a wave propagating in the radial direction:the

electric and magnetic elds are perpendicular to each other,

and to the direction of the wave (as we found for the case of

the plane wave).

Theory of EM Fields 29 Part I:Maxwell's Equations

Generating electromagnetic waves:the Hertzian dipole

The relative amplitudes of the electric and magnetic elds are

also as we found for a plane wave.

Note that the eld amplitudes fall o as 1=r;and that there is a

directional dependence on sin,so that the amplitudes are zero

in the direction of the current ( =0

,and =180

),and are

maximum in the plane perpendicular to the current ( =90

).

Theory of EM Fields 30 Part I:Maxwell's Equations

Fields around a Hertzian dipole

http://www.amanogawa.com

Theory of EM Fields 31 Part I:Maxwell's Equations

Fields around a Hertzian dipole

http://www.falstad.com

Theory of EM Fields 32 Part I:Maxwell's Equations

The Scream (Edvard Munch,1893)

Theory of EM Fields 33 Part I:Maxwell's Equations

Waves in conducting media

There are some important dierences between the behaviour of

electromagnetic waves in free space,and the behaviour of

electromagnetic waves in conductors.

One signicant dierence is that the electric eld in the wave

drives a ow of electric current in the conductor:this leads to

ohmic energy losses,and results in attentuation of the wave.

A key concept is the skin depth:this is the distance over which

the amplitude of the wave falls by a factor 1=e.

We shall derive an expression for the skin depth in terms of the

properties of the conductor.

Theory of EM Fields 34 Part I:Maxwell's Equations

Waves in conducting media

To start with,we dene an ohmic conductor as a material in

which the current density is proportional to the electric eld:

~

J =

~

E:(54)

The constant is the conductivity of the material.

In practice, depends on many factors,including (in the case

of an oscillating electric eld) on the frequency of oscillation of

the eld.However,we shall regard as a constant.

Theory of EM Fields 35 Part I:Maxwell's Equations

Waves in conducting media

In an ohmic conductor with absolute permittivity",absolute

permeability ,and conductivity ,Maxwell's equations take

the form:

r

~

E =0 r

~

B=0

r

~

E =

@

~

B

@t

r

~

B=

~

E +

1

v

2

@

~

E

@t

where 1=v

2

=".

We can derive wave equations for the electric and magnetic

elds as before;but with the additional term in ,the wave

equation for the electric eld takes the form:

r

2

~

E

@

~

E

@t

1

v

2

@

2

~

E

@t

2

=0:(55)

The rst order derivative with respect to time in equation (55)

describes the attenuation of the wave.

Theory of EM Fields 36 Part I:Maxwell's Equations

Waves in conducting media

We can write a solution to the wave equation (55) in the usual

form:

~

E(~r;t) =

~

E

0

e

i(

~

k~r!t)

:(56)

But now,if we substitute this into equation (55),we nd that

the dispersion relation is:

~

k

2

+i! +

!

2

v

2

=0:(57)

In general,the wave vector

~

k will be complex.We can write:

~

k =~ +i

~

;(58)

where ~ and

~

are real vectors,that we shall assume are

parallel.

Theory of EM Fields 37 Part I:Maxwell's Equations

Waves in conducting media

In terms of the real vectors and ,the electric eld (56) can

be written:

~

E(~r;t) =

~

E

0

e

~

~r

e

i(~~r!t)

:(59)

The amplitude of the wave falls by a factor 1=e in a distance

=1=. is known as the skin depth.

The dispersion relation (57) can be solved to nd the

magnitudes of the vectors ~ and

~

.The algebra is left as an

exercise!The result is:

=

!

v

0

@

1

2

+

1

2

s

1 +

2

!

2

"

2

1

A

1

2

;(60)

and:

=

!

2

:(61)

Theory of EM Fields 38 Part I:Maxwell's Equations

Waves in conducting media

Theory of EM Fields 39 Part I:Maxwell's Equations

Waves in conducting media

Equations (60) and (61) are exact expressions for the real and

imaginary parts of the wave vector for an electromagnetic wave

in an ohmic conductor.

In the case that !"(a good conductor),we can make the

approximations:

!

v

r

2!"

;(62)

and (using v =1=

p

"):

=

1

s

2

!

:(63)

Note that the skin depth is smaller for larger conductivity:the

better the conductivity of a material,the less well an

electromagnetic wave can penetrate the material.This has

important consequences for RF components in accelerators,as

we shall see in the next lecture.

Theory of EM Fields 40 Part I:Maxwell's Equations

Energy in Electromagnetic Fields:Poynting's Theorem

Electromagnetic waves carry energy.

The energy density in an electric eld is given by:

U

E

=

1

2

"

~

E

2

(64)

The energy density in a magnetic eld is given by:

U

H

=

1

2

~

H

2

(65)

The energy ux (energy crossing unit area per unit time) is

given by the Poynting vector:

~

S =

~

E

~

H (66)

These results follow from Poynting's theorem...

Theory of EM Fields 41 Part I:Maxwell's Equations

Energy in electromagnetic elds:Poynting's theorem

To derive Poynting's theorem,we start with Maxwell's

equations.First,we use:

r

~

E =

@

~

B

@t

(67)

Take the scalar product on both sides with the magnetic

intensity

~

H:

~

H r

~

E =

~

H

@

~

B

@t

(68)

Next,we use:

r

~

H =

~

J +

@

~

D

@t

(69)

Take the scalar product on both sides with the electric eld

~

E:

~

E r

~

H =

~

E

~

J +

~

E

@

~

D

@t

(70)

Theory of EM Fields 42 Part I:Maxwell's Equations

Energy in electromagnetic elds:Poynting's theorem

Now we take equation (68) minus equation (70):

~

H r

~

E

~

E r

~

H =

~

E

~

J

~

E

@

~

D

@t

~

H

@

~

B

@t

(71)

which can be written as:

@

@t

1

2

"

~

E

2

+

1

2

~

H

2

=r

~

E

~

H

~

E

~

J (72)

Equation (72) is Poynting's theorem.Using Gauss'theorem,it

may be written in integral form:

@

@t

Z

V

(

U

E

+U

H

)

dV =

I

A

~

S d

~

A

Z

V

~

E

~

J dV (73)

where the closed surface A bounds the volume V,

U

E

=

1

2

"

~

E

2

U

H

=

1

2

~

H

2

(74)

and:

~

S =

~

E

~

H (75)

Theory of EM Fields 43 Part I:Maxwell's Equations

Energy in electromagnetic elds:Poynting's theorem

Poynting's theorem in integral form is equation (73):

@

@t

Z

V

(U

E

+U

H

) dV =

I

A

~

S d

~

A

Z

V

~

E

~

J dV

We note that the last term on the right hand side represents

the rate at which the electric eld does work on electric

charges within the bounded volume V.It is then natural to

interpret the rst term on the right hand side as the ow of

energy in the electromagnetic eld across the boundary of the

volume V,and the left hand side as the rate of change of the

total energy in the electromagnetic eld.

With this interpretation,Poynting's theorem expresses the local

conservation of energy.

Theory of EM Fields 44 Part I:Maxwell's Equations

Energy in an electromagnetic wave

As an example,let us calculate the average energy density and

the energy ux in a plane electromagnetic wave in free space.

From equations (24) and (27),the (real) electric and magnetic

elds are given by:

~

E =

~

E

0

cos(

~

k ~r !t);(76)

~

H =

~

H

0

cos(

~

k ~r !t);(77)

where:

E

0

B

0

=c =

1

p

0

"

0

;)

E

0

H

0

=Z

0

=

s

0

"

0

:(78)

Note (in passing) that Z

0

is the impedance of free space.Impedance will

play an important role when we come to consider waves on boundaries.

Theory of EM Fields 45 Part I:Maxwell's Equations

Energy in an electromagnetic wave

The energy densities in the electric and magnetic elds are:

U

E

=

1

2

"

0

~

E

2

;U

H

=

1

2

0

~

H

2

:(79)

But since H

0

=E

0

=Z

0

,we nd:

U

H

=

1

4

"

0

~

E

2

=U

H

:(80)

In other words,the energy density in the magnetic eld is equal

to the energy density in the electric eld (for a plane

electromagnetic wave in free space).

Theory of EM Fields 46 Part I:Maxwell's Equations

Energy ux

The Poynting vector (which gives the energy ow per unit area

per unit time)

~

S is dened by:

~

S =

~

E

~

H (81)

Since

~

E

0

and

~

H

0

are perpendicular to each other and to

~

k (the

direction in which the wave is travelling),and the amplitudes of

the elds are related by the impedance Z

0

,we nd that:

~

S =

E

2

0

Z

0

^

k cos

2

(

~

k ~x !t):(82)

We see that (as expected) the energy ow is in the direction of

the wave vector.

The amount of energy carried by the wave depends on the

square of the electric eld amplitude,divided by the impedance.

Theory of EM Fields 47 Part I:Maxwell's Equations

Summary

Maxwell's equations describe the constraints on physical

electric and magnetic elds.

In free space,electromagnetic waves can propogate as

transverse plane waves.For such waves,the wave vector

~

k

denes the\mode"of the electromagnetic elds.

In conductors,electromagnetic waves are attenuated

because the energy is dissipated by currents driven by the

electric eld in the waves.

Electromagnetic waves can be generated by oscillating

electric charges.

Poynting's theorem provides expressions for the energy

density and energy ux in an electromagnetic eld.

Theory of EM Fields 48 Part I:Maxwell's Equations

Appendix A:curl in spherical polar coordinates

In spherical polar coordinates,the curl of a vector eld is given

by:

r

~

A

1

r

2

sin

^r r

^

r sin

^

@

@r

@

@

@

@

A

r

rA

r sinA

(83)

Theory of EM Fields 49 Part I:Maxwell's Equations

Appendix B:Exercises for the student

1.Derive the wave equation for the electric eld in a

conductor.Show that the real and imaginary parts of the

wave vector have magnitudes given by equations (60) and

(61).

2.Estimate the skin depth for microwaves in copper.

3.Find an expression for the total power radiated by a

Hertzian dipole.

Theory of EM Fields 50 Part I:Maxwell's Equations

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