A. Marconi

A Review of Electromagnetic Field Properties

Lecture 1Radiative Processes in AstrophysicsCourse on Relativistic Astrophysics AA 2016/2017

MAXWELL 'S EQUATIONS

Maxwell 's equations m the

InternationalSystem ( is ) take the Well known

form :

T.

Et- :

% .

BEa o

E- B- no ( Tt Effi ) - o

T×F + 21- o

Jt

E is the electric field ,

$ the magneticfield

,s is the charge density

,

T is the change current check is

related to the density byT - g F with to change velocity

E a 8.854 × 1512 Food m

''

is the

vacuum permittivity

µ° is the magnetic constant or

vacuum permeability Chuh molded

to E by the Well know ulotwn

No 6 -ch C sneed of light

In Astrophysics it is common to

use c.gs .nuts m klmch

the permittivity of vacuum

becomesE - 1¥ henceno

-

HEThis result is the consequence

of Writing the Coulomb forcebetween two point - like charges

cos ( m module ) :

For a

9£92

noted of7

cgs

Foul a kg 9%1 1.

s.

It also turns out that thec.g.es .

change unit is

1 l.

5. µ

.

= 3.3356×1510 (

e.s.ie .means Electrostatic Unit

,also

known os Stotculmb.

The election charge here is

@ - 4.8×510 E. S.

µ.

Maxwell'

equations m Cg

s nuts

then assume the from

8.

LF- at

E.B- °

8×8- E f¥ - he 5

Fx E + £ green - o

Maxwell 's equations include

the change continuity equotwn .

Indeed it is to obtain that

2£ + Bjs a o

Chuh, with T -

g8

,

her the form ofthe cloned continuity equationceheh is

also Well known fwmfluid mechanics

.

ENERGY CARRIED BY THE E. M,

FIELD

Considering or system of changesand currents Cluck one subjected

only t.oe.mn. forces,

we can

Consider .

to total mechanicalenergy

perunit volume

u em. field energy density

M a t# ( E2+ BYHis possible to show that

bond u follow the so - called

Poyntmy Theorem according to

Cch

;z( Wtu ) +8.8 - o

cohere 5 is the Poyntmy vector

defined as

5- at EII

|§| represents the e.

m. energy

Cheh is flowing her unit time

through a met surface,

placed

perpendicularly to the direction

of 5 ( direction along check the

e. m. fueled is propagating ) .

Indeed the dimensions of$1ore those of

151 -DE←

-

At

DE energy ,Dst Surface t

At time intervals

151 is then the energy flux ofthe em

. held .

MOMENTUM CARRIED BY THE EM.

FIELD

If £ is defined as the momentum

perund volume of the system

of charges and currents duck

interacts only through e. m.

forces ,

it can be shown that

Z,(£+tas→ ) - EE '

Ghere I is Maxwell 's Tensor

given by

I - tgu ( EEHBB ) - { ( E 2+56TU

isthe unit tenor which m

Cmponerito is Sij ( Krone deer 'S

delta, Sij - I iej

, Sg- o ttj ) .

Maxwell 's tensor is symmetricand ,m components

,on be widen

=ij

a Tji e at ( Eitj + Bi Bj ) -

- ft ( EH BY Sij

Going back to theprevious

equationIt ( £ +

EE) - FI

We can note that the divergenceof a tensor is

, of course,

d vector.

1g 5 Zefresento The momentum

per unit volume carried

over by the e.

m. field .

Sf we new consider DE,

as

on infinitesimal surfacewith normal unit vector m→

it twins out that

IF - -re

.

' T.DE'

is the force acting on the systemand re .

'

I is the momentum

flux through DE.

ELECTROMAGNETIC POTENTIALS

Ttmpombleto define the solar

and vector pattntials to and at

such that

B=

Fx £

E - -8$ - { JAIat

Inserting into Maxwell 's equationswe can obtain that

02£ taste . Be .E+±t#= . hers

0210 - ⇒Y÷+⇒÷(eF¥s¥)=#swith

given boundary conditions,

it

is ramble to obtain A and to

hence E→ and B,

E. and B- have a

gougeinvariance

for the transformations :

E. - At - E. -8x

.

to .→ to - to

. to

3¥this means that Es and B ore

the some if one uses Ro,

to a

A→ and to .

thus We have the

freedom of selecting theX.function . One possible ,Convenient

choice is to select a functionX. such that

onto - tastes - 8 .F+±dq÷in this case

, for £ and to we have

T.es + to 3¥ - o

and the equations for it and to

simplify to

02h - tames = - are0210 - ta YI = - uts

The equation 8£ + to 3¥ - o

is the so - called Loansgauge

( it is WREMZ not WREMTZ ) .

In the vacuum,

with no free charges ,

We can make on additional

tzosfrmetum within the Loans

gouge with a new Xz fun down

such that 10 - o;

we them have

part - sgajtee - °

HE:Othis is vohd only for vacuum

.

GENERAL SOLUTION OF MAXWELL 'S

EQUATIONS IN VACUUM

In vacuum f - o

,J - o them

,

elaborating Maxwell 's equations we

obtoaEs- tastes -

the general solutions of this equationCan be expressed os the superposition

If phone waves of the kind

Eu - de Ee e

:( 6.8 - at )

Be a da Be e:C Lie

.at )

As,

£z are versos fumet vectors )which are perpendicular To the

propagation direction,

and nerves

dealer amongthemselves

the direction along cluck the phoneWave prorogates is

m→ -

@Than fy

El

or,

x

DIRECTION

OF PROPAGATION

£ z

We can them show that.

B - n→ ×E

hen

;BI - IETwe then find that

µ a }p ( E2

+ BY a

ten EZ

5a f- Ex I

- £E2m→the momentum density g→ is

g→ - Is - EI m→- acne

Finally the momentum fluxthrough the arbitrary infinitesimalsurface d£ with normal versos

A'

is

me'

m→ g

: E

Ffi '

) - -a

'

. I'

- 1g Eyni'

.

ri )n→

always directed along themoneys

Tian direction but with a projectionf.eduna

'

.

m→ ease

The momentum received

from d {'

in time intenddtnequal to

des - Fti '

) dsidt

hence We can obtain the zodotum

pressure

Pros - §9÷ . dn÷ - a¥ase=uas2o

SPECTRUM OF THE RADIATION

& monochromatic phone wave such

as the one we have just seen us

on idealisation.

In zeethty the

electric fiend E ( givenE Ion

always obtain B ) is limited in

space and time.

In general Es will be houeetensed

by a generictime dependence

of outs modulegiven by

1E→/ - Ect )

chuck I can measure for esuffaently

long time.

Ooowmmy that

Ect ) → o for t→±a

We can take the Fourier transformof ECD

EH . saftFg eiotdt

and the inverse transform is

EE ) - £t¥Cw ) e- iotdu

If ECH vows with time,

also

theenergy flux of the

electromagneticfilled varies

FC 'D - f= EYD

asgiven by the Poyntmy vector

module .

The total amount of energy per

unit surface whichgoes

through e unit surfaceperpendicularto the propagation direction

-

F - Eta dt . §±ft¥G) dt

tokmy advantage of Personal 's

theoremF4Ddt= ¥ 'TEwPdw

we can them show that

I - of 'TEwPdw

cletwjpm the totalenergy

Gluck flaws through the unit

surface in the spectral zonyedw dummy the time interval

between t - -o and E - too

.

1 Ecu ) Prs The spectrum of the

electromagnetic radiation

If the"

observing" time of the

e. m. field is not too, to )

but is limited to

thenmtevol of mon

e,

roamingthd e is much longer than

the tgnmadvouabihty timescales

of ECD we on write

2/2

ECW , e) - ⇒ Eat eiwtdt

- 42

and

Fe- off Ecu , e) Pdw

The bddhon flux is then

F - fafadw - tee - Effetqdpdwand the monochromatic flux is

Fa a { / Ease ) 12

ELECTROMAGNETIC POTENTIALS FROM

CHARGES Ab CURRENTS

We have found that

Be- 8×£

Ea - 810 - § 2£←-

Fn the static Case,

it is Well

known that

loci ) .

Head"

'

Chen T is the volume occupied

by the charge density It ) .

An general , starting from the

static equation andusing

the

Loansgouge ,

it is noonbee to

obtain that

0210 - sa 3¥ -- ars

and that

KFH =

f ,sfEif→-d3i'

with t'

retarded time definedas

t'

a t -

lorry

this means that

'

the value ofthe potential to in I at time

t depends upon the chargedistribution m all I

'

noinntobut"

seen

"

at the time the

light left I '

to reach r→.

This means that at time tm

8 one sees the charge m

re'

ot time tminus theme

needed fr the nynol topropagate

from L'

to 8.

Smloly we have

ACAD . { FREIf

trzsy

9321

10,

A so defined one called the

retooled potentials .

St is mosableto verily that

they satisfy the condition

set by the Lorena gouge

F.Fttattoo

C at

THIS 15 THE EMD OF THE REVISION

OF THE TOPICS WHICH WERE

STUDIED DURING THE LECTURE OM

OM ELECTROMAGNETIC FIELD ( FISICAII ) .