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16 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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Lecture 18 Maxwell’s equations:
electromagnetic waves

Introduction

One of the major achievements of Maxwell
was the correct prediction of electromagnetic
waves from considerations of the forms of the
four equations which bear his name.

Maxwell’s equations

The general form of Maxwell’s equations (in
terms of
B

and
E
), valid in the presence of
dielectrics, magnetic materials, free charges
and currents is:

0
0 0 0
//
0/
C
E E B
B B J E
f r
r r r
t
t
 
 
     
      
We wish to consider the possibility of
electromagnetic waves in a vacuum where there
are no dielectrics, magnetic materials, free
charges and currents. Hence we have
J
C
=

f
=0

and

r
=

r
=1

and the form of Maxwell’s equations
can be considerably simplified to:

0 0
0/
0/
E E B
B B E
t
t

     
     
E

and
B
.
B

can be
eliminated by first taking the curl of the top RHS
equation

0 0
0/
0/
E E B
B B E
t
t

     
     
2
0 0
2
( ) ( )
B E
E B
t t t

  
 
        
 
  
 
where the final term is obtained by using the
lower RHS Maxwell equation to eliminate

B
.

The next step is to simplify the initial term

(


using the identity


(

䔩E

(


-

2
E

but the top LHS Maxwell equation gives

E=
0

so


(

䔩E
-

2
E

and hence we have finally

2
2
0 0
2
E
E
t


 

B

can also be eliminated to obtain a similar
equation

2
2
0 0
2
B
B
t


 

The identity

2
A

involves the application of the
Laplacian to a vector. For a scalar this gives
three components, for a vector there are nine
components, three for each spatial direction. For
example the
x
-
component of the resultant vector

2
A

has terms

2 2 2
2 2 2
x x x
A A A
x y z
  
 
  
with similar terms for the
y

and
z
-
components.

2
2
0 0
2
E
E
t


 

The form of the above equations for
E

and
B

is
easier to see if we simplify them. For example if
the
y
-
component

of the
E
-
field

(
E
y
) depends only
on one spatial co
-
ordinate (e.g.
x
) then
E
y

must
satisfy the equation

2 2
0 0
2 2
y y
E E
x t

 

 
but this is a one
-
dimensional wave equation
that describes waves which propagate with a
velocity
v
=(

0

0
)
-
½
.

Worked Example

Show that a function of the form

0
sin( )
A A kx t

 
represents a wave travelling along the x
-
axis
and that it satisfies the 1D wave equation

2 2
2 2 2
1
A A
x v t
 

 
Hence both
E

and
B

fields may exist in the
forms of waves

electromagnetic waves.

However if we calculate the value of
(

0

0
)
-
½

we
find it has a value equal to
c
, the speed of light.

Hence the electromagnetic waves predicted by
Maxwell’s equations propagate at the speed of
light, suggesting that light itself is a form of

Specific properties of electromagnetic

The wave equations for
E

and
B

tell us that
electromagnetic waves propagate at the speed of
light.

Further information on the form of these waves
can be obtained by noting that the
E
-

and
B
-
fields

must also satisfy Maxwell’s equations in addition
to the wave equation.

We consider a specific type of wave, although the
results derived are general.

This wave is said to be an unbounded, plane one

Plane indicates that there exist planes (or
wavefronts), which are perpendicular to the
direction of propagation, over which all quantities
of the wave are constant.

Unbounded indicates that the wave fronts can be
considered to be of infinite extent.

Consider an unbounded plane wave propagating
along the
x
-
direction
. The wavefronts for this wave
are
y
-
z

planes. Because all quantities must be
constant over a given wavefront none of the
components of the
E
-

and
B
-
fields

can be a
function of
y

or
z
.

Therefore all terms with

/

z

or

/

y

must be zero
and Maxwell’s equations become

0 0
E
y
x x
z
E
E E
E
x y z x

 

      
   
Similarly from

B=0

0
x
B
x

0
B
E
y
x
z
y
x
z
y
z
t
E
B
E
x component
y z t
B
E
E
y component
z x t
B
E
x t

  

 

    
 
  
 

 
   
 
  
 

 
 
y
x
z
y
z
E
E
B
z component
x y t
E
B
x t

 

   
 
  
 

  
 
Finally from

0 0
0 0 0 0
0,,
E
B
y y
x
z z
t
E B
E
B E
t t x t x

 

 

 

 
    
    
0
x x x x
E E B B
x t x t
   
   
   
From the above terms we immediately have

therefore the components of
E

and
B

along the
direction of wave propagation (
x
) neither vary in
time or space. These components must hence be
zero or at most constant and therefore can not be
part of the waves.

Hence for an electromagnetic wave there are no
components of
B

and
E

in the direction of
propagation, the wave is therefore
transverse
.

We now assume that the wave is polarised so that
the
E
-
field

lies along the
y
-
direction

(i.e.
E
y

0
,
E
z
=0
). The above terms now give

0
y y
B B
x t
 
 
 
therefore if
E

is polarised along
y

then
B

can have no
component in this direction

B

must be polarised
along
z
.

Hence for an electromagnetic wave
E
,
B

and the
direction of propagation are mutually perpendicular.

A wave for which both
E

and
B

are transverse is
known as a
TEM

wave (transverse electric and
magnetic).

Finally we determine the relative sizes of
E

and
B
. As
E

is polarised along
y

and
B

along
z

we
can write their spatial and temporal
dependencies as (monochromatic waves):

0 0
sin( ) sin( )
y y z z
E E kx t B B kx t
  
    
where the prefactors are constants and

is a
phase factor allowing for a possible phase
difference between
E

and
B
.

From above so

y
z
E
B
x t

 
 
0 0
cos( ) cos( )
y z
kE kx t B kx t
   
   
This equation can only be satisfied if

=0

(i.e.
E

and
B

are in phase) and

0 0 0 0 0
(/)
y z y z z
kE B E k B cB
 
   
Hence

at

any

point

and

time

the

amplitudes

of

E

and

B

are

in

the

ratio

c
.

The Electromagnetic Spectrum has no limits in terms of frequency (or
wavelength

=
c
/
f
).

In practical terms it runs from radio waves ~10
3
Hz to gamma waves
~10
20
Hz.

All frequencies obey the physics described in this lecture.

Different parts of the Electromagnetic Spectrum differ in terms of how
the waves are produced, how they are deselected and the effect they
have on physical and biological systems.

Conclusions

Maxwell’s equations in a vacuum and in the
absence of dielectrics and magnetic materials

Derivation of the wave equations for
E

and
B

Existence of electromagnetic waves
propagating at the speed of light

Transverse nature of electromagnetic waves

Relationship between amplitudes of
E

and
B

2
2
0 0
2
E
E
t


 

2
2
0 0
2
B
B
t


 