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Lecture 1
Electromagnetic waves. The electromagnetic spectrum
Waves: disturbances propagating in space and time. Categorizing:
According to the nature of the disturbance
• Mechanic
• Electromagnetic
• Matter
According to the direction of the disturbance relative to the direction of propagation
• Transverse
• Longitudinal

Figure 1. Transverse and longitudinal waves

Electromagnetic waves:
The disturbance is a superposition of oscillating electric and magnetic fields
Transverse - the fields oscillate perpendicularly to the direction of propagation of the
wave
m
They can propagate in empty space at a speed c= 1 / ε 0 µ0 = 3 × 108
s

A bit of history: The existence of the electromagnetic waves was predicted by Robert
Maxwell, the Scottish physicist who conceived a unified theory of electrical and
magnetic phenomena. His theory is based on four fundamental equations which
connect the sources of the fields to the physical quantities which describe their space-
time evolution. I will present here only the differential, local form of these equations.
When you say “local” form it means that these equations are valid at any position and
     
at any t, that is: E = E ( r , t ) and B = B ( r , t ) .
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
  ρ (r ,t )
(1) ∇ ⋅ E (r ,t ) =
ε0
 
( 2) ∇ ⋅ B (r ,t ) =0
 
  ∂B ( r , t )
( 3) ∇ × E (r ,t ) = −
∂t
 
    ∂E ( r , t )
( 4) ∇ ×=B ( r , t ) µ0 J ( r , t ) + ε 0 µ0
∂t
  
In the above equations E = E ( r , t ) is the electric field intensity, the vector quantity
V   
which describes the electric field (measured in volt per meter, ) and B = B ( r , t ) is
m
the magnetic field induction, the vector quantity which describes the magnetic field
(measured in tesla, T ).

The sources of the two fields are: the volume charge density ρ = ρ ( r , t ) measured in
C   
- for the electric field, and the electric current density J = J ( r , t ) , measured in
m3
A
- for the magnetic field. Note that the volume charge density is defined as the
m2
electric charge per unit volume which generates the electric field. The electric current
density is the electric current per unit area. If the electric charge and current do not
change in time, the field is described as a stationary field.
Here the medium in which the two fields coexist and generate each other is vacuum,
described by the material constants ε 0 - the dielectric permittivity, also named the
electric constant of vacuum and µ0 - the magnetic permeability, also named the
magnetic constant of vacuum.
∂  ∂  ∂ 
The del operator (also called nabla)= ∇ u x + u y + u z is applied to the
∂x ∂y ∂z
vector quantities of the electric and magnetic fields in a scalar (dot) or vector (cross)
operation to give the divergence and the curl of the vector quantities. So:
   ∂  ∂  ∂      ∂Ex ∂E y ∂Ez
∇= ⋅ E div = (
E  u x + u y + u z  Ex u x + E y u y + Ez u= z ) + +
 ∂x ∂y ∂z  ∂x ∂y ∂z
and
  
 ux u y uz 
 
  ∂ ∂ ∂
∇ × E curl
= = ( )
E 
 ∂x ∂y ∂z 
.
 
 Ex E y Ez 
Equation (1) is known as Gauss law and it connects the divergence of the
electric field to the charge density which is the electric field source; also, it shows that
the electric field lines are open, starting from or ending into that source (electric
charge).
Equation (2) is known as Gauss law for magnetic fields, and it shows that there
exist no magnetic charges, and the magnetic field lines are closed.
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Equation (3) is known as Faraday law, and it shows that a time changing
magnetic field will generate an electric field. This is also known as the law of
electromagnetic induction.
Finally, equation (4) is known as Ampere’s law. It states that the magnetic field
 
is generated by a current ( the term µ0 J ( r , t ) ) and/or by time changing electric fields
 
∂E ( r , t )
- the term ε 0 µ0 . The second term is also known as the displacement current
∂t
and was added to Ampere’s law by Maxwell himself.
If we want to retrieve the electromagnetic wave equation for waves
propagating in vacuum, starting from Maxwell equations, we need to write those
equations in the absence of the sources. This does not mean that the sources are
missing altogether, but it means that we are looking at a region in space where there
are no field sources. Thus, Maxwell equations become:
 
(1) ∇ ⋅ E (r ,t ) =0
 
( 2) ∇ ⋅ B (r ,t ) =0
 
  ∂B ( r , t )
( 3) ∇ × E ( r , t ) = −
∂t
 
  ∂E ( r , t )
( 4 ) ∇ × B ( r , t ) = ε 0 µ0 .
∂t
We are also going to use an identity known from vector analysis:
  
( ) ( )
∇ × ∇ × V = ∇ ∇ ⋅ V − ∇ 2V ,

where V is an arbitrary vector into Faraday’s law. We apply the curl operator ( ∇ × )
to equation (3). Keep in mind that curl and div act only on space coordinates, so they
∂
can change places with the time derivation operator, .
∂t

  ∂B 
(
∇× ∇× E = ∇×−  )
 ∂t 
  ∂ 
(

)
∇ ∇ ⋅ E − ∇2 E = −
∂t 
(
∇× B
 
)

=0 ∂E
ε 0 µ0
∂t

We obtained an equation in the electric field intensity vector E :
 
2
 ∂ ∂E  ∂2 E
−∇ E = −  ε 0 µ0  = −ε 0 µ0 2
∂t  ∂t  ∂t
Or
 
  ∂ 2
E (r ,t )
∇ 2 E ( r , t ) − ε 0 µ0 0
= (5)
∂t 2
This is a differential wave equation with partial derivatives of the second order which
describes the space-time evolution of the electric field vector in the electromagnetic
wave.
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In a similar way, by applying the curl operator to equation (4), we get the wave
equation which describes the space-time evolution of the magnetic field vectot in the
electromagnetic wave
 
  ∂2 B ( r , t )
∇ B ( r , t ) − ε 0 µ0
2
0
= (6)
∂t 2
In equations (5) and (6), the squared del operator is the Laplace operator (nabla) which
is a scalar differential operator whose action is given by the following:
 ∂  ∂  ∂    ∂  ∂  ∂  
∆= ∇2 =  ux + u y + uz  ⋅  ux + u y + uz  =
 ∂x ∂y ∂z   ∂x ∂y ∂z 
∂2 ∂2 ∂2
= + +
∂x 2 ∂y 2 ∂z 2
Then, the wave equations of an electromagnetic wave which propagates in a vacuum
can be written as
       
∂2 E ( r , t ) ∂2 E ( r , t ) ∂2 E ( r , t ) ∂2 E ( r , t )
+ + − ε 0 µ0 0,
=
∂x 2 ∂y 2 ∂z 2 ∂t 2
       
∂2 B ( r , t ) ∂2 B ( r , t ) ∂2 B ( r , t ) ∂2 B ( r , t )
+ + − ε 0 µ0 0.
=
∂x 2 ∂y 2 ∂z 2 ∂t 2

These are differential equations with partial derivatives, of the second order, in the
   
unknown functions E ( r , t ) and B ( r , t ) . They look exactly like an arbitrary wave

equation in which the disturbance is denoted by ψ ( r , t ) , for example:
   
∂ 2ψ ( r , t ) ∂ 2ψ ( r , t ) ∂ 2ψ ( r , t ) 1 ∂ 2ψ ( r , t )
+ + − 2 0,
= (6)
∂x 2 ∂y 2 ∂z 2 v ∂t 2

where v is the speed of the wave, and ψ ( r , t ) is a generic denotation for the wave
disturbance. This led R. Maxwell to the conclusion that there should be such a wave,
called electromagnetic wave, in which the disturbance is a superposition of variable
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electric and magnetic fields propagating at a speed v = . Such waves had not
ε 0 µ0
been experimentally obtained at the time he published his work! The surprise was that
by taking the values of the electric and magnetic constants of vacuum into that
m
expression a value of 3 × 108 is obtained! Some of the physicists of the time
s
(Young, Fresnel) used this observation as a proof that light is an electromagnetic wave.
This allowed for a fast development of the wave optics.
The spectrum of the electromagnetic waves covers a huge range in
wavelength/frequency (see Figure 2 and 3 below). Note that the frequency f is
connected to the wavelength λ by the equation
c
f = .
λ
The spectrum is customarily divided into regions which do not have very well-
defined boundaries. The individual regions (domains) are determined by the method to
obtain those waves. Their frequency values are like a “fingerprint” of the wave, they
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do not change when the waves propagate through different media. The wavelength
does depend on the medium in which travels. So, when we say that the wavelength in
the visible region is approximately in the range [400 nm, 700 nm] we consider that
light is propagating in vacuum (see Figure 4). Please note that the visible range is a

tiny fraction of the electromagnetic spectrum.

Figure 2. The domains of the electromagnetic spectrum.

Figure 3. The electromagnetic spectrum domains – type, production,

Figure 4. The visible range of the electromagnetic spectrum.