Download Chapter 15 ELECTROMAGNETIC WAVES IN VACUUM

Chapter 15 ELECTROMAGNETIC WAVES
IN VACUUM
• Introduction
• Maxwell equations for vacuum
• Transverse plane wave
• Relation of  and  fields
• Wave equation
• Generation of electromagnetic waves
• Electromagnetic spectrum
• Energy and momentum in electromagneitc waves
• Summary
INTRODUCTION
Last lecture presented the Maxwell’s equations in the
final form.
Gauss’s Law for electric field:
Φ =
I
→
− −
→
1
E · S =

0
 
Z



Maxwell’s equations display a symmetry between electric and magnetic fields. However, this symmetry is
broken in that there are no magnetic monopoles and
no current of magnetic monopoles corresponding to
charges and electric current due to moving charges.
Maxwell used vector differential calculus to prove,
in a few steps, that his equations predict the existence of electromagnetic waves. Maxwell was astonished to discover that his predicted electromagnetic
waves propagated with the known velocity of light,
leading him to conclude that light was electromagnetic
radiation. It was not until after the death of Maxwell
that Hertz demonstrated that electromagnetic waves
are generated by oscillating electric fields. This chapter will show that Maxwell’s equations are consistent
with the existence of electromagnetic waves. This will
be proved using only integral calculus.
To simplify the mathematics, the discussion will
be restricted to discussion of plane waves in vacuum
which are the simplest type of wave to consider. A
plane wave is a function of only one direction, e.g. z,
and is uniform as a function of the other orthogonal
directions. As discussed last lecture all waves satisfy
the Wave Equation. In one dimension a travelling wave
in the  direction with velocity  satisfies the equation
2Ψ
1 2Ψ
= 2 2
2

 
Gauss’s law for magnetism
Φ =
I

 
→
− −
→
B · S = 0
Faraday’s Law
I
→
− −
→
 E · l = −


Z
 



−→
→
B −
· S

Ampère-Maxwell law:
I
→
− −
→
 B · dl = 0


Z
→
−
→
−
→
E −
(
j
+

) · S
0



This wave equation in one dimension is independent
of the sign of the velocity. There are an infinite number of possible shapes of waves travelling in one dimension, all of these must satisfy this one-dimensional
wave equation. The converse is that any function that
satisfies this one dimensional wave equation must be a
wave in this one dimension.
One example of a solution of this one-dimensional
wave equations is the sinusoidal wave
Ψ() =  sin([ −
where  =  =  The wave number  = 2
  where 
is the wavelength of the wave, and angular frequency
 = 2. Note that this satisfies the above wave equation where the wave velocity equals  =  =  
The Wave Equation in three dimensions is

Maxwell equations have had a profound impact on science and technology. The addition of the displacement current, needed to satisfy charge conservation,
is the key to the prediction of electromagnetic waves.

]) =  sin( − )

∇2 Ψ ≡
2Ψ 2Ψ 2Ψ
1 2Ψ
+
+
=
2
 2
 2
2 2
There are an infinite number of possible solutions Ψ to
this wave equation, any one of which corresponds to a
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wave motion with velocity . In the subsequent discussion of electromagnetic waves, the Maxwell Equations
will be used to deduce a wave equation. This is equivalent to proving the existence of electromagnetic waves
of any wave form.
MAXWELL’S EQUATIONS IN
VACUUM
Maxwell’s Equations in vacuum, that is, assuming zero
→
−
current and charge densities, j =  = 0 are;
I
→
− −
→
E · S = 0
I
→
− −
→
B · S = 0

 

 
I
→
− −
→
 E · l = −
I


Z
→
− −
→
 B · dl = 0 0


 



Z
−→
→
B −
· S

→
−
→
E −
· S




Note the symmetry both for the flux and circulation equations. The right-hand side of the last equation comes from Maxwell’s displacement current. The
only non-symmetric aspects are the negative sign in
Faraday’s law, and the product of the constants, 0 0 
TRANSVERSE PLANE WAVE
An especially simple form of wave motion to handle
mathematically, is a plane wave which travels in one
direction,  and is uniform in the two other orthogonal
directions,  . Such plane waves will be considered
in this chapter to simplify the mathematics. Let us
consider a plane wave travelling in the  direction, that
is, it is uniform everywhere in the  and  directions.
That is:
→ −
−
→

E = E0 sin([ − ])

b diThis corresponds to a wave propagating in the k
rection with velocity v.
Wave motion can be transverse or longitudinal to
the direction of propagation. Waves on a string, and
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Figure 1 Flux out of an infinitessimal box of size ∆
∆ ∆ at x,y,z.
water waves are oscillations transverse to the direction
of propagation, whereas sound waves in air are longitudinal oscillations. Earthquakes excite both longitudinal and transverse waves in the earth. Transverse
waves can have polarization, that is the orientation of
the transverse oscillation can be primarily in one direction. The first question is can both transverse, and
longitudinal electromagnetic plane waves exist. To address this question consider the flux for an infinitesimal
rectangular box shown in figure 1. For the plane wave
chosen,  = +∆  and  = +∆  Therefore the
net flux out of the box in the x and y directions is zero
since the inward flux cancels the outward flux in each
of these directions. But from Gauss’s law in vacuum
we have that;
I
→
→ −
−
E · S = 0

 
Thus, since for a plane wave the net flux in the x and
y directions in zero for the box, then the net flux in
the z direction also must be zero, that is  = +∆
implying that the z component of the electric field,
  propagating in the z direction, is uniform, that is,
there is no oscillatory E field pointing in the direction
of propagation. A uniform E field in the direction of
propagation means that the component in the direction of propagation is not a travelling wave. That is,
the oscillating electric field is transverse to the direction of propagation.
Consider a plane magnetic wave propagating in the
z direction. For this plane magnetic wave the magnitude of B is uniform everywhere in the x and y directions. That is,  = +∆ and  = +∆ .
therefore the net flux out of the box in the x and y
directions is zero. From Gauss’s Law we have that:
I
→
→ −
−
B · S = 0

 
Thus since the net flux in the x and y directions is
zero then the net flux in the z direction must be zero,
that is,  = +∆ implying that the z component
propagating in the z direction must be uniform, that is,
there is no oscillatory B field pointing in the direction
Figure 2 A rectangular line integral in the x-z plane.
of propagation. Thus the oscillating magnetic field is
transverse to the direction of propagation.
The above proofs show that both the electric and
magnetic fields of an oscillatory travelling wave are
transverse to the direction of propagation. That is,
since we have chosen z as the direction of propagation,
then the transverse E and B fields must be in the x-y
plane.
RELATION OF  AND  FIELDS
Maxwell’s two flux equations led to the conclusion that
electromagnetic waves are transverse, not longitudinal,
waves. For simplicity, let us assume that the electromagnetic wave, propagating in the z direction, has the
electric vector pointing in the  direction. That is, the
travelling wave is assumed to be of the form:
−
→

E = 0bi sin([ − ])

Let us now use Maxwell’s two circulation equations to
derive the relation between the transverse electric and
magnetic waves.
Maxwell’s expression of Faraday’s law states that:
I
→
− −
→
 E · l = −


Z
 



−→
→
B −
· S

Figure 3 Rectangular line integral in the y-z plane.
Canceling the area ∆∆ on both sides gives:


=−


(1)
Thus we have that there is a relation between the spatial derivative of  and the time derivative of  
Thus, the rate of change of magnetic flux in the y direction has created a  field varying in the z direction.
The fact that there can be a non-zero  field,
makes it useful to make use of the Maxwell-Ampère
circulation equation:
→
−
I
Z
→
→
→ −
−
E −
B
·
dl
=


· S
0 0 






The right-hand term is due to Maxwell’s displacement
current.
Consider the line integral for  taken in a clockwise
direction about the +bi direction as shown in figure 3.
The contributions from the legs are, − ( + ∆)∆bj
b + ()∆bj + ∆ k
b Thus Maxwell’s equa− ∆ k
tion gives:
[− ( + ∆) +  ()]∆ =  

∆∆

where the right-hand side is the rate of change of electric flux for the area ∆∆ The left-hand side can be
rewritten as:
−[

 ( + ∆) −  ()
]∆∆ =  
∆∆
∆

and the area ∆∆ on both sides cancel leaving:
Consider the line integral for E around the loop in the
x-z plane taken in a clockwise direction about the +bj
direction as shown in figure 2. The contributions from
b
each leg of the loop are + ( + ∆)∆bi − ∆ k
b
b
− ()∆i, and + ∆ k Thus Faraday’s law gives:
[ ( + ∆) −  ()]∆ = −

∆∆

The right-hand term is the rate of change of magnetic
flux for the area ∆∆ pointing in the +y, bj direction.
The left-hand side can be rewritten as:
[
 ( + ∆) −  ()

]∆∆ = −
∆∆
∆

−


=  


(2)
This relates the spatial derivative of  to the time
derivative of   Thus the rate of change of electric
flux in the x direction has created a  field varying
in the z direction.
Equations 1 and 2 describe a system of coupled
electric and magnetic fields where an oscillating 
field generates a travelling  wave and an oscillating
 field generates a travelling  wave. Now it is
necessary to show that the non-zero derivatives of 
and  with respect to the  direction do correspond
to a travelling wave in the z direction.
115
WAVE EQUATION
As already discussed, a travelling wave moving in the
+ direction of the form
Φ( ) =  ( − )
moves with velocity  =
of the wave equation:


Moreover, this is a solution
1 2Φ
2Φ
=
 2
2 2
(Wave equation)
All possible travelling waves in the ± direction satisfy
this equation and they travel with velocity . Thus if
we derive such an equation for electromagnetism, then
immediately it tells you that electromagnetic waves
exist travelling with velocity .

Differentiate equation (1) by 
and equation (2)

by  give the following;
2
2
 
 
=  

2
Combining these gives the equation:
−
 2 
 2 
=  
2

2
(3)

and equaSimilarly, differentiating equation (1) by 

tion (2) by  and combining these equations gives:
 2 
 2 
=




 2
2
(4)
Equations (3) and (4) are wave equations for both
the electric and magnetic waves travelling at a common
velocity in vacuum, called c, of:
1
≡= √
 
Inserting the known values  = 885410−12  2  2
and  = 410−7 2 gives that the velocity of an
electromagnetic wave in vacuum is:
 = 299792458 × 108 
This equals the measured velocity of light and radio
waves in vacuum. Note that  is now chosen to be
a fundamental constant of nature, and in the SI system of units, the constant 0 was chosen to be  =
410−7 2 exactly. Thus the permittivity of free
space is given in terms of these two constant by:
 =
116
1
 2
This equation is a manifestation of the interrelation
of the electric and magnetic fields given by Einstein’s
theory of relativity.
Assuming that the electric field is:
−
→

E = 0bi sin([ − ])

then equations (1) and (2) give:
−
→
0 b

B=
j sin([ − ])


 2 
 2 
=
−
 2

and:
Figure 4 The electric and magnetic field vectors in a
plane polarized electromagnetic wave.
Thus the coupled magnetic field has a maximum am→ −
−
→
plitude  =   This wave travels in the E × B direction and the phase of the perpendicular transverse
E and B fields is as shown in figure 4.
The important points of figure 4 are that the electric and magnetic fields are in phase, perpendicular to
each other and perpendicular to the direction of propagation. That is, they are transverse waves.
GENERATION OF
ELECTROMAGNETIC WAVES
Generation of electromagnetic waves requires accelerating electric charges or currents to radiate the electromagnetic wave. Thus one has to include the free
charges and currents that were neglected in the prior
discussion. The frequency  of the radiated electromagnetic wave equals the frequency of oscillation of
the accelerating charges. The resultant wavelength 
of the wave in vacuum is given by the relation  = 
The electric dipole antenna is the most frequently
encountered radiator. As illustrated in figure 5, the
radiated electric field oscillates in sign with the electric
field vector in the plane of the antenna. This antenna
has maximum radiation perpendicular to the axis of
the dipole and is zero along the axis of the dipole and
the electric field vector is parallel to the axis of the
electric dipole. Note that the wave always points in
→ −
−
→
the E × B direction. The radiation from the electric
dipole of the Tesla coil lights a fluorescent tube illustrating electromagnetic radiation. Electromagnetic radiation was first shown by Hertz confirming Maxwell’s
predictions.
Figure 5 The electric field vector due to an electric dipole antenna. The magnetic field vector is perpendicular
to the electric field vector.
Figure 6 Electric and magnetic dipole antenna for
detecting EM waves.
Figure 7 The electromagnetic spectrum.
Electric dipole or magnetic dipole antenna also can
be used for detecting electromagnetic waves. The sensitivity for reception from an electric dipole antenna
has the same properties as the radiation from an electric dipole transmitter. Thus the electric dipole antenna is most sensitive when the electric field vector is
parallel to the axis of the electric dipole and incident
perpendicular to the dipole axis. The 10 microwave
demonstration illustrates these properties. Note that
the electromagnetic wave is polarized and the receiver
detects a maximum signal when the transmitting and
receiving electric dipoles are parallel.
The magnetic field component of the electromagnetic wave is detected using a magnetic dipole coil.
The oscillating magnetic flux through this coil induces an oscillating emf in the coil as illustrated in the
demonstration. The magnetic dipole antenna gives a
maximum signal when the normal of the coil is parallel
to the magnetic field vector.
ELECTROMAGNETIC SPECTRUM
The electromagnetic wave equation admits solutions
for any frequency and shape of travelling waves. The
collection of all frequencies is known as the electromagnetic spectrum. Figure 7 shows the 1021 range of
frequencies and wavelengths of electromagnetic waves.
Remember that wavelength  and frequency  are re-
lated by:
 = 
Radio waves can range in wavelengths from 107 down
to sub-millimeter microwaves. The visible region is
quite narrow, it ranges from 400-700 nanometers (10−9 )
The shorter wavelengths range through X-rays to 
rays. A frequency decomposition can be made of any
travelling waveform just as it is possible to decompose
into harmonics the sounds from a musical instrument.
A wide range of the electromagnetic spectrum is
encountered in life, from radio waves for radio and television, microwaves for ovens, infrared for body scans
and heat lamps, the visible region, ultraviolet to identify atomic structure, X rays for medical diagnostics,
and  rays for cancer treatment.
ENERGY AND MOMENTUM IN
ELECTROMAGNETIC WAVES
A remarkable feature of electromagnetic waves is that
energy is transmitted through vacuum, that is the travelling electric and magnetic fields carry energy. The
existence of life on earth is a direct result of the transmission of energy from the Sun to the Earth via electromagnetic waves in vacuum. The energy transmitted
via an electromagnetic wave can be related to the E
and B fields in the wave.
117
The energy density for electromagnetic fields in
vacuum was shown to equal:
=
1
1 2
0  2 +
2
2 0
where the first term is due to the electric field and the
second term due to the magnetic field. Inserting the
relation  =  between the E and B fields for an
electromagnetic wave gives that:
 =
1 2
1 2
1
=
= 0  2 = 
2 0
2 2 0
2
that is, the same energy is stored in the magnetic and
electric fields.This energy density can be written as:
=
Figure 8 An electromagnetic wave incident upon a
point charge that initially is at rest.
9. The radiation pressure due to the sun causes the
tail of the comet to stream away from the sun, that is,
it does not trail behind the path of the nucleus of a
comet.
1
1 2

=
0  2 +
2
2 0
0 
The average power flowing through a surface per
unit area equals the energy density times the velocity of propagation. That is, the instantaneous power
transmitted is:
instantaneous =  =

0
The instantaneous power transmitted by the wave usually is written as the Poynting vector S where:
−
→
→ −
→
1−
S ≡
E×B
0
→
−
The Poynting vector S gives the power, in watt/m2 
→ −
−
→
radiated in the direction E × B
Consider a harmonic EM wave of the form:
→
−
→
−
E = 0bi sin( − ) and B = 0bj sin( − )
Inserting these into the Poynting vector and using the
fact that the average of a sin2 function equals 1/2,
gives:
|S| =
1 1
1
0 0 =
 
2 0
0
0
0
where  = √
and  = √

2
2
An electromagnetic wave carries momentum as well
as energy. The momentum carried by an electromagnetic wave can be seen by considering the force on a
charge in the EM wave and remembering that force
is rate of change of momentum. A charge q, mass
−→
m, will experience a force F = bi in the electric
−
→
→
field leading to a velocity −
v = E  at a time t. The
moving charge then will experience a magnetic force
2 −
−−→
→ −
→
→
−
→
F =  −
v × B =  E × B That is, the net force is
in the direction of the electromagnetic field. This force
per unit area is called radiation pressure. A demonstation of radiation pressure is illustrated by the tail of
the comet Ikeya-Seki seen in 1965 and shown in figure
118
SUMMARY
P114 has introduced the laws of electricity and magnetism culminating in Maxwell’s equations that summarize all the laws of electromagnetism. Maxwell’s
equations have been used to predict the existence of
transverse electromagnetic radiation. This is the radiation that is responsible for light, vision, and the
transmission of energy from the nuclear reactor at the
center of the Sun to Earth ultimately leading to the
existence of life on Earth.
Reading assignment: Giancoli, Chapter 32.