Download CHAPTER 32: ELECTROMAGNETIC WAVES • For those of you who

CHAPTER 32: ELECTROMAGNETIC WAVES
 For those of you who are interested, below are the differential, or point, form
of the four Maxwell’s equations we studied this semester. The version of
Maxwell’s equations that we studied are known as the integral form. They
involve vector differentiation (the first two use a divergence and the second
two use a curl). Maxwell’s original presentation (1861) of these equations
involved 20 equations instead of 4. The vector calculus notation that you see
below was developed by Heaviside and Gibbs around the late 1800s and the
original 20 equations were reduced to these four by Heaviside around 1885.
 Faraday’s law says that a time-varying magnetic field acts as a source of an
electric field, and Ampere’s law says that a time-varying electric field acts as
a source of a magnetic field. These two phenomena can actually sustain one
another, creating an electromagnetic wave that propagates thru space.
 Much of what we learned in Chapter 15 applies to electromagnetic waves,
even though, physically, they are very different from mechanical waves.
 One huge difference is that electromagnetic waves do not require a material
medium in order to propagate thru space – this idea was so foreign to
physicists at the time that many postulated the existence of a “luminiferous
aether,” an invisible medium which visible light and other electromagnetic
waves needed in order to be able to propagate thru space.
 Maxwell’s equations predict that stationary charges produce electrostatic
fields. They also predict that charges moving at constant velocity produce




magnetostatic fields. These static fields can be analyzed independent of one
another. We need for one of these fields to vary in time (not static) in order
to induce the other, so that these fields are “coupled” – meaning the behavior
of one field influences that of the other.
It turns out that, according to Maxwell’s equations, an accelerating charge
will produce a time-dependent, self-sustaining electromagnetic field that
propagates thru space as a transverse wave. The electric and magnetic fields
oscillate perpendicular to the direction the wave is traveling, and the
amplitudes of these oscillations are the maximum values of the fields
themselves.
Thus, electromagnetic waves are not completely divorced from matter – you
do need an electric charge in order to generate an electromagnetic wave, but
sustaining this type of wave does not require the acceleration of any adjacent
charges. It propagates thru space whether or not there is any matter present
other than the accelerating charge which produced it.
As with mechanical waves, one way to generate electromagnetic waves (I
will abbreviate this as EM waves for the remainder of the lecture) is thru
simple harmonic motion, which means the displacement of the source from
equilibrium oscillates sinusoidally (in time) at a well-defined frequency. The
term electromagnetic radiation is used interchangeably with electromagnetic
waves since the fields radiate away from the source (some accelerating
distribution of charge).
Maxwell’s equations also predict a wave equation for both the electric and
the magnetic fields where the speed c (see the discussion concerning the
wave equation in the CH 15 notes) is given by
1
𝑐2
= 𝜀0 𝜇0 . As with
mechanical waves, the constant wavespeed c obeys the relationship c = λf.
For EM waves propagating thru empty space (vacuum), c = 3.00 x 108 m/s.
 EM waves span a wide range of frequencies (and, thus, a wide range of
corresponding wavelengths) and what we can visually perceive is called
“visible light” and this occupies a very narrow band of the electromagnetic
spectrum shown in the figure below. Note the wavelengths/frequencies are
depicted using a logarithmic scale (as opposed to a linear scale).
Plane Electromagnetic Waves
 A plane wave is one whose properties do not vary in directions
perpendicular to the direction that the wave travels
 For plane EM waves this means that, at any instant in time, on any plane
perpendicular to the EM wave’s direction of propagation, the electric and
magnetic fields are uniform
 It is worthwhile to examine Maxwell’s equations as they apply to a plane
wave and see if they are satisfied, meaning that plane waves are consistent
with the mathematical framework
 As I mentioned earlier, we will be discussing the propagation of EM waves
in empty space (a vacuum). This means that Qenc = 0 in Gauss’s law for
electric fields and Ienc = 0 in Ampere’s law.
 To apply the first two of Maxwell’s equations we choose as our gaussian
surface a closed rectangular box shown in the figure below on the right.
⃗⃗ ⋅ 𝑑𝑨
⃗⃗ = 0 is always true for any magnetic field. The box does not
∮𝑩
⃗⃗ ⋅ 𝑑𝑨
⃗⃗ = 0. 𝑬
⃗⃗ ⊥ 𝑑𝑨
⃗⃗ everywhere on the gaussian
enclose any charge, so ∮ 𝑬
⃗⃗ is uniform so the flux thru the top
surface except the top and bottom. 𝑬
surface is +EA and thru the bottom surface is –EA. The net flux over the
entire gaussian surface is zero and this equation is satisfied. If we enclosed
the wave front shown below on the left such that part of the box lies in the
region where E = 0, then if the electric field has a component parallel to the
x-axis (i.e., Ex ≠ 0), there will be zero flux thru the right surface but not thru
⃗⃗ ⋅ 𝑑𝑨
⃗⃗ = 0 will not be satisfied. Same goes for the
the left and therefore ∮ 𝑬
magnetic field. This means that the electric and magnetic fields must be
perpendicular to the direction of propagation (in this case, the x-axis) – EM
waves are transverse
⃗ ⋅ 𝑑𝒍 = −
 Next consider Faraday’s law written out explicitly: ∮ ⃗𝑬
−
𝑑
𝑑𝑡
𝑑Φ𝐵
𝑑𝑡
=
⃗⃗ ⋅ 𝑑𝑨
⃗⃗
∫𝑩
 The closed path that is chosen for the line integral defines the boundary of
the surface used to compute the magnetic flux (note that this flux is not
computed over a closed surface). According to the RHR, curl the fingers of
your right hand in the direction used to compute the closed path integral
(either CW or CCW) and your thumb will point in the general direction of
⃗⃗ vectors on the surface.
the 𝑑𝑨
 Consider the closed loop efgh shown in the figures below where at the
instant shown, the wave front is located somewhere between the sides
perpendicular to the x-axis. This rectangle has length Δx and width a. E = 0
to the right of the wave front and so does not contribute to the line integral.
⃗⃗ ⋅ 𝑑𝒍 = 0 everywhere where the path is
To the left of the wave front, 𝑬
⃗⃗ and 𝑑𝒍
parallel to the x-axis and only along gh is the line integral nonzero. 𝑬
are in opposite directions on this segment of length a, therefore the line
⃗ ⋅ 𝑑𝒍 = −𝐸𝑎.
integral along gh is –Ea and so ∮ ⃗𝑬
 Now let’s compute the right hand side of Faraday’s law, the magnetic flux.
We have assumed a magnetic field that points in the z-direction at this
⃗⃗ also points in the z-direction (out of
particular instant and by the RHR, 𝑑𝑨
⃗⃗ is everywhere uniform and parallel to 𝑑𝑨
⃗⃗ on the flat surface
the page). So, 𝑩
defined by efgh. However, we don’t need the flux but, rather, the change in
flux. During a time-interval dt the wave front moves a distance c∙dt in the xdirection and the magnetic flux has increased by an amount 𝑑Φ𝐵 = 𝐵𝑎(𝑐 ∙
𝑑𝑡), so the rate of change of magnetic flux is
𝑑Φ𝐵
𝑑𝑡
= 𝐵𝑎𝑐.
 For a plane wave, Faraday’s law becomes –Ea = –Bac. So, for a plane wave
to satisfy Faraday’s law, it must obey the relationship E = cB, where c is the
speed of an electromagnetic wave.
 Finally, let’s see if there are any further restrictions placed on plane EM
waves by Ampere’s law. There is no physical conduction current (Ienc = 0) so
⃗ ⋅ 𝑑𝒍 = 𝜀0 𝜇0 𝑑Φ𝐸 =
Ampere’s law written out explicitly is: ∮ ⃗𝑩
𝑑
𝑑𝑡
⃗⃗ ⋅ 𝑑𝑨
⃗⃗ . In order to calculate both integrals, we move our
∫𝑬
rectangular loop so it’s in the xz-plane as shown in the figures below.
𝜀0 𝜇0
𝑑𝑡
⃗ and
 Using the same arguments as for the integrals in Faraday’s law, with ⃗𝑬
⃗𝑩
⃗ reversed, we have ∮ ⃗𝑩
⃗ ⋅ 𝑑𝒍 = 𝐵𝑎 since this integral is zero everywhere
except along gh, which has length a. The line integral is taken CCW around
⃗⃗ points out of the page in (b) and is parallel to 𝑬
⃗⃗ .
the loop so by the RHR, 𝑑𝑨
In a time-interval dt, the electric flux increases by 𝑑Φ𝐸 = 𝐸𝑎(𝑐 ∙ 𝑑𝑡) and so
𝑑Φ𝐸
𝑑𝑡
= 𝐸𝑎𝑐
 For a plane wave, Ampere’s law becomes 𝐵𝑎 = 𝜀0 𝜇0 𝐸𝑎𝑐 and so 𝐵 =
𝜀0 𝜇0 𝐸𝑐 is another condition that plane EM waves must satisfy in addition to
𝐸 = 𝑐𝐵 required because of Faraday’s law. Combining the two fixes the
speed that plane EM waves must travel in a vacuum: 𝑐 =
1
√ 𝜀0 𝜇 0
.
Numerically, c = 3.00 x 108 m/s.
⃗⃗ ⊥ 𝑩
⃗⃗
 One other assumption that we made for this plane EM wave was that 𝑬
⃗⃗ × 𝑩
⃗⃗
and it turns out that the direction of propagation is given by the vector 𝑬
 In the previous discussion, having the electric field point in the y-direction
was completely arbitrary. The term polarization refers to the orientation of
the EM fields as they oscillate. By convention, the electric field determines
the polarization of an EM wave. If the oscillation of this field is along a
particular direction, then the wave is said to be linearly polarized. If this
direction rotates as the wave propagates thru space, then the wave is said to
be circularly polarized or elliptically polarized.
The Electromagnetic Wave Equation
 In CH 15, we derived a differential equation (the wave equation) that
describes how the amplitude 𝑦 of a mechanical wave varies in space and
time:
𝜕2 𝑦
𝜕𝑥 2
=
1 𝜕2 𝑦
𝑣 2 𝜕𝑡 2
, where 𝑣 is the (constant) propagation speed of the wave
 It turns out that using Maxwell’s equations for empty space, a wave equation
can also be derived for the electric field as well as one for the magnetic field

𝜕2 ⃗𝑬
𝜕𝑥 2
= 𝜀0 𝜇0
𝜕2 ⃗𝑬
𝜕𝑡 2
and
⃗
𝜕2 ⃗𝑩
𝜕𝑥 2
= 𝜀0 𝜇0
⃗
𝜕2 ⃗𝑩
𝜕𝑡 2
. And this implies that all EM waves
(not just plane waves) must travel at speed 𝑐 =
1
√ 𝜀0 𝜇 0
in empty space
Sinusoidal Electromagnetic Waves
 A sinusoidal EM wave mathematically behaves just like the sinusoidal
transverse mechanical wave on a string that we studied in CH 15
⃗ and ⃗𝑩
⃗ at a particular point in space are sinusoidal functions of time and, at
 ⃗𝑬
a particular instant in time, the fields vary sinusoidally in space (the figure
below shows a snapshot of a linearly polarized wave at some instant in time)
 Many sinusoidal EM waves are plane waves – at any instant in time, the
fields are uniform over any plane perpendicular to the direction of
propagation but not uniform from one plane to the next
 For sinusoidally varying plane waves, the electric and magnetic fields
⃗ is zero where ⃗𝑩
⃗ is zero and ⃗𝑬
⃗ is maximum where ⃗𝑩
⃗ is
oscillate in phase: ⃗𝑬
maximum and, as before, the direction of propagation is given by the vector
⃗𝑬
⃗ × ⃗𝑩
⃗.
 If we take the direction of propagation to be in the +x-direction and let the yaxis correspond to the direction that the electric field is oscillating, then the
magnetic field oscillates along the z-axis. Let 𝐸𝑚𝑎𝑥 and 𝐵𝑚𝑎𝑥 denote the
amplitudes (maximum field strength) of the oscillations of the two fields.
Then the wavefunctions for the fields comprising the EM wave are:
𝐸𝑦 (𝑥, 𝑡) = 𝐸𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡) and 𝐵𝑧 (𝑥, 𝑡) = 𝐵𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡). Now
that the fields are no longer uniform in space, it is the amplitudes that must
satisfy the restriction imposed by Faraday’s law: 𝐸𝑚𝑎𝑥 = 𝑐𝐵𝑚𝑎𝑥
⃗⃗ (𝑥, 𝑡) =
 Written as vectors, the wavefunctions above would be written as 𝑬
̂𝐵𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡). 𝑬
⃗⃗ (𝑥, 𝑡) = 𝒌
⃗⃗ ×
𝒋̂𝐸𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡) and 𝑩
̂) = 𝒊̂, which is the direction of propagation.
⃗⃗ ~ (𝒋̂ × 𝒌
𝑩
 For an EM wave traveling in the –x-direction, the argument of the cosine
function changes to 𝑘𝑥 + 𝜔𝑡, as we learned in CH 15 for sinusoidal
transverse mechanical waves. For transverse EM waves, we also have to
̂) = −𝒊̂ and (−𝒋̂) × 𝒌
̂ = −𝒊̂.
⃗⃗ × 𝑩
⃗⃗ ~ − 𝒊̂. By the RHR, 𝒋̂ × (−𝒌
ensure that 𝑬
̂ OR ⃗𝑬
̂
⃗ (𝑥, 𝑡) ~ 𝒋̂ and ⃗𝑩
⃗ (𝑥, 𝑡) ~ − 𝒌
⃗ (𝑥, 𝑡) ~ − 𝒋̂ and ⃗𝑩
⃗ (𝑥, 𝑡) ~ 𝒌
So having ⃗𝑬
will work.
 The theory behind the behavior of EM waves in the presence of matter (as
opposed to empty space) can be considerably more complicated and I am not
going to cover this
**EXAMPLE 32.1 1061**
Electromagnetic Energy Flow and the Poynting Vector
 Just like mechanical waves, EM waves transport energy. In Chapter 24, we
derived the energy density in an electric field and in Chapter 30, we derived
the energy density in a magnetic field. So in a region devoid of matter, but
⃗ and ⃗𝑩
⃗ fields are present, the total electromagnetic energy density
where ⃗𝑬
1
1
2
2𝜇0
is 𝑢 = 𝑢𝐸 + 𝑢𝐵 = 𝜀0 𝐸 2 +
 Since 𝐵 =
𝜀0 𝐸 2 =
1
𝜇0
𝐸
𝑐
and 𝑐 =
1
√ 𝜀0 𝜇 0
𝐵2
1
, 𝑢 = 𝜀0 𝐸 2 +
2
1
2𝜇0
𝑐2
1
1
2
2
𝐸 2 = 𝜀0 𝐸 2 + 𝜀0 𝐸 2 =
𝐵2
 This shows that in a vacuum, the energy transported by an EM wave is split
equally between the energy stored in the electric field and the energy stored
in the magnetic field. Also, the fields oscillate in both position and time,
which means the energy density does too.
 In Chapter 15, we discussed both traveling waves, which transport energy
from one region of space to another, as well as standing waves, which do
not. For traveling waves, we quantified the “flow of energy” by defining the
intensity of the wave. This is the amount of energy transferred thru a cross-
sectional area perpendicular to the direction of propagation per unit time. In
other words, the wave’s power per unit area.
 The variable S will be used to represent intensity and has units W/m2. We
can compute the intensity of a plane EM wave by considering a rectangular
box of thickness dx and cross-sectional area A, where the rectangular face is
perpendicular to the direction of propagation.
 If dx is sufficiently small, the fields are approximately uniform within the
box and the total energy inside is the sum of the electric and magnetic
energy densities multiplied by the volume of the box.
1
𝐵2
 So, 𝑑𝑈 = (𝑢𝐸 + 𝑢𝐵 )𝐴𝑑𝑥 = (𝜀0 𝐸 2 + ) 𝐴𝑑𝑥
2
𝜇
0
 The energy is moving at the speed of light, so all of the energy contained in
the box exits it in a time dt = dx/c. Therefore, the rate at which energy is
transferred thru the cross-sectional area A is
𝑑𝑈
𝑑𝑡
𝐵2
𝑐
1
𝐵2
0
(𝜀0 𝐸 2 + 𝜇 ) 𝐴. The intensity (rate of energy flow per unit area,
2
0
𝑐
𝐴𝑑𝑥
= (𝜀0 𝐸 2 + )
=
2
𝜇 𝑑𝑥/𝑐
𝑑𝑈/𝑑𝑡
𝐴
) is
𝐵2
𝑆 = (𝜀0 𝐸 2 + ) = 𝑐𝜀0 𝐸 2 .
2
𝜇
0
 Since 𝐸 = 𝑐𝐵 and 𝑐 =
𝑐𝜀0 𝐸 2 = 𝜀0 𝑐 2 𝐵𝐸 =
𝐸𝐵
𝜇0
1
√ 𝜀0 𝜇 0
, we can also express the intensity as 𝑆 =
. The direction of this energy flow is in the direction
⃗⃗ × ⃗𝑩
⃗ ) and the intensity can be defined as a vector quantity.
of propagation (𝑬
This is called the Poynting vector:
⃗𝑺 =
1
⃗⃗ × ⃗𝑩
⃗)
(𝑬
𝜇0
⃗ “Poynts” (actually, Poynting is the name of the physicist who
 The vector 𝑺
proposed this idea) in the direction of propagation of the EM wave and has
𝐸𝐵
⃗ ⊥ ⃗𝑩
⃗.
magnitude equal to since ⃗𝑬
𝜇0
 The instantaneous intensity is an oscillating function of time but often we
are not interested in this rapid oscillation but, instead, want the average
intensity 𝑆𝑎𝑣 . Since the instantaneous intensity is a product of sinusoidally
varying terms that are in phase, the average intensity is just half the value of
the maximum (or peak) intensity.
 Thus, 𝑆𝑎𝑣 =
1
𝜇0
(𝐸𝐵)𝑎𝑣 =
𝐸𝑚𝑎𝑥 𝐵𝑚𝑎𝑥
2𝜇0
=
𝐸𝑚𝑎𝑥 2
2𝜇0 𝑐
=
𝑐𝐵𝑚𝑎𝑥 2
2𝜇0
, where the relationship
𝐸 = 𝑐𝐵 was used to obtain the last two equalities
**EXAMPLE 34-4 905**
**EXAMPLE 34-5 906**
Electromagnetic Momentum Flow and Radiation Pressure
 In addition to transporting energy, electromagnetic waves also carry
momentum with them. It will take us a bit far afield to derive the following,
so you will have to, for now, accept the fact that the energy and momentum
in an EM wave are related as: 𝑈 = 𝑝𝑐 and therefore,
definition, the rate of energy flow per unit area is 𝑆 =
momentum flow per unit area is
1 𝑑𝑝
𝐴 𝑑𝑡
𝑆
𝐸𝐵
𝑐
𝑑𝑝
𝑐𝜇0
= =
 From Newton’s 2nd law, we know that
𝑑𝑡
𝑑𝑈
=𝑐
𝑑𝑡
1 𝑑𝑈
𝐴 𝑑𝑡
𝑑𝑝
𝑑𝑡
. By
so the rate of
.
= 𝐹 and pressure is defined as a
force per unit area. The average force per unit area for an EM wave is called
1 𝑑𝑝
radiation pressure. That is, 𝑝𝑟𝑎𝑑 = ( )
𝐴 𝑑𝑡
𝑎𝑣
 If the EM wave is completely absorbed, then 𝑝𝑟𝑎𝑑 =
𝑆𝑎𝑣
𝑐
 However, if the wave is completely reflected, then the change in momentum
is twice as great and 𝑝𝑟𝑎𝑑 = 2
𝑆𝑎𝑣
𝑐
 Lasers exert enough electromagnetic pressure to levitate small objects.
Radiation pressure has been suggested as a means of driving spacecraft by
using “solar sails”. IKAROS, launched in 2010 by JAXA (Japan’s space
agency), was the first practical solar sail vehicle. As of 2015, it was still
under acceleration, proving the practicality of a solar sail for long-duration
missions.
 The idea that EM waves carry momentum played an important role in
Einstein’s development of his famous equation 𝐸 = 𝑚𝑐 2
**EXAMPLE 32.5 1067**