Electromagnetic Fields and Waves
Recommended class days:
There’s little research on student difficulties with
more technical and mathematical aspects of
electromagnetism. In an investigation of light waves,
oup at the University of Washington
found that students often interpret a “picture” of a
light wave as if the wave has spatial extent, like a
water wave. Thus a light wave that is “too big” for an
aperture is truncated as it passes through. If you ask
her or not it is possible for a light wave
to pass through an aperture of
, many students reply, “No, because it won’t fit through the opening if
is less than
These responses suggest that most students do not have a corre
ct mental image of an
electromagnetic wave and cannot interpret the standard textbook picture of a transverse
A study of students’ organization of knowledge in the domain of electromagnetism (Bagno and
Eylon, 1997) found that most stu
at the end of instruction
could not identify the main ideas
of electromagnetism (a large fraction identified Ohm’s law as one of the most important ideas), nor
could they identify what the primary equations (Maxwell’s equations and the Lorentz force
us. While students may become adept at manipulating equations, for most it is a mathematical
“game” with little or no connection to physical phenomena.
Student Learning Objectives
To understand that electric and magnetic fields are interdepende
nt. There’s just a single
electromagnetic field that presents different faces, in terms of
to different observers.
Electromagnetic fields obey four general laws, called Maxwell’s equations.
ectromagnetic fields can exist without source charges or currents in the form of a self
sustaining electromagnetic wave.
Maxwell’s equations predict that all electromagnetic waves travel at the same speed.
Electromagnetic waves can be polarized.
Chapter 35 assumes that you’ve covered Gauss’s law and Ampère’s law. Much of this chapter is
fairly standard. Earlier chapters emphasized electric and magnetic phenomena and a conceptual
understanding of electric and magnetic fields. Now it
’s time for mathematical sophistication.
The one non
standard topic in this chapter is the Galilean field transformation for electric and
magnetic fields. The magnetic force
is a velocity
dependent force, and better stu
e often troubled by recognition that velocity depends on the choice of reference frame. By
ignoring this issue, most textbooks present electromagnetism in an outdated 19th century mode.
Unlike mechanics, where relativistic effects are of order
, the f
ield transformations are of
. Thus field transformations are important at
speeds. Indeed, the fields
to reconcile the observations of two observers moving at an arbitrarily slow relative velocity. The
Galilean field transform
ation equations, valid for
are no more complex than the Lorentz
force equation, and they provide an important opportunity for showing that
order equations allow students to recognize that the Biot
for the magnetic field of a moving charge is simply the Coulomb electric field of a static charge
transformed to a moving reference frame. They also give you a modern perspec
tive on Faraday’s
law. More detail can be found in Galili and Kaplan (1997).
Since Maxwell’s equations are invariant under the Lorentz transformation, not the Galilean
transformation, the Galilean field
transformation equations do break down if pushed too
rule for their use at low speeds is to ignore any
terms that might arise. On the other
hand, this provides you an opportunity to say a few words about the origins of Einstein’s theory of
relativity, a topic about which nearly all stude
nts are curious. And this is a tremendous motivator
you plan to cover relativity in Chapter 37.
Much of the chapter is involved with assembling bits and pieces of prior information into
Maxwell’s equations. Although this is fairly standard, I do emphasi
ze the physical meaning that is
embedded in the four Maxwell equations and the Lorentz force law:
: Charged particles create an electric field.
: An electric field can also be created by a changing magnetic field.
: There are no magnetic monopoles.
Maxwell law, first half
: Currents create a magnetic field.
Maxwell law, second half
: A magnetic field can also be created by a changing electric
Lorentz force law
: An electric force
is exerted on a charged particle in an electric field. A
magnetic force is exerted on a moving charged particle in a magnetic field.
This is the big picture of what electromagnetism is all about, but few students gain this perspective
without your help.
here are various ways to show that Maxwell’s equations predict electromagnetic waves. For an
introductory course, I’ve adopted what seems to me the most straightforward approach: assume that
transverse periodic waves exist, then show that such a wave is co
nsistent with Maxwell’s equations
if and only if
. The rest of the chapter, with the Poynting vector
and polarization, is straightforward.
Chapter 35: Electromagnetic Fields and Waves
Using Class Time
Because this chapter is mathematically rather sophisticated and b
ecause there are few relevant
demos, this does become a good candidate for expository lectures.
Much of the chapter is a recapitulation, and then extension, of ideas introduced earlier.
been several weeks since you covered Gauss’s law, so this
is worthy of a review before intro
ducing Gauss’s law for magnetism. The Lorentz force law is a combination of the now
electric and magnetic forces, but it looks “scarier” when written out, so students need reassurance
that they already know this
Most of day 1 will be devoted to developing and using the Galilean field transformations.
want to begin with a brief review, from Chapter 4, of Galilean relativity. After pointing out
that the magnetic force is a velocity
dependent force, you can
ask the discussion question “velocity
measured by whom?” In the particle’s reference frame, for example,
so how does the particle
explain the fact that it experiences a force? The students’ naive assumption that
(i.e., that the individual fields have an independent existence) quickly leads to conflict, and the
paradox is resolved only by understanding that the fields are different in different reference frames.
The main p
oint, as noted in the textbook, is that there is a single electromagnetic field that presents
different faces, in terms of
to different viewers. This is a powerful but subtle message.
alilean field transformations give you a new perspective on Faraday’s law and the
role of induced electric fields. Induced electric fields are now joined by the displacement current,
induced magnetic fields, and the Ampère
Maxwell law, all treated fairly c
induced magnetic fields are gone through quickly, it’s important to emphasize the symmetry
between induced magnetic fields and induced electric fields.
Day 2 concludes nicely with the full set of Maxwell’s equations. Although studen
ts have seen all
the pieces before (except for the displacement current), these equations still look quite formidable.
Note that we have only one use for them in this course
the “discovery” of electro
but that these equations are the startin
g point for more advanced treatments
electromagnetic fields that many students will see in later classes. The more important task is to
convey the “big picture,” discussed in the Pedagogical Approach section, that these equations are a
ary of the key ideas of Part VI.
The final day can be spent on electromagnetic waves. The detailed use of Faraday’s law and
Maxwell law is messy and best left to reading. It’s better to give an
Maxwell’s equations are used
, then jump to the conclusion that electromagnetic waves can exist if
and only if
. Another important characteristic is that
are transverse and are oriented such that
is in the direction of propagation. Since this is an
intro class, these vector properties are assumed rather than proved.
Intensity and radiation pressure are treated conventionally. They provide an opportunity for such
worked examples as the electric field strength in a laser beam or the possibility of usi
pressure to accelerate spacecraft.
Polarization is the only aspect of optics (as taught at this level) that depends on light being an
electromagnetic wave. This topic is easily dealt with here, allowing the rest of optics to fall more
y with the chapters on waves and interference. Relatively few students seem familiar with
polarization, and this is the one topic in the chapter that readily lends itself to demonstrations.
Although it takes some time to explain, students are truly perplex
ed by the reappearance of light
when a third polarizer is introduced between two crossed polarizers. Some departments have
“microwave optics” that use metal grids as polarizers. These can be useful, but they lack the “seeing
is believing” nature of polariz
ation demonstrations with visible light.
Sample Reading Quiz Questions
Maxwell’s equations are a set of how many equations?
Maxwell introduced the
as a correction to
The law that characterizes polarizers is called
Experimenter A creates a magnetic field in the laboratory. Experimenter B moves relative to
rimenter B sees
just the same magnetic field.
just an electric field.
a magnetic field of different strength.
both a magnetic and an electric field.
a magnetic field pointing the opposite direction.
Sample Exam Questions
Sample exam question
s for Chapters 35
36 are at the end of Chapter 36.