Electromagnetic Fields and Waves

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18 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

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35
-
1

3
5

Electromagnetic Fields and Waves


Recommended class days:
3

Background Information

There’s little research on student difficulties with
the

more technical and mathematical aspects of
electromagnetism. In an investigation of light waves,
McDermott’s gr
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
whet
her or not it is possible for a light wave
with

wavelength


to pass through an aperture of
width
a




, many students reply, “No, because it won’t fit through the opening if


is less than
a
.”
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
electromagnetic wave.

A study of students’ organization of knowledge in the domain of electromagnetism (Bagno and
Eylon, 1997) found that most stu
dents

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
law) tell
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

and

to different observers.



Electromagnetic fields obey four general laws, called Maxwell’s equations.



El
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.

35
-
2

Instructor Guide

Pedag
ogical Approach

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
dents
ar
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
v
2
/
c
2
, the f
ield transformations are of
order
v
/
c
. Thus field transformations are important at
all

speeds. Indeed, the fields
have

to transform
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

and

are not
independent entities.

These first
-
order equations allow students to recognize that the Biot
-
Savart law
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
far. An
ad

hoc

rule for their use at low speeds is to ignore any
v
2
/
c
2

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
if

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:



Gauss’s law
: Charged particles create an electric field.



Faraday’s law
: An electric field can also be created by a changing magnetic field.



Gauss’s law
for magnetism
: There are no magnetic monopoles.



Ampère
-
Maxwell law, first half
: Currents create a magnetic field.



Ampère
-
Maxwell law, second half
: A magnetic field can also be created by a changing electric
field.



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.

T
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
E
0



v
em
B
0

and
v
em



c



(

0

0
)

1/2
. The rest of the chapter, with the Poynting vector
and polarization, is straightforward.

Chapter 35: Electromagnetic Fields and Waves

35
-
3

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.

DAY 1:
Much of the chapter is a recapitulation, and then extension, of ideas introduced earlier.
It’s

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
-
familiar
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.
You’ll

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

and

(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

and

to different viewers. This is a powerful but subtle message.

DAY 2:
The G
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
onventionally. Although
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
magnetic
waves

but that these equations are the startin
g point for more advanced treatments
of

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
mathematical summ
ary of the key ideas of Part VI.

DAY 3:
The final day can be spent on electromagnetic waves. The detailed use of Faraday’s law and

the Ampère
-
Maxwell law is messy and best left to reading. It’s better to give an
overview

of how
Maxwell’s equations are used
, then jump to the conclusion that electromagnetic waves can exist if
and only if
E
0



v
em
B
0

and
v
em



c



(

0

0
)

1/2
. Another important characteristic is that

and

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
ng radiation
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
naturall
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.
35
-
4

Instructor Guide

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


1.

Maxwell’s equations are a set of how many equations?


2.

Maxwell introduced the
displacement current

as a correction to

a.

Biot
-
Savart’s law.

d.

Faraday’s law.

b.

Gauss’s law.

e.

Co
ulomb’s law.

c.

Ampère’s law.


3.

The law that characterizes polarizers is called

a.

Malus’s law.

c.

Poynting’s law.

b.

Maxwell’s law.

d.

Lorentz’s law.


4.

Experimenter A creates a magnetic field in the laboratory. Experimenter B moves relative to
A. Expe
rimenter B sees

a.

just the same magnetic field.

d.

just an electric field.

b.

a magnetic field of different strength.

e.

both a magnetic and an electric field.

c.

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.