Electrodynamics - A Motivational Overview
Abstract Electrodynamics is one of the main courses in physics studies. Electrodynamics enables us to understand electromagnetic phenomena based on Maxwell's Equations, Ď (r, t) Îľ0 ∂B (r, t) ∇ Ă— E (r, t) = − ∂t ∇ ¡ E (r, t) =
, and
∇ Ă— B (r, t) = Âľ0 j (r, t) + Âľ0 Îľ0
∂E (r, t) ∂t
∇ ¡ B (r, t) = 0 .
In this article, we shall discuss the importance of electrodynamics not only for physicists and why it is worth to take the course. Remarks on the importance of the uni cation of electric and magnetic elds are given. The general structure of electrodynamic courses to teach the main aspects of the theory is outlined. Corresponding phenomena are linked to the theory and references to a freely available online course are provided.
Keywords
: electrodynamics, electromagnetism, electrostatics, magnetostatics, circuit theory,
electromagnetic radiation
Contents
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1
Electrodynamics is Everywhere . . . . . . . . . . . . . . . . . . .
1.1
Uni cation of Electric and Magnetic Forces
. . . . . . . . . . . .
2
1.2
Electrodynamics and the Golden Age of Physics . . . . . . . . .
3
2
Classical Electrodynamics Overview
5
2.1
Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Magnetostatics
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Circuit Theory
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Full Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . .
6
. . . . . . . . . . . . . . . .
1
Electrodynamics is Everywhere
It is rarely the case that people know the principles behind the phenomena and e ects they encounter or even use in their everyday lives.
This is sad since
school should give us a general feeling for the mechanisms that are responsible
1
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1 Electrodynamics is Everywhere
for these phenomena. In this document we shall try to connect the course of classical electrodynamics to the phenomena the theory may explain. Some of these are lightning due to a huge electric discharge, the movement of a compass caused by the magnetic eld of the earth, the occurence of rainbows caused by di raction and re ection in water droplets and the function of blow-dryers based on an electric motor and ohmic heating. It is clear that the connection between theory and phenomena can never be even remotely complete, so we may just try to explain the main topics of electrodynamics and give some intuitive examples that everyone should be familiar with. We hope that this connection may be a source of motivation for one or the other student. Let us begin our journey towards an understanding of electrodynamics from a historical perspective: electrodynamics in terms of its forces. We will see that naturally humans were thinking in terms of electric and magnetic forces. Their combination and the introduction of the eld concept later lead to the Golden Age of Physics that we will discuss afterwards.
1.1
Uni cation of Electric and Magnetic Forces
Thousands of years passed by before we nally understood that electric charges exist, humans were fascinated by the possibility of amber to attract other objects. Now we know that if we rub this material, charges are transferred from the surface to the rubbing material causing an electric eld between the piece of amber and the rubbing material, i.e.
some fur.
In fact, amber in Greek
is written ÎŽÎťÎľÎşĎ„Ď Îż and spelled electro - amber gave the electron its name. Very important concepts arise already in the simple experiment of rubbing amber: there are charges that may be transferred from one body to the other, the charged bodies may attract (or repell) each other and this force also depends on the distance of the bodies. Scientists concluded that charges
q
Eq (r).1 This q 0 , F (r) = q 0 Eq (r, t) which gets q and q 0 . What they found was the
cause an electric eld
electric eld causes a force on another charge weaker by the squared distance between so-called
Coulomb force,
Fel (r)
=
q0 q r0 − r 1 ¡ 0 . 2 4πξ0 |r0 − r| |r − r|
The force is named after Charles Augustin de Coulomb who found it in 1785. Here, in SI-units, vacuum.
Îľ0 ≈ 8.85 Ă— 10−12 As/Vm
is the so-called permittivity of
Note that charges cause an electric eld and electric elds in turn
render a force on electric charges. In this way, electric charges interact via an electric eld. mass.
But of course, electrons, the carrier of charges, have a certain
Then, if the electrons are free, the Coulomb force will set electrons in
motion.
1 See the sections charges and their electric elds and forces and movement for more information such as the calculation for the electric eld of a point charge.
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1 Electrodynamics is Everywhere
However, the Coulomb force may not be the only present force acting on electric charges. The whole situation changes if a magnetic eld
B (r) is present.
Then, the actual motion of the charges have to taken into account.
Roughly
speaking, moving charges are nothing else but currents that interact with the magnetic eld - just in the same manner as charges interact with the electric eld. The magnetic force on a single charge is given by
= qv Ă— B (r) .
Fma (r)
This force was found by Thomson and Heaviside around 1880. If both electric and magnetic elds are present, the force on a charge must be combination of both forces. This combination is called the after Hendrik Lorentz
2
:
F (r, t)
=
Lorentz force
q (E (r, t) + v Ă— B (r, t)) ,
where we now also incorporate the time explicitly as a coordinate to emphasize possible time-dependencies in both elds, i.e. caused by other moving charges. Historically it is not clear who the actual inventor of the force is - Maxwell, Heaviside or Lorentz.
The important point is that it combines electrical and
magnetic forces. In the present it seems almost trivial to consider both forces together:
we just added them up.
But there is a huge implication in this
addition. We know that electric charges cause electric elds. However, if they are moved, they also cause magnetic elds.
That means, electric elds and
magnetic elds have to transform into each other if just the coordinate system is moved. Electric and magnetic forces, and hence electric and magnetic elds are then natrually just two di erent parts of the same medal.
At the end
of the 19th century, these thoughts were groundbreaking and the beginning of a Golden Age of Physics - it turned out that Maxwell's electrodynamics, naturally combining electric and magnetic elds, was just the beginning.
1.2
Electrodynamics and the Golden Age of Physics
At the beginning of the 20th century, it was common knowledge that there exists a medium called the ether in which light can propagate just as acoustic waves can propagate in air or water. But as we know now, this is not the case: light does not need a medium to propagate. This is a consequence of Maxwell's equations, which are, mathematically speaking, invariant under Lorentz transformations, a strong contradiction against the ether theory. Furthermore, Maxwell's equations incorporate an ultimate limit for the speed of electromagnetic wave propagation, a combination of the magnetic permeability of vacuum, and
Âľ0 ≈ 10.57Ă—10−7 Vs/Am
ξ0 : c ≥ =
1 Âľ0 Îľ0 299, 792, 458 m/s , √
2 Not to be confused with Ludvig Lorenz, who found the Lorenz gauge which is very important in i.e. relativistic electrodynamics.
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1 Electrodynamics is Everywhere
the speed of light.
It was generally believed that the Lorentz transformation
behaviour is wrong and scientists tried to incorporate the ether through modi cations of Maxwell's equations. It was until 1905 when a 26-year-old physicist working in a patent o ce in Bern argumented in Zur Elektrodynamik bewegter KĂśrper (engl.: On the Electrodynamics of Moving Bodies ) that everything is alright with Maxwell's equations and what consequences arise. Albert Einstein's theory of relativity and his other invaluable contributions in the same year such as the quantum hypothesis
3
kicked of a physics golden age which
endured roughly until the 70's of the 20th century with the discovery of the renormalizability of Yang-Mills theories by scientists around Gerard 't Hooft and Martinus Veltman.
4
One of the consequences was that all quantum eld
theories and general relativity are gauge eld theories that have a geometrical interpretation. In such an interpretation, forces appear because of curvatures. The di erence is that in general relativity space-time is curved, whereas in all quantum eld theories, roughly speaking, so-called groups attached to spacetime are curved. Classical electrodynamics can be considered as the gauge eld theory that is the easiest accessible.
The reason is that the corresponding group is the
group of rotations on a circle, the group
U (1).
This group has the special
property: two rotations on the circle can be performed in an arbitrary order and will always have the same result -
U (1)
is abelian . During the study of
electrodynamics students will learn why this correspond to the linearity of the Maxwell equations whereas, for example, the theory of the strong interaction, quantum chromodynamics, belongs to
SU (3),
a group that can be thought of
as closely related to rotations in three dimensions which are non-abelian. Thus, the eld equations of quantum chromodynamics are inevetibly nonlinear and render the theory much more complicated than electrodynamics. So even if electrodynamics and all of its related elds are not in the focus of a student's interest, she/he is strongly advised to understand electrodynamics on a solid basis since it is the rst gauge theory that will be introduced during the study and provides a natural feeling for the other theories in limiting cases. It is for example not a coincidence that Newton's theory of gravitation is, at least from the mathematical point of view, equivalent to electrostatics: corresponding potentials follow Poisson's equation,
the
âˆ†Ď†grav,el (r) âˆ? Ď grav,el (r).
Now we went all the way from electrical forces towards a grand uni ed theory, the uni cation of all of the mentioned forces, the fundamental goal of physics in the 21st century. We have seen how much of an impact electrodynamics had on science in general as the rst discovered gauge theory. In the following we will see how electrodynamics is tought and what insights the main topics provide.
3 4
Nobel prize 1922, but not for his theories of relativity. Nobel prize 1999.
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Classical Electrodynamics Overview
There are di erent approaches to teach electrodynamics. We will refer to the standard way that is historically motivated and also used in the freely available course promoted by this article, problemsinelectrodynamics.com. Let us motivate the di erent topics by the phenomena we can understand and why hese topics are important during later courses.
2.1
Electrostatics
In the historic approach, students rst discover how electric charges cause electric elds due to Gauss's law,
∇ ¡ E (r)
=
Ď (r) /Îľ0 .
Also, the concept of an electrostatic potential is be introduced using
0.
âˆ‡Ă—E (r) =
This introduction can be seen as a bridge between highschool physics and
more advanced theoretical approaches. The electrostatic potential is not only a very powerful concept, i.e. to introduce multipole moments, it also allows to build up very useful intuition for electrostatics. Soon after, boundary condition 5
problems can be solved and the concept of eigenfunctions will be introduced.
Naturally, this concept will help to understand quantum mechanics in later courses. Furthermore, the notion of, say, locally xed dipoles will lead to a
P (r) and the Îľ (r) = Îľ0 Îľr (r) relating electric D linearly via D (r) = Îľ (r) E (r).6
description of dielectric media in terms of a polarization density closely related understanding of a permittivity eld
E
and the electric displacement eld
The generalization to a permittivity that varies not only in space, but also depends on the frequency will later be crucially important to understand optical phenomena such as refraction. Electrostatics provides a starting point for a thorough understanding of electric phenomena starting from the nanoscale, i.e. to understand interactions of molecules. But electrostatics is also useful on other length scales such as the attraction of hair to a rubbed balloon, the physics of capacitors or even charge transfer processes during lightning. Usually, electrostatics makes up for more than one third of the whole electrodynamics course.
2.2
Magnetostatics
A lot of the concepts from electrostatics can be directly applied to magnetostatics, at least in vacuum.
In magnetostatics, we learn how currents cause
magnetic elds and which tricks exist to calculate them using Ampère's law
∇ Ă— B (r)
=
Âľ0 j (r)
5 even if the fundamental solutions to the di erential equation problems might not be called eigenfunctions 6 Please note that this constitutive relation is an approximation valid under certain assumptions.
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2 Classical Electrodynamics Overview
in integral and di erential forms. Again, the introduction of a potential using
∇ ¡ B (r) = 0
will be extremely helpful to understand magnetic phenomena in
terms of approximations, for example couplings of magnetic elds to magnetic dipoles.
Just as in electrostatics, locally attached dipoles will be used to in-
troduce a so-called permeability
B (r)
and magnetic induction
Âľ (r) = Âľ0 Âľr (r) relating magnetic eld H (r) B (r) = Âľ (r) H (r). This notion will allow to
via
explain magnetic media. Magnetostatics will be necessary to understand induction, superconductivity, but also large scale phenomena like the formation of galaxies. The theory is also used later to understand quantum bits and the Ising model, one of the working horses in quantum mechanics.
2.3
Circuit Theory
One of the main reasons for the success of electronics is that electronic circuits may be written in a modularized form, i.e.
containing simple elements with
certain, well-de ned e ects. During electrodynamics, a basic circuit theory is introduced based on slowly varying elds. That means, electric and magnetic elds may now be coupled via the Maxwell-Farady equation
∇ Ă— E (r, t)
= −∂t B (r, t) ,
or, simply, the law of induction. Usually, basic circuits consisting of resistors, capacitors and inductors are discussed. Even though nonlinear elements such as transistors are omitted, a lot of interesting phenomena can be explained like the oscillatory behaviour of RLC-circuits or re ection of waves in transmission lines. It is needless to say that electrical engineering and its applications hugely rely on circuit theory concepts. Electrical engineering is a huge topic in itself involving power engineering, electronics itself, microelectronics, telecommunication and so on. However, a lot of other topics rely on the simpli ed concepts of circuit theory, for example a deeper understanding of processes in the life sciences.
2.4
Full Electrodynamics
Up to now, electrodynamics is tought using di erent approximations.
In the
last part of the course, this limitation is entirely lifted. This is accomplished by completing Ampère's law with Maxwell's correction to couple electric and magnetic elds symmetrically:
∇ Ă— B (r, t)
=
Âľ0 j (r, t) +
1 ∂E (r, t) . c2 ∂t
The now complete set of Maxwell's equations forms the necessary starting point to understand: all radiation phenomena, including electromagnetic waves and multipolar radiation, electromagnetic modes to later understand their quantization in quantum electrodynamics, optics, waveguides, light-matter-interactions,
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relativistic electrodynamics and its generalization to a curved space-time, and, if you wish, microwave heating. The lists we have presented are far from being complete. Electrodynamics is one of the main courses in physics and electrical engineering; it introduces the student to the world of electromagnetic elds and their use for society. Electrodynamics is omnipresent: in the speakers of your mobile, in the superconductors of the Large Hadron Collider (LHC) or in the signals from distant galaxies teaching us about our own limitations.
And, it's far from being understood
completely. The number of scienti c publications in electrodynamics or related elds like photonics, quantum electrodynamics, nanotechnology, solar energy, atomic physics and so on is just countless. Isn't it time to learn electrodynamics right now? Take your chance and visit problemsinelectrodynamics.com now! Thank you for reading
Your ProblemsInElectrodynamics.com team!
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