Classical **electrodynamics** is the first field theory students usually encounter. Let us just take a look what kind of **impact** the discovery of Maxwell's equations had **on science**. Also: What phenomena will students understand taking the course? If you want, a little **motivational overview** of electrodynamics.

**Electrodynamics** is one of the main physics courses. It enables us to understand electromagnetic phenomena based on **Maxwell's Equations**,\[\begin{eqnarray*} \nabla\cdot\mathbf{E}\left(\mathbf{r},t\right)=\frac{\rho\left(\mathbf{r},t\right)}{\varepsilon_{0}}&,&\nabla\times\mathbf{B}\left(\mathbf{r},t\right)=\mu_{0}\mathbf{j}\left(\mathbf{r},t\right)+\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}\left(\mathbf{r},t\right)}{\partial t}\\ \nabla\times\mathbf{E}\left(\mathbf{r},t\right)=-\frac{\partial\mathbf{B}\left(\mathbf{r},t\right)}{\partial t}\ &\mathrm{and}&\nabla\cdot\mathbf{B}\left(\mathbf{r},t\right)=0\ . \end{eqnarray*}\]In this article, we will discuss the **importance of electrodynamics** not only for physicists. We will see that electrodynamics provides a unique understanding of nature and is often the starting point for advanced studies. In this respect, some remarks on the historical importance of the **unification of electric and magnetic** fields are given. The general structure of electrodynamic courses to teach the main aspects of the theory is outlined. Since this approach is also used on this site, the article may also be seen as a motivational introduction to the problemsinelectrodynamics.com aside from our about page.

## Electrodynamics is Everywhere

It is rarely the case that people know the principles behind the phenomena and effects they encounter in their everyday lives. This is sad since school should give us a general feeling for the mechanisms that are responsible for these phenomena. In this article 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 field of the earth, the occurence of rainbows caused by diffraction and reflection 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 field concept later lead to the “Golden Age of Physics” that we will discuss afterwards.

## Unification of Electric and Magnetic Forces

Thousands of years passed by before mankind finally understood the existance of electric charges, 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 field between the piece of amber and the rubbing material, i.e. some fur. In fact, amber in Greek is written “ήλεκτρο” 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. These charged bodies may attract (or repell) each other. Furthermore, this force also depends on the distance of the bodies.

Scientists concluded that **charges** \(q\) cause an **electric field** \(\mathbf{E}_{q}\left(\mathbf{r}\right)\).^ This electric field causes a force on another \(charge q^{\prime}\), \(\mathbf{F}\left(\mathbf{r}\right)=q^{\prime}\mathbf{E}_{q}\left(\mathbf{r},t\right)\) which gets weaker by the squared distance between \(q\) and \(q^{\prime}\). What the scientists deduced was the so-called **Coulomb force**,\[\begin{eqnarray*} \mathbf{F}_{\mathrm{el}}\left(\mathbf{r}\right)&=&\frac{1}{4\pi\varepsilon_{0}}\frac{q^{\prime}q}{\left|\mathbf{r}^{\prime}-\mathbf{r}\right|^{2}}\cdot\frac{\mathbf{r}^{\prime}-\mathbf{r}}{\left|\mathbf{r}^{\prime}-\mathbf{r}\right|}\ . \end{eqnarray*}\]The force is named after Charles Augustin de Coulomb who found it in 1785. Here, in SI-units, \(\varepsilon_{0}\approx8.85\times10^{-12}\mathrm{As/Vm}\) is the so-called **permittivity of vacuum**. Again: charges cause an electric field and electric fields in turn render a force on electric charges. In this way, electric charges interact via an electric field. But of course, electrons, the carrier of charges, have a certain mass. Then, if the electrons are free, the Coulomb force will set electrons in motion.^

However, the Coulomb force may not be the only present force acting on electric charges. The whole situation changes if a **magnetic field** \(\mathbf{B}\left(\mathbf{r}\right)\) is present. Then, the actual motion of the charges has to be taken into account. Roughly speaking, moving charges can be seen as **currents** that cause a magnetic field and interact with it - just in the same manner as charges interact with the electric field.^ The magnetic force on a single charge is given by\[\begin{eqnarray*} \mathbf{F}_{\mathrm{ma}}\left(\mathbf{r}\right)&=&q\mathbf{v}\times\mathbf{B}\left(\mathbf{r}\right)\ . \end{eqnarray*}\]This force was found by Thomson and Heaviside around 1880.^{1}

If both electric and magnetic fields are present, the force on a charge must be combination of both forces. This combination is called the **Lorentz force** after Hendrik Lorentz^{2}\[\begin{eqnarray*}\mathbf{F}\left(\mathbf{r},t\right)&=&q\left(\mathbf{E}\left(\mathbf{r},t\right)+\mathbf{v}\times\mathbf{B}\left(\mathbf{r},t\right)\right)\ ,\end{eqnarray*}\]where we now also incorporate the time explicitly as a coordinate to emphasize possible time-dependencies in the electromagnetic field, 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 unification. We know that electric charges cause electric fields. However, if they are moved, they also cause magnetic fields. That means, electric fields and magnetic fields have to transform into each other if just the coordinate system is moved. Electric and magnetic forces, and hence electric and magnetic fields are then naturally just two different 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 fields, was just the beginning.

## 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, \(\mu_{0}\approx10.57\times10^{-7}\mathrm{Vs/Am}\) and \(\varepsilon_{0}\):\[\begin{eqnarray*} c&\equiv&\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}\\&=&299,792,458\ \mathrm{m/s}\ , \end{eqnarray*}\]**the speed of light**. It was generally believed that the Lorentz transformation behaviour is wrong and scientists tried to incorporate the ether through modifications of Maxwell's equations. It was until 1905 when a 26-year-old physicist working in a patent office in Bern argumented in “Zur Elektrodynamik bewegter Körper”, engl.: “On the Electrodynamics of Moving Bodies”, that everything is alright with Maxwell's equations and what consequences arise, i.e. the principle of relativity. Albert Einstein's theory of relativity and his other invaluable contributions in the same year such as the quantum hypothesis 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. One of the consequences was that all quantum field theories and general relativity are **gauge field theories** that have a geometrical interpretation. In such an interpretation, forces appear because of curvatures. The difference is that in general relativity space-time is curved, whereas in all quantum field theories, roughly speaking, so-called groups “attached” to space-time are curved.

Classical electrodynamics can be considered as the gauge field theory that is the easiest accessible. The reason is that the corresponding group is the group of rotations on a circle, the group \(U\left(1\right)\). 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 - \(U\left(1\right)\) 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 \(SU\left(3\right)\), a group that can be thought of as closely related to rotations in three dimensions which are non-abelian. Thus, the field equations of quantum chromodynamics are inevetibly nonlinear and render the theory much more complicated than electrodynamics.

So even if electrodynamics and all of its more closely related fields 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 first 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: the corresponding potentials follow Poisson's equation, \(\Delta\phi_{\mathrm{grav,el}}\left(\mathbf{r}\right)\propto\rho_{\mathrm{grav,el}}\left(\mathbf{r}\right)\).

Now we went all the way from electrical forces towards a grand unified theory, the unification 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 first discovered gauge theory. In the following we will see how electrodynamics can be tought and what insights the main topics provide.

## Classical Electrodynamics Overview

There are different approaches to teach electrodynamics. We will refer to the standard way that is historically motivated and also used on this website. Let us motivate the different topics by the phenomena we can understand and why these topics are important during later courses.

### Electrostatics

In the historic approach, students first discover how electric charges cause electric fields due to **Gauss's law**,\[\nabla\cdot\mathbf{E}\left(\mathbf{r}\right) = \rho\left(\mathbf{r}\right)/\varepsilon_{0}\ . \]Also, the concept of an electrostatic potential is introduced using \(\nabla\times\mathbf{E}\left(\mathbf{r}\right)=0\). 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 problems can be solved and the concept of eigenfunctions is introduced. Eigenfunctions will help to understand quantum mechanics in later courses. The notion of, say, “locally fixed dipoles” then leads to a description of dielectric media in terms of a polarization density \(\mathbf{P}\left(\mathbf{r}\right)\) and the closely related understanding of a permittivity \(\varepsilon\left(\mathbf{r}\right)=\varepsilon_{0}\varepsilon_{\mathrm{r}}\left(\mathbf{r}\right)\) relating electric field \(\mathbf{E}\) and the electric displacement field \(\mathbf{D}\) linearly via \(\mathbf{D}\left(\mathbf{r}\right)=\varepsilon\left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)\).^{3} 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 dispersion.

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.

### 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 fields and which tricks exist to calculate them using **Ampère's law **\[\nabla\times\mathbf{B}\left(\mathbf{r}\right) = \mu_{0}\mathbf{j}\left(\mathbf{r}\right) \]in integral and differential forms. Again, the introduction of a potential using \(\nabla\cdot\mathbf{B}\left(\mathbf{r}\right)=0\) will be extremely helpful to understand magnetic phenomena in terms of approximations, for example couplings of magnetic fields to magnetic dipoles. Just as in electrostatics, locally attached dipoles will be used to introduce a so-called permeability \(\mu\left(\mathbf{r}\right)=\mu_{0}\mu_{\mathrm{r}}\left(\mathbf{r}\right)\) relating magnetic field \(\mathbf{H}\left(\mathbf{r}\right)\) and magnetic induction \(\mathbf{B}\left(\mathbf{r}\right)\) via \(\mathbf{B}\left(\mathbf{r}\right)=\mu\left(\mathbf{r}\right)\mathbf{H}\left(\mathbf{r}\right)\). This notion will allow to 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.

### 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-defined effects. During electrodynamics, a basic circuit theory is introduced based on slowly varying fields. That means, electric and magnetic fields may now be coupled via the **Maxwell-Farady equation** \[ \nabla\times\mathbf{E}\left(\mathbf{r},t\right) = -\partial_{t}\mathbf{B}\left(\mathbf{r},t\right)\ , \]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 reflection 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 simplified concepts of circuit theory, for example a deeper understanding of processes in the life sciences.

### Full Electrodynamics

Up to now, electrodynamics is tought using different 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 fields symmetrically: \[ \nabla\times\mathbf{B}\left(\mathbf{r},t\right) = \mu_{0}\mathbf{j}\left(\mathbf{r},t\right)+\frac{1}{c^{2}}\frac{\partial\mathbf{E}\left(\mathbf{r},t\right)}{\partial t}\ . \]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, 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 fields 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 is far from being understood completely. The number of scientific publications in electrodynamics or related fields like photonics, quantum electrodynamics, nanotechnology, solar energy, atomic physics and so on is just countless. Isn't it time to learn electrodynamics right now? You are in the right place!

^{1}Please note the difference between electric and magnetic force formulations. The first one could be expressed in terms of a given charge whereas the latter is only given in terms of a magnetic field which is caused by currents. The reason is that the first-order-interaction for the electric field is due to the monopole, the net charge, whereas the first order magnetic interaction is caused by magnetic dipole which would complicate the formulation here.)

^{2}Not to be confused with Ludvig Lorenz, who found the Lorenz gauge which is very important i.e. in relativistic electrodynamics.

^{3}Please note that this constitutive relation is an approximation valid under certain assumptions.