Electrodynamics

In the study of physics, the science of electromagnetism is referred to as electrodynamics. The word "dynamics" takes into account the fact that it is about the description of all electrical and magnetic interactions, whereby a temporal change (dynamics) of the electrical and magnetic fields is also considered. The special case of (static) electric and magnetic fields that do not change over time is referred to as electrostatics or magnetostatics.

Electromagnetism, in turn, refers to the phenomenon of magnetic and electrical phenomena per se.

Electrodynamics, on the other hand, refers to the physics theory used to describe electromagnetism.

Maxwell's equations: the fundamental equations of electrodynamics

The fundamental equations of electrodynamics are Maxwell's equations. Every physics student should know Maxwell's equations. They describe the magnitude of magnetic and electric fields subject to currents and charges. The time-dependent Maxwell equations take into account that time-varying electric fields are the cause of magnetic fields and that time-varying magnetic fields are accompanied by electric fields. Material-specific parameters can also be factored into Maxwell's equations. This allows the behaviour of electric and magnetic fields in matter to be calculated.

At first, the theory of electrodynamics may seem like a physical theory that is only applied to specific electrotechnical problems.

However, almost all phenomena in our world can be traced back to electrical and magnetic forces and are therefore explained by electromagnetism and electrodynamics. The fundamental stability of matter, from the structure of the hydrogen atom to molecules, cells, organisms and the forces in our biosphere are all driven by electromagnetic forces. Only below the size of atoms, in the atomic nucleus, do atomic forces play a role, and only the structure of planets and stars is influenced by gravitational forces. Everything else is electromagnetism.

Deriving the wave equation from Maxwell's equations

How the description of electrodynamic phenomena works with the help of the mathematics of Maxwell's equations will be used as an example for the description of electromagnetic waves.

Typically, the four time-dependent Maxwell equations are written as follows:

\(1) \nabla\cdot\vec{E} = \frac\rho\epsilon_0\)
\(2) \nabla{\times{\vec{E}}}+\dot{\vec{B}} = 0\)
\(3) \nabla\cdot\vec{B} = 0\)
\(4) \nabla{\times{\vec{B}}} =\mu_0\cdot\vec{j}+\frac1{c^2}\dot{\vec{E}}\)
Equation 1) states that the charges are the sources of the electric field Ed. Strictly speaking, a charge density ρ, which must be divided by the permittivity of the vacuum ε₀, acts as the source of the electric field (the fact that we are talking about the sources is taken into account by the so-called divergence of the electric field, which is the expression \(\nabla\cdot\vec{E}\)).

Equation 2) states that time-varying magnetic flux densities \(\dot{\vec{B}}\) (the dot over the magnetic flux density denotes the change in this quantity over time) cause vortices in the electric field (vortices of an electric field are expressed by\(\nabla{\times{\vec{E}}}\)).

Equation 3) says that there are no sources of magnetic flux density and equation 4) indicates that vortices of the magnetic flux density \(\nabla{\times{\vec{B}}}\) are always accompanied by current densities j and time-varying electric fields \(\dot{\vec{E}}\), which must be scaled according to the 4th equation with the magnetic permeability of the vacuum μ₀ or the speed of light c.

You can now temporally differentiate the 4th equation and then insert the 2nd equation into the time derivative of the 4th equation:

\(\nabla{\times{\vec{B}}} =\mu_0\cdot\vec{j}+\frac1{c^2}\dot{\vec{E}}\)
\(\Rightarrow\nabla{\times{\dot{\vec{B}}}} =\mu_0\cdot\dot{\vec{j}}+\frac1{c^2}\ddot{\vec{E}}\) \(\Rightarrow{-{\nabla{\times{(\nabla{\times{\vec{E}}}})}}}=\mu_0\cdot\dot{\vec{j}}+\frac1{c^2}\ddot{\vec{E}}\)
Without delving any further into mathematics, it should still be mentioned at this point that the last expression in the case of vanishing current densities and charges j=0, ρ=0 in the form

\({-{\nabla{\times{(\nabla{\times{\vec{E}}})}}}} =\frac1{c^2}\ddot{\vec{E}}\)
is an equation that is solved by waves. A solution is possible, for example, if a mathematical expression that describes a plane wave is used for the electric field. Such as a sine or cosine function, for example. The above equation is therefore also referred to as a wave equation.

A wave equation can therefore be derived from Maxwell's equations. Physicists and mathematicians concluded from these calculations that there must be electromagnetic waves that propagate in a vacuum. This is theoretically necessary if Maxwell's equations are correct and complete, which is what we assume to this day.

An interesting success of electrodynamics at this point is that, even before they were proven, the existence of electromagnetic waves was deduced only from the conversion of Maxwell's equations. If we now take an expanded form of Maxwell's equations in matter and insert known material parameters into these matter equations, we can calculate how electromagnetic waves behave in contact with matter.

Since radio and television waves, mobile phone radiation, microwaves, thermal radiation, light with its various colours, UV radiation, X-rays and gamma radiation are all electromagnetic waves that differ from each other only by the wavelength of the radiation, many phenomena and applications become understandable and predictable when you consider electrodynamics. Maxwell's equations have contributed greatly to identifying many of the mentioned phenomena as electromagnetic waves in the first place. Without electrodynamics, radio and television, mobile phones, microwaves, computers and much more would never have existed.

However, electrodynamics does not only deal with electromagnetic waves. Through the concept of measurable electron motion in the material, values such as electrical conductivity, colour, light refraction, luminescence or thermal conductivity are accessible.

Electrodynamics is therefore not only helpful in the development of electronic circuits but also in the design of surface coatings, spectacle lenses and materials for thermal insulation, to name just a few examples.