Definition of Maxwell’s Equations

Maxwell’s equations are a set of mathematical expressions that manage to unify electrical and magnetic phenomena into one called “electromagnetism”. These elegant and sophisticated equations were published by the mathematician James Clerk Maxwell in 1864.

Prior to these equations, it was said that electric and magnetic forces were “forces at a distance”, no physical means was known by which this type of interaction occurred. After many years of research on electricity and magnetism, Michael Faraday intuited that there would have to be something physical in the space between charges and electric currents that would allow them to interact with each other and manifest all known electrical and magnetic phenomena, he in at first he referred to these as “lines of force”, which led to the idea of ​​the existence of an electromagnetic field.

Based on Faraday’s idea, James Clerk Maxwell develops a field theory represented by four partial differential equations. Maxwell referred to this as “electromagnetic theory” and was the first to incorporate this type of mathematical language into a physical theory. Maxwell’s equations in their differential form for vacuum (that is, in the absence of dielectric and/or polarizable materials) are the following:

\(\nabla \cdot \vec{E}=\frac{\rho }{{{\epsilon }_{0}}}\)
\(\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\)
\(\nabla \cdot \vec{B}=0\)
\(\nabla \times \vec{B}={{\mu }_{0}}\vec{J}+{{\mu }_{0}}{{\epsilon }_{0}}\frac {\partial \vec{E}}{\partial t}\)
Maxwell’s equations for vacuum in its differential form

Where \(\vec{E}~\)is the electric field, \(\vec{B}~\)is the magnetic field, \(\rho ~\)is the electric charge density, \(\vec{ J}~~\)is a vector associated with an electric current, \({{\epsilon }_{0}}~\)is the electric permittivity of vacuum and \({{\mu }_{0}}~ ~\) is the magnetic permeability of the vacuum. Each of these equations corresponds to a law of electromagnetism and has a meaning. Next, I will briefly explain each of them.

Gauss’s law

\(\nabla \cdot \vec{E}=\frac{\rho }{{{\epsilon }_{0}}}\)
Gauss’s law for electric field

What this first equation tells us is that the electric charges are the sources of the electric field, this electric field “diverges” directly from the charges. Furthermore, the direction of the electric field is dictated by the sign of the electric charge that produces it, and how close together the field lines are indicates the magnitude of the field itself. The image below somewhat summarizes what was just mentioned.

Illustration 1. From Studiowork.- Diagram of the electric fields generated by two point charges, one positive and the other negative.

This law owes its name to the mathematician Johann Carl Friedrich Gauss who formulated it based on his divergence theorem.

Gauss’s law for the magnetic field

\(\nabla \cdot \vec{B}=0\)

This law does not have a specific name, but it is called that because of its similarity with the previous equation. The meaning of this expression is that there is no “magnetic charge” analogous to “electric charge”, that is, there are no magnetic monopoles that are the source of the magnetic field. This is the reason why if we break a magnet in half we will still have two similar magnets, both with a north pole and a south pole.

Faraday’s Law

\(\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\)
Faraday’s law of induction

This is the famous law of induction formulated by Faraday when in 1831 he discovered that variable magnetic fields were capable of inducing electric currents. What this equation means is that a magnetic field that changes over time is capable of inducing an electric field around it, which in turn can cause electric charges to move and create a current. Although this may sound very abstract at first, Faraday’s law is behind how motors, electric guitars, and induction cooktops work.

Illustration 2.- Manifestation of Faraday’s Law. An ammeter registers an electric current in a closed circuit when a magnet moves relative to it.

Ampere–Maxwell’s Law

\(\nabla \times \vec{B}={{\mu }_{0}}\vec{J}+{{\mu }_{0}}{{\epsilon }_{0}}\frac {\partial \vec{E}}{\partial t}\)

The first thing that this equation tells us is that electric currents generate magnetic fields around the direction of the current and that the magnitude of the generated magnetic field depends on its magnitude. This was what Oersted observed and what Ampère was later able to formulate. However, there is something curious behind this equation, and that is, the second term on the right side of the equation was introduced by Maxwell because this expression was originally inconsistent with the others, particularly, it led to a violation of the law of conservation of electric charge. To avoid this Maxwell simply introduced this second term so that all of his theory would be consistent, this term was called “displacement current” and at the time there was no experimental evidence to support it.

Illustration 3. De Rumruay.- An electric current circulating through a cable generates a magnetic field around it according to Ampère’s Law.

The meaning of displacement current is that, in the same way that a changing magnetic field induces an electric field, an electric field that changes with time is capable of generating a magnetic field. The first experimental confirmation of displacement current was Heinrich Hertz’s demonstration of the existence of electromagnetic waves in 1887, more than 20 years after Maxwell’s theory was published. However, the first direct measurement of the displacement current was made by MR Van Cauwenberghe in 1929.

Light is an electromagnetic wave

One of the first mind-blowing predictions made by Maxwell’s equations is the existence of electromagnetic waves, but not only that, they also revealed that light had to be such a wave. To somehow see this we will play a bit with Maxwell’s equations, but before that, the form of any wave equation is presented below:

\({{\nabla }^{2}}u=\frac{1}{{{v}^{2}}}\frac{{{\partial }^{2}}u}{\partial {{ t}^{2}}}\)
General form of a wave equation in three dimensions.

Where \({{\nabla }^{2}}\) is the Laplacian operator, \(u\) is a wave function, and \(v\) is the velocity of the wave. We will also work with Maxwell’s equations in empty space, that is, in the absence of electric charges and electric currents, only electric and magnetic fields:

\(\nabla \cdot \vec{E}=0\)

\(\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}\)

\(\nabla \cdot \vec{B}=0\)

\(\nabla \times \vec{B}={{\mu }_{0}}{{\epsilon }_{0}}\frac{\partial \vec{E}}{\partial t}\)

And we will also use the following vector calculus identity:

\(\nabla \times \left( \nabla \times \vec{A} \right)=\nabla \left( \nabla \cdot \vec{A} \right)-{{\nabla }^{2}} \vec{A}\)

If we apply this identity to electric and magnetic fields using Maxwell’s equations for empty space above, we get the following results:

\({{\nabla }^{2}}\vec{E}={{\mu }_{0}}{{\epsilon }_{0}}\frac{{{\partial }^{2} }\vec{E}}{\partial {{t}^{2}}}\)

\({{\nabla }^{2}}\vec{B}={{\mu }_{0}}{{\epsilon }_{0}}\frac{{{\partial }^{2} }\vec{B}}{\partial {{t}^{2}}}\)

Note the similarity of these equations with the wave equation above, in conclusion, electric and magnetic fields can behave like waves (electromagnetic waves). If we define the speed of these waves as \(c\) and compare these equations with the wave equation above, we can say that said speed is:

\(c=\frac{1}{\sqrt{{{\mu }_{0}}{{\epsilon }_{0}}}}\)

\({{\mu }_{0}}\) and \({{\epsilon }_{0}}\) are the magnetic permeability and electrical permittivity of the vacuum, respectively, and both are universal constants whose values ​​are \({{\mu }_{0}}=4\pi \times {{10}^{-7}}~~T\cdot m/A\) and \({{\epsilon }_{0} }=8.8542\times {{10}^{-12}}~{{C}^{2}}/N\cdot m~\), substituting these values, the value of \(c\) is \ (c=299,792,458\frac{m}{s}\approx 300,000~km/s\) which is exactly the speed of light.

With this small analysis we can obtain three very important conclusions:

1) Electric and magnetic fields can behave like waves, that is, there are electromagnetic waves that are also capable of propagating through a vacuum.

2) Light is an electromagnetic wave whose speed depends on the magnetic permeability and electrical permittivity of the medium through which it propagates. In empty space, light has an approximate speed of 300,000 km/s.

3) Since magnetic permeability and electrical permittivity are universal constants, then the speed of light is also a universal constant, but this also implies that its value does not depend on the frame of reference from which it is measured.

This last statement was highly controversial at the time. How is it possible that the speed of light is the same regardless of the movement of who measures it and the movement of the light source? The speed of something has to be relative, right? Well, this was a watershed for physics at the time and this simple but profound fact led to the development of the Special Theory of Relativity by Albert Einstein in 1905.

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