Revisiting the 1865 Maxwell’s equations#
In this page, I revisit the equations for electromagnetism expressed by Professor James Clerk Maxwell in 1865 using a structure strongly inspired by Hermann Minkowski’s 1908 paper, The Fundamental Equations for Electromagnetic Processes in Moving Bodies. Starting from the widely recognized vector formulation introduced by Mr. Oliver Heaviside, I rephrase this equations to adheres to the spirit of the 1865 formulation, albeit with modern notation and structuring.
My aim is to argue that the 1865 formulation not only remains relevant today but also highly useful. With simple reordering and adoption of modern notation, we reveal its underlying structure and gain valuable insights. These insights are not immediately obvious using the commonly used vector formulation.
This page can be read on its own with no reference to other content on this site. I assume the reader possesses a strong grasp of vector calculus, including the vector dot and cross products.
Vector formulation of Mr. Heaviside#
Mr. Maxwell’s groundbreaking work in 1865, A Dynamical Theory of the Electromagnetic Field (pdf), utilized differential expressions rather than the modern vector formulation proposed by Mr. Heaviside. Indeed in 1865, the concept of vectors had not yet been introduced.
Mr. Heaviside proposed both the concept of vectors as well as the vector form of the Maxwell equations. This is the most widespread formulation today. I therefore start from there and unpack into a form closer in spirit to the 1865 equation. The equations consist of two inhomogeneous (Gauss’s law and Ampère’circuital law) and two homogeneous vector equations (Gauss’s law for magnetism and Faraday’s law of induction), which is how I organize the equations.
While these equations are undoubtedly intriguing, within the context of this derivation, a deep comprehension of the physics behind them is unnecessary. I simply present them as the initial basis for the derivation. With the exception of Maxwell’s modification to Ampère’s circuital law, these equations represent the mathematical expression of empirical observations. Therefore, they can be regarded as established experimental facts.
Gauss’s Law
Ampère’s Circuital Law
Faraday’s law of induction
Gauss’s Law for Magnetism
With the electric field \(\overrightarrow{E} = \begin{bmatrix} E^x \\ E^y \\ E^z \end{bmatrix}\), magnetic field \(\overrightarrow{B} = \begin{bmatrix} B^x \\ B^y \\ B^z \end{bmatrix}\), and operator \(\overrightarrow{∇} = \begin{bmatrix} \frac{∂}{∂x} \\ \frac{∂}{∂y} \\ \frac{∂}{∂z} \end{bmatrix}\)
The equations of Mr. Maxwell#
Unpacking the vector equations into their component form, we obtain in spirit the 1865 Maxwell formulation, albeit with modern notation and conventions.
Gauss’s Law
Ampère’s Circuital Law
Gauss’s Law for Magnetism
Faraday’s Law of Induction
Underlying structure#
Gathering and reordering the terms, a clear structures becomes apparent:
Inhomogenous equations: Gauss’s law and Ampère’s circuital law
Homogenous equations: Guauss’s law and Faraday’s law of induction
Ordered equations#
Recognizing the emerging structure, we slightly modify the expressions. These modifications are not intricate. The objective is merely to present a compact and symmetrical form, where all terms are aligned.
To eliminate the factor \(1/c\), we introduce \(\tilde{E}^x = E^x / c\), \(\tilde{E}^y = E^y / c\), and \(\tilde{E}^z = E^z / c\). Additionally, we define for the time dimension \(∂_t = \frac{∂}{∂(ct)}\), and for the spatial dimensions \(∂_x = \frac{∂}{∂ x}\), \(∂_y = \frac{∂}{∂y}\), as well as \(∂_z = \frac{∂}{∂z}\). The equations are now:
Inhomogenous equations
Homogenous equations
For readers well-versed in the tensor formulation of electromagnetism, the presence and nature of the Faraday tensor and its dual are likely evident. Moreover, for those acquainted with matrix multiplication principles, it should be apparent that we can employ matrices operations.
Note
Although beyond our current discussion’s scope, defining \(∂_μ\) unifies all dimensions to a unit of inverse distance. \(\frac{1}{c}\frac{∂}{∂t} =\frac{∂}{∂(ct)}\) has the units of an inverse distance, exactly like the partial derivative with respect to the spatial dimensions \(\frac{∂}{∂x}\), \(\frac{∂}{∂y}\), and \(\frac{∂}{∂z}\).
Note
The experimental relation between the speed of light \(c\), the permitivity of free space \(ε_0\), and and the permeability of free space \(μ_0\) is used:
Beyond the standard formulation#
The resulting equations and their symmetries are not only satisfying in themselves, but also useful. They permit to reframe electromagnetism in terms of the Faraday tensor, or alternatively in terms of differential forms. To dive deeper down that rabbit hole, I direct you to my articles: