Deriving the Faraday Tensor from the 1865 Maxwell’s Equations#
In this article, I present a straightforward and natural derivation of the electromagnetic field tensor. This derivation draws strong inspiration from Minkowski’s 1908 paper, The Fundamental Equations for Electromagnetic Processes in Moving Bodies. What sets this derivation apart is that it avoids the often-seen backward proof that the tensor formulation is equivalent to the Maxwell equations. Notably, it doesn’t rely on the widespread vector formulation introduced by Mr. Heaviside, but rather adheres to the original 1865 formulation by Mr. Maxwell, albeit presented with modern notation and the benefit of hindsight.
This derivation might not be widely known. If I am mistaken and you are aware of freely available resources, open an issue and I will include a reference. If you find mistakes, don’t hesitate to open an issue or directly provide corrections by sending a merge request to my Github repository.
I assume the reader possesses a strong grasp of vector calculus, including familiarity with the vector dot and cross products, as well as matrix multiplication. While an understanding of Maxwell’s equations and tensor calculus certainly can enrich the content, they are not essential to follow this derivation.
The 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
The 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
The 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:
The Tensor of Mr. Faraday#
From matrix multiplication rules, we infer the ordered equations are equivalent to:
We have thus obtained the Faraday tensor (inhomogenous equations) and its dual (homogeneous equations).
The Tensor Equations#
The flat left hand side is a covector, which we note in tensor notation with lower indices \(∂_μ\). The right hand side is also flat and therefore is a covector \(J_ν\). The rank 2 tensors in the expressions are necessarily one time contravariant and one time covariant. We multiply each column of \(∂\) with each row of \(F\), and repeat for all columns of \(F\). With the first index of \(F\) being the row \(μ\), and \(ν\), this means \(∂_μ F^μ{}_ν\). We then write in tensor notation \(F^μ{}_ν\) for the Faraday tensor, and \(G^μ{}_ν\) for its dual:
Maxwell’s equations are then:
To double-check the result, you can have a look at this alternative derivation of the mixed electromagnetic tensor.