Electromagnetic field

It is the fundamental notion for Optics.

Classically

Electric and magnetic fields are connected by Maxwell's equations. They are a summary of a lot of relations worked out during the 19th century to explain how the electric and magnetic fields evolve with time, one influenced by the other. This set of equations was a big achievement for science.

\begin{aligned} \nabla \cdot E & = \frac{ρ}{ε_{0}} & (Gauss’s Law) \\ \nabla \cdot B & = 0 & (Gauss’s Law for Magnetism) \\ \nabla \times E & = - \frac{\partial B}{\partial t} & (Faraday’s Law) \\ \nabla \times B & = μ_{0} j + μ_{0} ε_{0} \frac{\partial E}{\partial t} & (Ampère’s Law with Maxwell’s correction) \end{aligned}

Here, $ρ$ and $j$ are data about the distribution and movement of the charges affecting the fields. See four-current.

On the other hand, it was discovered by Lorentz that a particle with electric charge $q$ subjected to an electric field $E$ and a magnetic field $B$ experiences a force (Lorentz force law):

\vec{F} = q [E + v \times B]

where $v$ is the speed of the particle.
Observe that to determine the electric field $E$ at a point $x$ and at an instant $t$ , we measure the force on a stationary unit charge placed at $x$ . Once $E$ is determined, we can measure the magnetic field $B$ by observing the forces on unit charges at $x$ moving with velocity vectors $\hat{i}$ , $\hat{j}$ , and $\hat{k}$ . These measurements help fully determine $B$ since $E$ has already been obtained. Thus the Lorentz force law serves to define $E$ and $B$ ! See @frankel2011geometry page 119.

Differential forms approach

Indeed, since forces are 1-forms (see axiomatic Newtonian mechanics), ant taking into account Lorentz force law above, we should consider that the electric field is indeed a 1-form, and the magnetic field is a 2-form, so that $v \times B$ is, instead, the interior product

v \times B = - v ⌟ B .

From this point of view, and assuming the existence of the standard metric $g$ in 3D space, Maxwell's equations can be rewritten as

\begin{aligned} d_{s} ⋆ E & = \frac{ρ}{ε_{0}} {vol}_{g} & (Gauss’s Law) \\ d_{s} B & = 0 & (Gauss’s Law for Magnetism) \\ d_{s} E & = - \frac{\partial B}{\partial t} & (Faraday’s Law) \\ ⋆ d_{s} ⋆ B & = μ_{0} j + μ_{0} ε_{0} \frac{\partial E}{\partial t} & (Ampère’s Law with Maxwell’s correction) \end{aligned}

where $⋆$ is the Hodge star operator for this metric. Here, the symbol $d_{s}$ stands for the exterior derivative in $R^{3}$ , with the subindex $s$ to distinguish from the exterior derivative in 4D below.
In this guise, Maxwell's equations are invariant through any transformation affecting only spatial coordinates. But, what about time? We should be able to transform not only through space but also through the time coordinates (for example when an observer is moving with respect to the other). That is, we can consider that $E, B$ are defined on a 4D space $(t, x, y, z)$ , and we have transformations involving these four variables. With this approach, Lorentz force ca be written as

\vec{F} = - q u ⌟ F,

where $u$ is the four-velocity for a suitable 4D metric related to the provided 3D metric $g$ , and with

F := E \land d t + B .

Or, in components

F_{μ ν} = (\begin{array}{cccc} 0 & - E_{x} & - E_{y} & - E_{z} \\ + E_{x} & 0 & + B_{z} & - B_{y} \\ + E_{y} & - B_{z} & 0 & + B_{x} \\ + E_{z} & + B_{y} & - B_{x} & 0 \end{array})

Observe that this is a spacetime version of the previous Lorentz's force $\vec{F} = q [E - v ⌟ B]$ , so it has one more component.

Now, observe that Gauss's law for magnetism and Faraday's law can be simply written as

d F = 0,

since

\begin{aligned} d F & = d B + d E \land d t \\ = d_{s} B + d t \land \partial_{t} B + (d_{s} E + d t \land \partial_{t} E) \land d t \\ = d_{s} B + (\partial_{t} B + d_{s} E) \land d t \end{aligned} .

The other pair of equations arises from considering the four-vector $J = (ρ, j^{1}, j^{2}, j^{3})$ and use the 4D metric to obtain the 1-form $J^{♭}$ (see musical isomorphism), and the Hodge dual of $F$ , $⋆ F$ . Those equations can be summarized as

⋆ d ⋆ F = J^{♭} .

Gauss's law and Ampere's law can be recovered from this expression, if we assume that the 4D metric is the Minkowski metric.

In conclusion, given a Lorentzian manifold representing spacetime, an electromagnetic field is, therefore, a 2-form $F$ satisfying Maxwell equations

\begin{aligned} d F & = 0, \\ ⋆ d ⋆ F & = J^{♭}, \end{aligned}

where $J$ is a vector field called current density, which represents the source of the field. Observe that the continuity equation $d ⋆ J^{♭} = 0$ is automatically satisfied. See also evolution of current density and EM field.

Maxwell equations are equivalently given in index notation as

\partial_{μ} F_{ν σ} + \partial_{ν} F_{σ μ} + \partial_{σ} F_{μ ν} = 0,

\partial_{ν} F^{μ ν} = J^{μ},

or in Penrose abstract index notation as

\nabla_{[a} F_{b c]} = 0,

\nabla_{a} F^{a b} = 4 π J^{b} .

The vector potential

The first Maxwell equation $d F = 0$ implies that, by Poincare lemma, there exists a 1-form $A$ such that $F = d A$ . Indeed, there is no only such an $A$ , but infinite: given $A$ with $d A = F$ , any $A^{'} = A + d ϕ$ is admissible, where $ϕ$ is any smooth function. When we take a particular $ϕ$ is is said that we are choosing a gauge. It turns out that this has to do with connections in gauge theory, but I don understand yet.

The Lagrangian

We can think that Maxwell's equations are the analogous of the equation of motion of a particle but for the electromagnetic field. So it would be great if they, instead of being obtained empirically (we postulate their existence from the experiments), they were derived from a Lagrangian (also postulated, of course) with the action principle (Euler-Lagrange). That would confer a more compact approach to electromagnetism, giving the same treatment to particles and fields.

The first half of Maxwell's equations are deduced only from math. Susskind (@susskind2017special) calls it the Bianchi identity (page 300).

For the second half, the Lagrangian turns out to be

L = - \frac{1}{4} F_{μ ν} F^{μ ν}

when we are in the vacuum. Euler-Lagrange equations from here lead to

\partial_{ν} F^{μ ν} = 0

It can be guessed that, if we have the moving charges in the ambient, the Lagrangian would be the same but with the additional term

J^{μ} (x) A_{μ} (x)

This modified Lagrangian leads to an action (which is gauge invariant) that yields the equation

\partial_{ν} F^{μ ν} = J^{μ}

In summary, the action given by:

S = \int d^{4} x \sqrt{- g} (- \frac{1}{4} F_{μ ν} F^{μ ν} + J^{μ} A_{μ})

leads to Maxwell's equations:

\partial_{μ} F^{ν μ} = J^{ν}, \partial_{[μ} F_{ν κ]} = 0