It is the fundamental notion for Optics.
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 that explain how these fields evolve with time, one influenced by the other. This set of equations was a big achievement for science.
$$ \begin{align} \vec{\nabla} \cdot \vec{E} &= \frac{\rho}{\varepsilon_{0}} \\ \vec{\nabla} \cdot \vec{B} &= 0 \\ \vec{\nabla} \times \vec{E} &= -\frac{\partial \vec{B}}{\partial t} \\ \vec{\nabla} \times \vec{B} &= \mu_{0} \vec{j} + \mu_{0} \varepsilon_{0} \frac{\partial \vec{E}}{\partial t} \end{align} $$Here, $\rho$ 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 subjected to an electric field $E$ and a magnetic field $B$ experiences a force (Lorentz force law):
$$ F = e[E + v \times B] $$where $v$ is the speed of the particle.
But there were two facts that led to a modification of all this stuff.
One was the development of Lagrangian mechanics. The perspective was changed: forces are not the fundamental objects that one acquires from experimentation; instead, the least action principle is the key. Physicists look for Lagrangians whose Euler-Lagrange equations explain the phenomena. Moreover, the evolution of the fields can be explained in a similar way, from a Lagrangian and the least action principle (classical field theory).
The other fact was the development of special relativity. All scientific laws have to be adapted to be Lorentz invariant, with this new paradigm.
The result was that physicists realized the existence of two new objects, more fundamental than $E$ and $B$. One is the vector potential $A_{\mu}$ (a 1-form) and the other is the electromagnetic tensor $F_{\mu \nu}$ (a 2-form).
The vector potential $A_{\mu}$: When formulated in terms of $E$ and $B$ there is no way to express the dynamics of charged particles as an action principle. For that, the vector potential is essential. We invent an action that involves a mysterious $A_{\mu}$. Then we write the Euler-Lagrange equations and compare with Lorentz force law to obtain that we can define $A_{\mu}$ such that:
$$ \vec{E} = -\frac{\partial \vec{A}}{\partial t} + \vec{\nabla} A_{0} $$and
$$ \vec{B} = \vec{\nabla} \times \vec{A} $$There is a point here. Is $A_{\mu}$ anything physical or is it only a mathematical trick to sum up electric and magnetic field? Both perspectives are valid.
It turns out that the Lagrangian for the particle inside the electromagnetic field is
$$ \mathcal{L} = -m \sqrt{1 - \dot{x}^{2}} + e A_{0}(t, x) + e \dot{X}^{p} A_{p}(t, x) $$and the equation of motion, under special relativity, is
$$ m \frac{d}{dt} \frac{\dot{X}_{p}}{\sqrt{1 - \dot{x}^{2}}} = e\left(\frac{\partial A_{0}}{\partial X^{p}} - \frac{\partial A_{p}}{\partial t}\right) + e \dot{X}^{n}\left(\frac{\partial A_{n}}{\partial X^{p}} - \frac{\partial A_{p}}{\partial X^{n}}\right) $$or
$$ m \frac{d}{dt} \frac{\dot{X}_{p}}{\sqrt{1 - \dot{x}^{2}}} = e(\vec{E} + \vec{v} \times \vec{B})_{p} $$It can be written in pure relativistic form (4-vectors) in the following way:
$$ m \frac{d U_{\mu}}{d \tau} = e\left(\frac{\partial A_{\nu}}{\partial X^{\mu}} - \frac{\partial A_{\mu}}{\partial X^{\nu}}\right) U^{\nu} $$The electromagnetic tensor: On the other hand, the electromagnetic tensor is
$$ F_{\mu \nu} = \frac{\partial A_{\nu}}{\partial X^{\mu}} - \frac{\partial A_{\mu}}{\partial X^{\nu}} $$This tensor encodes $E$ and $B$:
$$ F_{\mu \nu} = \left( \begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ +E_{x} & 0 & +B_{z} & -B_{y} \\ +E_{y} & -B_{z} & 0 & +B_{x} \\ +E_{z} & +B_{y} & -B_{x} & 0 \end{array} \right) $$It lets express the equation of motion for the charged particle as
$$ m \frac{d^{2} X_{\mu}}{d \tau^{2}} = e F_{\mu \nu} U^{\nu} $$and Maxwell's equations get reduced to
$$ \mathrm{d} \boldsymbol{F} = 0, \quad \mathrm{d}^{*} \boldsymbol{F} = 4 \pi^{*} \boldsymbol{J} $$or in index notation
$$ \partial_{\mu} F_{\nu \sigma} + \partial_{\nu} F_{\sigma \mu} + \partial_{\sigma} F_{\mu \nu} = 0 $$ $$ \partial_{\nu} F^{\mu \nu} = J^{\mu} $$or in Penrose abstract index notation
$$ \nabla_{[a} F_{b c]} = 0 $$ $$ \nabla_{a} F^{a b} = 4 \pi J^{b} $$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 (Theoretical Minimum 3) calls it the Bianchi identity (page 300).
For the second half, the Lagrangian turns out to be
$$ \mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} $$when we are in the vacuum. Euler-Lagrange equations from here lead to
$$ \partial_{\nu} F^{\mu \nu} = 0 $$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
$$ J^{\mu}(x) A_{\mu}(x) $$This modified Lagrangian leads to an action (that is gauge invariant) that yields the equation
$$ \partial_{\nu} F^{\mu \nu} = J^{\mu} $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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