| In which it is shown that electricity and magnetism can no more be separated than space and time. |
The aim of this article is to express the relativistic version of the classical Lorentz force law, which says that the force F on a test particule of charge q and velocity v is given by
F = q(E + v x B)
where E and B denote the ordinary 3-dimensional electric and magnetic fields (with magnitude E and B, respectively)
Invariant Lagragian
We should start with the simplest possible Lagragian, describing a free particle in the presence of a four-vector field Aμ(t,x).
We know already that the Lagragian of a free particle of mass m can be written as
choosing units where c = 1 and (X.i)2 stands for x.2 + y.2 + z.2
On the other hand, when coupling the particle to an electromagnetic field, it looks natural to consider a small part of the particule trajectory, the quadri-vector dXμ, and to combine it with the electromagnetic four-vector Aμ(t,x) = (V/c, -A) - where V is the scalar potential and A the magnetic vector potential - to bring up a scalar, infinitesimal quanity dXμAμ(t,x) - which as per special relativity principle, should remain invariant per Lorentz transformation.
Multiplying this action by the q electric charge yields to
If we bring together the two distinct parts, we get the whole action as
Dividing the second integral by dt yields to
As we know that the action is defined as the integral of the Lagrangian L for an input evolution between the two times a and b, we can easily identify the Lagrangian for our particule as:
The easiest thing to do now is guess what, resolve the Euler-Lagrange equation!
Euler-Lagrange equation
For your reference, let us rewrite the Euler-Lagrange equation:
This single equation actually represent three distinct equations, each one corresponding to a p distinct value (x,y, and z)
Let us start by evaluating dL/dX.p
Developping this derivation with respect to time - A(x,t) depends explicitly on time and implicitly as well via the position x gives
and we are all done with the left expression of the Euler-Lagrage equation.
Considering now the left expression of the equation gives
so that rendering the two sides of the equation equal and rearranging the terms yields to
If we recognize the left side as the mass x the relativistic acceleration - so equivalent to the Lorentz force, it is then easy to identity the p component of the electric field E as:
and we have retrived the first term of the Lorentz law, qE.
The second term containing the speed X.n should certainly be compared to the vector product v x B.
Let us try to confirm this by calculating for example the z component of v x B
By definition of the vector product
so replacing p by z in the second part of the equation gives
so that we can identify By and Bx as the following expressions
It is easy to show by setting p as y in our initial equation that the Bz component of the magnetic field can be expressed as
We can express this with a more concise notation, the rotational of A:
