Derivation of the Biot-Savart Law from Coulomb's Law?

[demoncore]

The title says it all--I'm working through The Feynman Lectures, and came across the assertion that a magnetic field can be thought of as a relativistic-transformation of an electric field (and vice-versa). This makes sense to me, since the magnetic field of a moving point-charge can easily be transformed away by working in the reference frame of the charge. The electric field seems more (classically) invariant, but I imagine length-contraction/time dilation play a role there. I was wondering if there is a possible derivation of the Biot-Savart Law by calculating the Lorentz transformation of the Coulomb force field, or something along those lines?

 

[Vailanor]

The answer is yes. In fact, this was actually first done by Lorentz himself in 1895. He derived the Biot-Savart Law from the Lorentz transformation of the Coulomb force field. This derivation is not as straightforward as one might expect, however. It involves some intricate algebraic manipulations and a clever use of vector calculus. Another approach is to use the relativistic form of the Lorentz force law, which states that the force on a particle in an electric and magnetic field is given by:F = q (E + v × B)where q is the charge of the particle, v its velocity and E and B are the electric and magnetic fields respectively. By comparing this equation to the classical form of the Lorentz force law (F = qE + qv × B), one can show that the magnetic field is simply the relativistic transformation of the electric field. This approach is much simpler than Lorentz's original approach.