Yes, this is how I learned it. Suppose you have a line of charges with density p moving up at velocity v, and want to find the force on a stationary particle wi

Yes, this is how I learned it. Suppose you have a line of charges with density p moving up at velocity v, and want to find the force on a stationary particle with charge q and distance r to the right. The classical Gauss' law gives F = (p / 2πrε) * q If we switch to a moving reference frame (up at velocity v), the charge density decreases to p' = p*sqrt(1-v^2/c^2) = p - pv^2/2c^2 + O(v^4/c^4) (from Taylor series) The force should stay the same, so (p / 2πrε) * q = (p' / 2πrε) * q + F' where F' is some other magnetic force created by the current. Solving, to second-order we have F' = pqv^2/4πrεc^2 If we introduce a new constant µ = 1 / εc^2, we get F' = (µ / 4π) * pqv^2 / r Now, we know F' = B * qv where B is the magnetic field, so B = dF'/d(qv) = µpv / 2πr = µI / 2πr (where I is the current in the wire). This agrees with the Biot-Savart Law for an infinitely long wire.