Edward M. Purcell's derivation of the magnetic field from the Coulomb field

In this blog we have been claiming that one cannot derive the magnetic field from the Coulomb field using special relativity alone. But Edward M. Purcell seemingly did derive the magnetic field in his classic book Electricity and magnetism (1965):

https://en.wikipedia.org/wiki/Electricity_and_Magnetism_(book)

What is going on?

https://physics.weber.edu/schroeder/mrr/mrrtalk.html

Daniel V. Schroeder (1999) presents the argument of Purcell. The key idea is to do the analysis in the comoving frame of the test charge.

                        ---->  v 

     

Purcell argues that the electric field of a moving charge is "squeezed" in the direction of its velocity v.

Above we have a wire where positive charges carry the electric current. The positive test charge approaches the wire directly from a normal direction, in the laboratory frame (upper picture).

In the comoving frame of the test charge, the positive carriers of electricity move obliquely down and slightly to the right. The deformation of their electric field produces a net force to the left, a "magnetic" force (lower picture).

              positive carriers

                 +   +   +   +   +

                           ^   V

                           |

                           |

                   v <-- • + test charge

                           | 

                           v   F

But something is wrong. In the comoving frame of the positive charge carriers, the test charge sees a force F which pushes it directly downward. The velocity of the test charge upward is slowed down. If the inertia of the test charge would decline with its kinetic energy, then the test charge would move faster than v to the left.

But the momentum p of the test charge in the horizontal direction does not change. And since the negative charges in the wire cancel the repulsion of the positive charges, the kinetic energy of the test charge does not actually change. Thus, there should be no acceleration in the horizontal direction.

The model of Edward M. Purcell probably fails to recognize that there is no change in the kinetic energy of the test charge. If there were, then we would get the right horizontal acceleration to match the calculated magnetic field B.

Conclusions

The model of Edward M. Purcell can be used to derive the magnetic field B from the Coulomb force, but then we must ignore the effect of the opposite charges on the test charge  –  and there is no obvious reason why we would be allowed to do so.