Electricity and magnetism/Magnetism/The magnetic force of the electric current is a consequence of the Fitzgerald contraction and the Coulomb force

Fitzgerald contraction is the contraction of all solid bodies in the direction of their motion. It is appreciable only for bodies whose speed approaches the speed of light. The contraction factor is . When an electrically charged body is contracted, its charge density increases. This effect is at the origin of the magnetic force of the electric current. The relative movement of positive and negative charges can give rise to differences in charge density and therefore electrostatic forces. These electrostatic forces in one frame of reference are magnetic forces in another frame of reference. We can make a model of a conducting wire carrying an electric current with two insulating wires which carry opposite charges and which slide over each other. Let be the charge density on the positive wire assumed to be stationary and the charge density on the negative wire which goes at speed . The current . The negative wire represents the conduction electrons, the positive wire represents the rest of the metal wire. In the frame of reference R where the metal wire is at rest, it is electrically neutral. So in this frame . Let R' be a frame of reference which goes at speed with respect to R, and the densities of negative and positive charge, measured in R'. The negative wire is therefore at rest in R'. , because from the point of view of R', the positive wire is contracted in the direction of its movement. is the inverse of the length contraction coefficient. , because from the point of view of R, the negative wire is contracted in the direction of its movement. So . From the point of view of R', the superposition of the two wires, positive and negative, is not electrically neutral, it is therefore the source of an electrostatic field. Consider a charge of speed in R at distance from the wire. The force F+ exerted by the positive wire on the charge is perpendicular to the wire: Measured in R, This result is proven from Gauss' theorem in the chapter on Maxwell's equations. The charge is at rest in R'. The electrostatic force exerted by the negative wire on the charge is perpendicular to the wire. Measured in R the force of the negative wire on the charge is equal to . The charge therefore experiences a force If we set we obtain is the magnitude of the magnetic field created by the current in a wire of infinite length. The Lorentz force is the magnetic force exerted on a charge which moves at speed in a magnetic field if its speed is perpendicular to the field. 380px We thus find that the magnetic field only acts on moving electric charges. In the frame of reference R there is no electrostatic force, because the wire carrying a current is electrically neutral. But in the frame of reference R' the negative charge density is lower than the positive charge density, because of the Fitzgerald contraction. The charge density is not zero and is the source of an electrostatic force field. This electrostatic force in R' is the magnetic force in R. The Fitzgerald contraction and the Coulomb force are therefore sufficient to explain the existence of the magnetic force produced by an electric current. A wire carrying a current repels negative charges that go in the conventional direction of the current (a current of positive charges) and attracts negative charges that go in the opposite direction. We thus find Ampère's law: two parallel wires carrying a current attract each other if the currents go in the same direction and repel each other if the currents go in the opposite direction. Reference: Feynman's physics course, Electromagnetism, chapter 13-6.