As we know for over 100 years magnetic force in an relativistic effect caused by the movement of charged particles. Let’s take the standard example - current in a wire. If a charged test particle (say electron) is moving relative to the wire it feels the Lorentz force. I can understand what happens when electron is moving parallel to the wire – electron feels the force which is perpendicular to the wire because of Lorentz contraction of concentration of electrons and/or ions in the wire. But I can not understand why is the Lorenz force parallel to the wire if the test particle is moving perpendicularly to the wire and why is the Lorentz force nil if the test particle is moving in the direction that we used to call “direction of the magnetic field lines”.