The Lorentz force law specifies the force of a magnetic field on a moving charged particle, and the Ampere force law specifies the force of a magnetic field on a current carrying wire. I am trying to understand the relationship between these laws. Suppose we have a horizontal current carrying wire and either a moving charged particle or a current carrying wire above it:
Currents are in orange, moving particles in black, and forces in blue.
Am I right in thinking that the force on the moving particle should be the same as the force on the wire, because the wire has moving charged particles? If so, I have run into a difficulty: according to the wikipedia page on Apere's force law, the force between two current elements is directed along the line between the two current elements, see this equation. This implies that the force is zero if the wires are perpendicular. However, the Lorentz force of a particle moving toward the wire seems to be directed parallel to the wire and nonzero.
Where am I going wrong?