Why is Lorentz's Force Equation Used for the BiLsinθ Force Derivation?

[cnh1995]

If a charge is moving in a magnetic field with velocity v, q(vXB) force acts on it. If a current carrying conductor is placed in a magnetic field, it experiences a force BiLsinθ
(all symbols have usual meaning). But the reason behind the later force is the interaction between two magnetic fields;the one already existing and the one created by current. In my textbook, the derivation of this BiLsinθ force starts with Lorentz's force equation. How can we use that equation?? The cause of that force is not Lorentz force but the interaction of two fields.
Also,why isn't the Hall effect seen in this case?? Why is it limited to specific materials only?

 

[Delta2]

No the reason of the Laplace force (BiLsinθ force) is not the interaction of two fields, it is the interaction of the one external field with the moving charged particles that make up the current i. A field cannot apply a force to another field, a field applies a force to a particle.

Ofcourse the particle that accepts the force can be generating a field of his own which field will interact with other particles and the source particle itself but that's another story.