Question

The free electrons in a conducting wire are in constant thermal motion. If such a wire, carrying no current, is placed in a magnetic field, is there a magnetic force on each free electron? Is there a magnetic force on the wire?

Answer 1

As the electrons are in motion, there is a magnetic force acting on them individually. But the current through the wire  represents the collective motion of all the electrons that are moving or vibrating very randomly; so, overall effect is negligible. Hence there is no net magnetic force on the wire.
Also, F = LLB sin (θ)
So, if the current in the wire is zero, then the force experienced by the wire will also be zero.