Final answer:
The problem involves physics and magnetism, where two parallel wires exert magnetic forces on each other. The current in the upper wire, for levitation, depends on whether the Earth's magnetic field is considered or not. The stability of the wire after displacement also becomes a point of consideration.
Step-by-step explanation:
In physics, the interaction between two parallel currents can result in magnetic force, which, depending on the direction of the currents, can be attractive or repulsive. When two wires carry currents in the same direction, they attract each other, while currents in opposite directions cause the wires to repel. For the wires described in the problem, we have a lower wire carrying a 100 A current and an upper 10-gauge copper wire which we want to support above it by means of magnetic repulsion.
For part (a), neglecting the Earth's magnetic field, the repulsive magnetic force between the two currents needs to be equal to the weight of the upper wire for it to levitate. The required current in the upper wire can be determined using Ampère's law and the equation for the magnetic force between two parallel currents.
For part (b), if the Earth's magnetic field is not neglected and is parallel to the ground, this external field will exert an additional magnetic force on the upper wire, altering the current required for levitation. The smallest current would need to counteract both the wire's weight and the Earth's magnetic influence.
Regarding stability, if the supported wire is displaced vertically, the magnetic forces will either increase or decrease depending on the direction of the displacement, leading to it returning to equilibrium or moving further away. A horizontal displacement will not change the vertical magnetic forces, but it could affect the wire's lateral stability.