A current element in a magnetic field experiences dF = i(dl x B). For a straight conductor in uniform B, the magnitude becomes F = BiL sin theta, and the force stays perpendicular to both current direction and field.

• Fleming's left-hand rule gives the force direction when current and field are perpendicular.
• If the conductor is parallel to the field, sin theta becomes zero and no magnetic force acts.
• Trap: a closed loop in a uniform field can have zero net force even though different segments still experience forces.

Two long parallel wires separated by distance a exert force per unit length mu0 i1 i2 divided by 2pi a on each other. Same-direction currents attract, while opposite-direction currents repel.

• Each wire first produces a magnetic field at the location of the other, and that field then acts on the current in the second wire.
• The force grows with both currents and falls as separation increases.
• Trap: students often remember the magnitude formula but reverse attraction and repulsion when currents are opposite.

Springs, suspended wires, and rods on rails reduce to magnetic-force balance with weight, tension, or the component of gravity along the incline. The method is always to write equilibrium first and then substitute the correct magnetic-force expression.

• A current-carrying spring contracts because adjacent turns carry current in the same direction and attract each other.
• For a tensionless string or suspended conductor, balance requires magnetic force to equal the weight of the rod.
• Trap: in inclined-rail problems, only the component of magnetic force along the motion balance matters, so geometry must be resolved before substitution.