Two wires carrying equal currents exert a force? on each other. (a) The current in each wire is doubled, while the distance of separation remaining constant. What is the magnitude of the force that one wire exerts on the other? (b) When a magnetic field is parallel to a current-carrying wire, what is the force on the wire?

a) The magnetic field created by a current-carrying wire is proportional to the current:

B ∝ I, B = magnetic field strength, I = current

The magnetic force acting on a current-carrying wire immersed in a magnetic field is proportional to the current and the magnetic field strength:

F ∝ IB, F = magnetic force, I = current, B = magnetic field strength

Let's focus on wire 1.

Since wire 2's current is doubled, wire 2 produces a magnetic field twice as strong as before.

Wire 1's current is also doubled, therefore we now have a wire having twice as strong a current immersed in twice as strong a magnetic field. The magnetic force on wire 1 (and you can make a similar argument for wire 2) will be four times as strong as before.

b) The general formula for the magnetic force acting on a current-carrying wire immersed in a magnetic field is given by:

F = IL*B

F = magnetic force vector

I = current

L = vector having a magnitude equal to wire length and representing direction of current

B = magnetic field vector

Note we are taking a cross product of the IL and B vectors, not the product of two scalar quantities.

The very nature of the cross product means that if L and B are parallel to each other, F = 0N