Why exactly does current carrying two current wires attract/repel?

When to parallel wires carrying currents in same direction $I_1$ & $I_2$.

The MIT Physics Demo Forces on a Current-Carrying Wire this video demonstrates that effect.

My question is, why exactly does this happen?

I know the reason, but I'm not convinced with it. One wire generates the magnetic field which is into the plane at another wire. Electrons are moving in the wire which experience Lorentz force $F = q(V \times B)$.

My arguments are,

1. this force is experienced by the electrons, not the nucleus. And these electrons that are in motion are the "Free electrons". So, when the experience force just they alone should be drifted towards/away from the wire but not the entire atom(s).

2. The only force binding an electron to the material/matter is the coulombic attraction force from the nucleus. If the Lorentz force is sufficiently large, then it should be able to remove electrons from atoms. I other words, they should come out of the material.

But I never heard/read of any thing like that happening. Why doesn't this happen?

In any case, atoms must not experience any force, then why is it that entire wire is experiencing a force of $i(L \times B)$?

You are right in both arguments. The thing is just, this "only force, ... the coulombic attraction" is incredibly much stronger than the Lorentz force due to the magnetic field of a single wire carrying current in the same direction.

As for "In any case, atoms must not experience any force", this is obviously wrong, as can be seen very plainly when you think of Newton's third law and the fact that the coulomb attraction occurs between the electrons and nuclei in the wire of question.

• Wait, so I can shoot electrons out of material with an electromagnet? How cool is that! Wait, that is already invented and called TV sets... – dbanet Jul 7 '16 at 16:23

Your question is assuming that the electrons are weakly interacting with the nucleus. The interaction with the nucleus is extremely strong. It is better to ask instead why do we have conductivity at all. Electrons are so tightly bound to nuclei of atoms, why should a tiny external electric field get them moving?

The answer is that quantum mechanical effects can spread out electrons over many atoms. This is responsible for chemical bonding. In metals, the electrons have a spread out wavefunction, and the energy-band of spread-out electron states is only partly filled, so it only takes a little bit of energy to push an electron into motion.

But for your original question, there is an easy way to see the answer. Consider two infinite charged wires 1 cm apart. You know that they repel, so they move apart. Now boost to a frame moving along the wires at a huge speed, near the speed of light. Relavistic time dilation slows down the rate at which they move apart. But the charge density has gone up in this frame, because of the length contraction. So there must be an additional attractive force due to the currents in the wires. In the limit that you are moving at the speed of light, the attractive like-current force must exactly cancel the repulsive like-charge electrostatic force.

• Consider two infinite charged wires 1 cm apart, held together by a series of springs spaced 1 cm apart along the wire (1 spring per cm). Now boost to a frame moving along the wires at .866 c. The Lorentz factor is 2, so charge density doubles in the new frame because of length contraction. The spring density also doubles (2 springs per cm) to exactly cancel the increase in the repulsive electrostatic force. If there is an additional attractive force-per-meter due to the current in this frame, there must also be an additional repulsive force-per-meter we haven't considered. Right? – Nick Jan 7 '14 at 10:23