What really is the Magnetic Force on a wire? I have a doubt regarding the significance of a force on a wire. Well, first of all, I know that if I have a particle and if there are several forces acting over it, then we can compute one total force $F$ that gives the same effect as the combination of the several forces, and this force is just the vector sum.
Well, this is pretty good: a vector depends on the point it's being applied (since it's an element of the tangent space at the point), so that since all the forces are on the same point (the same tangent space) we can take their sum and get another thing acting on the same point. This is pretty clear and simple to understand.
My doubt is when we have a wire for example. For a wire normally we use the relationship:
$$F=\int_\gamma i \ \gamma'(t)\times B(\gamma(t)) \ dt$$
In other words, we parametrize the wire with some curve $\gamma$, and we integrate $i \gamma' \times B$ over the wire to get the "force on the wire". But now this is confusing me, we are summing vectors at different points, and getting a vector that I don't know where's located. What I mean is: while when working with particles it makes sense to add the forces and use the total force on the same particle, with a wire this is kind of confusing, because it has length, so what should really mean "a force on a wire" since it's not just a point?
Thanks in advance for the help.
 A: The force law you show gives us the total force on the wire.  This force comes from the sum total of forces on all the electrons moving through the wire.  So imagine that your wire is supported at either end, and the magnetic field is strictly between the two supports, so the total force on the wire is – in some sense – between the two supports.  Then, if you want to keep the wire in place, the supports have to exert a total force that is equal and opposite to the electromagnetic force.
This answer might make you happy, but it's important to note that I've just swept your question under the rug a little.  We usually don't think of forces as being "located" at a precise spot.  The reason for this is pretty basic.
To simplify physics, we group particles together into collective objects, and frequently just assume that there are forces (that we ignore) within those objects to make sure that they don't get destroyed by the forces we're analyzing.  This allows us to just look at the sum of forces.  For example, we frequently deal with rigid bodies, in which all the particles are described completely by a single position (the position of the center of mass, for example) and the orientation of the body.  You might say that we ignore the fact that a rigid body is made up of smaller particles, and just treat it as one big particle.  Or, when talking about a rope, we might talk about the tension in that rope being transmitted between different sections of the rope.
So, in the case of the wire and its supports, those supports are extended objects, and the forces they exert are not located at a precise spot.  But we treat them as rigid bodies, and look at the sum of the forces exerted by each infinitesimal part of them.  We don't need to keep track of all those infinitesimal parts because we assume that the supports stay together, and maintain their shape.  Same thing with your wire.  We assume that the wire is continuous and doesn't break, so we can just treat it as a single object, and sum up all the forces.
A: The "magnetic force on the wire" means the magnetic force on the center of mass of the wire, which moves according to Euler's first law: $\vec F = M\frac d{dt}\vec v_{cm}$ where the terms are the total force, total mass and velocity vector of the center of mass.
