Force on a current carrying wire of arbitrary shape in a magnetic field

To obtain the net force by an external magnetic field on a current carrying wire, we divide the wire into small sections of infinitesimal lengths. The force due to the magnetic field on a given section is then given by dF=i(dl x B), where B is the external magnetic field at that section.

Consider one of these sections. If the wire is not a straight line(has some random shape), then won't the current in a different section of the wire exert a force on the considered section due to the magnetic field produced due to that section?

I am aware that when we consider any closed system, we are not concerned with the internal forces operating within the system, as internal forces, always occurring in pairs, impart no effect to the system as a whole(to its centre of mass, to be specific).

However, this reasoning is only valid as long as the internal forces obey Newton's third law. But, as I have learnt, magnetic force due to charged particles DOES NOT obey Newton's third law. So, if I were to consider the current carrying wire as my system, the external magnetic field will indeed exert a force on the wire. But what about the internal forces and why are they not considered?

To summarize: WHY DOES A CURRENT CARRYING WIRE OF RANDOM SHAPE NOT EXERT A FORCE ON ITSELF?

Please correct me where I am wrong.

• Newton's third law is basically a restricted formulation of momentum conservation for contact forces. Momentum conservation stays valid for electromagnetic fields, so there is, apart from a legalistic attitude, really no reason to make a big fuss about it. That internal forces are not being "considered" is far less correct than the application of Newton's third law in a wider view. Internal forces are not being considered in high school physics simply because the math needed to consider them is too advanced. In mechanical engineering internal forces are what gets people paid. – CuriousOne Jun 7 '16 at 15:04