Question:- Find magnetic force between wires as shown in figure. The infinte wire with current 'I' and the small finite wire of length 'l' with current 'i' at a distance of x from infinte wire (direction shown in figure).
I got force on small wire in negative x-axis direction. But as Newton's third law says, for every action there is an equal and opposite reaction. So according to this there must be same magnitude of force applying on infinite wire in positive x-axis (like that in second figure)
But how is this possible, because of magnetic field of finite wire, we have simple cross product for force so force must be perpendicular to both L and B but here it is parallel to L. (L is length vector and B is magnetic field vector)
Now if we go particularly for small wire we get something like this(in figure 3)
We get torque and F net on infine wire equals to zero. How is this even possible. Neither we are getting force in opposite direction nor of same magnitude. Does this question violates Newton's third law? But this can't. I am sure something I am missing. Please help me through this!
Are there more general cases like this where problem like this arises?
Edit:
I was asked in comments that how I solved for force, so here it is
Best Answer
I think @jensen paull is a bit dismissive of Newton 3. It is the case that the force between short current elements is not symmetric: $$K\mathbf{dl_2}\times(\mathbf{dl_1}\times\mathbf{r_{12})} =K[(\mathbf{dl_2}.\mathbf{dl_1})\mathbf{r_{12}}-(\mathbf{dl_2}.\mathbf{r_{12}})\mathbf{dl_1}]$$ is the force on element 2 ($K=I_1I_2\mu_0/4\pi r_{12}^3$ is a constant). The first term is equal and opposite if we interchange 1 and 2, but the second is not. So at the level of current elements Newton 3 is not true. But current elements are unphysical: they require creation of charge at one end and destruction at the other. If we integrate around a circuit the second term vanishes, so at the level of whole circuits in magnetostatics (ie constant current) Newton 3 is valid.
In a time-dependent situation we can recover conservation of momentum (which is the big prediction of Newton 3) only by introducing the momentum of the EM field.