# Validity of Newton's Laws in 3D: does the second law work for Lorentz force given the magnetic field is an axial vector?

I know the title is mostly disturbing, but when vectors are considered in 3D we can see that a vector quantity divides into 2 categories namely polar vectors and axial vectors based on parity transformation.

Now, Newton's second law $$\vec{F} = m \frac{\text{d}\vec{v}}{\text{d}t}$$ is polar vector. Now if $$\vec{F}$$ is replaced by Lorentz force is this still valid, because the magnetic field is itself not a polar vector.

Please help in understanding this concept.

## 1 Answer

While the magnetic field is an axial vector, the expression that goes into the Lorentz force is $$\vec{v} \times \vec{B}$$, which is the cross product of a polar vector with an axial vector, i.e., a polar vector. Hence, the Lorentz force is still a polar vector and there is no issue at the end of the day.

Let me point out, however, that the electromagnetic fields do carry momentum, and as a consequence of this fact Newton's Third Law no longer holds once electromagnetic phenomena are taken into consideration. Essentially, if you move a charge here on Earth, it takes a while for a charge at Andromeda to notice the change and feel the force in the new configuration.