Superposition; the right-hand rule
We will conclude this chapter by making two further points regarding the subject of magnetostatics. First, our basic equations for the magnetic field, \begin{equation*} \FLPdiv{\FLPB}=0,\quad\FLPcurl{\FLPB}=\FLPj/c^2\epsO, \end{equation*}  are linear in $\FLPB$ and $\FLPj$. That means that the principle of superposition also applies to magnetic fields. The field produced by two different steady currents is the sum of the individual fields from each current acting alone. Our second remark concerns the right-hand rules which we have encountered (such as the right-hand rule for the magnetic field produced by a current). We have also observed that the magnetization of an iron magnet is to be understood from the spin of the electrons in the material. The direction of the magnetic field of a spinning electron is related to its spin axis by the same right-hand rule. Because $\FLPB$ is determined by a “handed” rule—involving either a cross product or a curl—it is called an axial vector. (Vectors whose direction in space does not depend on a reference to a right or left hand are called polar vectors. Displacement, velocity, force, and $\FLPE$, for example, are polar vectors.) Physically observable quantities in electromagnetism are not, however, right- (or left-) handed. Electromagnetic interactions are symmetrical under reflection (see Chapter 52, Vol. I). Whenever magnetic forces between two sets of currents are computed, the result is invariant with respect to a change in the hand convention. Our equations lead, independently of the right-hand convention, to the end result that parallel currents attract, or that currents in opposite directions repel. (Try working out the force using “left-hand rules.”)  An attraction or repulsion is a polar vector. This happens because in describing any complete interaction, we use the right-hand rule twice—once to find $\FLPB$ from currents, again to find the force this $\FLPB$ produces on a second current. Using the right-hand rule twice is the same as using the left-hand rule twice. If we were to change our conventions to a left-hand system all our $\FLPB$ fields would be reversed, but all forces—or, what is perhaps more relevant, the observed accelerations of objects—would be unchanged. Although physicists have recently found to their surprise that all the laws of nature are not always invariant for mirror reflections, the laws of electromagnetism do have such a basic symmetry.