Physics Asked by Hitman Reborn on May 8, 2021
I have a little question about magnetic fields.
Suppose we have an uniform magnetic field $vec{B}$ and a metal wire immersed in $vec{B}$ crossed by a stationary corrent $i$ . I know, for the Second Elementary Laplace Law, that the infinitesimal force the wire is affected is: $$dvec{F}=idvec{s}timesvec{B}$$ microscopically speaking electrons in the wire are affected by the magnetic field and they change their motion and they exert shocks on the internal wire surface producing a mechanical force $vec{F}_1$ and so the wire moves.
But also protons are affected by the magnetic field and they should move in the opposite directions producing a mechanical force $vec{F}_2$ also bigger than $vec{F}_1$ because $m_e< m_p$.
So why does the wire go in the electrons direction? Am I forgetting something about electrons and protons behavior?
In a conventional (metallic) conductors, the charge carriers are the electrons in the conduction band. The protons in the nuclei form a static lattice which does not move, even if you apply an electric field: it might budge a bit, but it cannot continue moving the way that conduction electrons do.
(Note that there are other types of conductors, listed in this great answer, where the charge carriers are different.)
If you place a metallic wire, carrying a current, in a magnetic field, then the protons do feel the magnetic field, but they are stationary, so the magnetic Lorentz force on them is zero.
That said, your question holds even if we change the conductor for something with equivalent positive and negative charge carriers. (As an example, consider a plastic pipe full of water with dissolved table salt (NaCl), where the charge carriers are positive Na$^+$ and negative Cl$^-$ ions, whose masses are reasonably similar.*) There, the velocity of the two different carriers is opposite $-$ but it is important to keep in mind that they are still carrying current in the same direction, which means that the Lorentz force, $$mathrm dvec{F}=i:mathrm dvec{s}timesvec{B},$$ points in the same direction for both!
* Or, even better, potassium chloride, so the mass ratio between the carriers is $m_mathrm{K}/m_mathrm{Cl} = 1.10$
Correct answer by Emilio Pisanty on May 8, 2021
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