The relativistic solenoid

We close with a surprise that links relativity to electromagnetism in an elegant and almost elementary way. A solenoid with $N$ turns and length $\ell$, carrying current $i$, produces inside it a magnetic field

$|\vec{B}|_B = \mu_0\,i\,\frac{N}{\ell}$

measured in its own proper reference frame, SRB. Now consider an observer SRA who sees the solenoid moving with velocity $v$ along its own axis. By length contraction, in SRA the solenoid is shorter:

$\ell_A = \ell\,\sqrt{1 - v^2/c^2} = \frac{\ell}{\gamma}$

The crucial point is that the number of turns $N$ remains invariant: it is an objective count, because the atoms of the wire are the same in every reference frame and no turn can appear or disappear by changing observer. But if $N$ stays fixed while the length contracts by a factor $\gamma$, then in SRA the turn density $N/\ell_A$ is larger by that same factor. The field grows accordingly:

$\ev{|\vec{B}|_A = \mu_0\,i\,\frac{N}{\ell_A} = \gamma\cdot|\vec{B}|_B}$

The magnetic field is therefore more intense precisely because the contracted turns have been packed closer together. This is a particularly clean way to show that $\vec{E}$ and $\vec{B}$ “mix” under Lorentz transformations: the very concept of a “pure magnetic field” is relative to the reference frame, and the same wire, viewed from a different SR, produces a different field. Einstein used exactly this kind of argument — geometric, never invoking tensor formalism — to convince himself that electromagnetism and relativity were inseparable, two faces of the same structure.

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Topics: Relatività ristretta Concepts: Contrazione delle lunghezze Objects: Solenoide

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