Magnets on parallel strings Suppose you have two simple straight bar magnets,
each hanging by their north poles from the ceiling by a thread,
a distance d apart,
and let their south poles likewise be attached to the floor by two threads.
You thus have two straight bar magnets on two parallel lines a distance d apart.
They tend to repel each other with a force f in this configuration.
Now twist one thread rapidly in a positive sense and twist the other thread in a negative sense so that each magnet is spinning with its thread as its axis.
How does the force f change as a function of the speed of this twisting and why?
 A: Minimally, if at all.  If the surface of the magnets was moving at the speed of sound in air, this would lead to a fractional change in the force of about one part in $10^{12}$.
A moving magnetic dipole $\vec{m}$ moving perpendicular to its axis will acquire an electric dipole moment given by $\vec{v}\times \vec{m}/c^2$.  Since the two magnets are counter-rotating, this will lead to them acquiring equal and opposite electric dipole moments in the laboratory frame;  these electric dipole moments will be given by $\vec{p} = \pm \alpha v \vec{m} /c^2$, where $v$ is the greatest linear speed of the magnets (at its surface) and $0< \alpha < 1$ is a geometric factor related to the fact that not all the points in the magnets will be moving at the same speed.
The magnetic moment of a moving dipole, meanwhile, is multiplied by $\gamma$.  So the magnetic force will change by something like $(\beta v)^2$, where $0< \beta < 1$ is another dimensionless factor depending on the geometry.
So to lowest relativistic order, the force between the magnets will become
$$
F = \frac{\mu_0}{4 \pi r^3} \gamma^2 m^2 - \frac{1}{4 \pi \epsilon_0 r^3} p^2 = \frac{\mu_0}{4 \pi r^3} m^2 \left(\gamma^2 - \frac{\alpha^2 v^2}{c^2} \right) \approx  \frac{\mu_0}{4 \pi r^3} m^2 \left(1 + (\beta^2 - \alpha^2) \frac{v^2}{c^2} \right)
$$
where we have used the fact that $c^2 = 1/\mu_0 \epsilon_0$.
This means that overall, we would expect the force between the two magnets to change (fractionally) by a factor on the order of $v^2/c^2$.  This would be a very small number;  for example, if $v$ was the speed of sound, then $v^2/c^2 \approx 10^{-12}$.  Whether or not the force would increase or decrease depends on the relative size of the factors $\alpha$ and $\beta$, which would require a bit of effort to determine.  It is possible that these two factors would be exactly equal, in which case the force might vary by factors of $\mathcal{O}(v^4/c^4)$ instead, which would be even smaller.
