There’s a very neat demonstration of how relativity is ingrained in electricity and magnetism, and it’s particularly nice because it involves length contraction. Of the two complimentary effects of relativity, time dilation is easier to see because of our ability to measure time precisely. Length contraction has the muon decay experiment, where it is observed that muons in the atmosphere survive to reach the ground, which is far longer than they should if time and length are not relative, and also in certain relativistic collisions which show that the nuclei must be length contracted for the dynamics to make sense.

But this one is simpler. Consider a charge moving along a path parallel to a current-carrying wire. (I’m going to use positive charges so that there are no pesky minus signs). For simplicity we’ll model this a series of charges with some spacing, all moving at a speed v, and a test charge moving with that same speed. Using the right-hand rule, the magnetic field is into the page below the wire, and the cross product right-hand rule tells us that the force will be toward the wire.

We also know there is no electrostatic force in this frame of reference — whatever charge densities we have, they are balanced. But now let’s look at what happens in other frames. Specifically, let’s look at the rest frame of the positive test charge: it is at rest, and the current is now due to the negative charges moving to the left.

Since the charge is at rest, there can be no magnetic force — it only acts on moving charges. And yet the frames have to agree on the attraction between the charge and the wire: it can’t just disappear in this frame. So where does it come from?

It’s an electrostatic force. The charge densities were equal in the lab frame, but now the negative charges are moving and the positive charges at at rest. This introduces length contraction for the negative charges, increasing their charge density relative to the positive ones, and we now have a net charge. So what we really have is this

This gives rise to an electrostatic attraction, which is exactly the force we had in the lab frame. (If you’re interested in the details of the derivation, one example is here. Sorry. Writing it out in Latex is a bit of a pain.)

In any intermediate frame, we would see a mix of the two forces; as we move into a frame with some speed other than v, we will see a smaller value for both the current and the speed of the charge, meaning the magnetic force falls off (and quadratically), but the electrostatic force appears in the right amount and gives us the net force we expect. And, of course, it’s not magic. It’s relativity.