By the 1860s, the classical theory of electricity and magnetism was on a very solid theoretical footing. Maxwell’s equations describing the interplay of charges and currents with electric and magnetic fields were on paper by 1862, and with some changes in notation they’re the exact same today. Relativity wouldn’t be invented for another half-century or so, and that makes it all the more remarkable that Maxwell’s equations don’t actually need to be modified at all to work in a relativistic framework. Lorentz covariance is built right in, though it’s a bit hidden.
But Maxwell and Faraday and Ampere and the rest didn’t know that. There were some tantalizing hints though, and in fact it was the exploration of classical electrodynamics that led Einstein to the theory of special relativity. It’s entertaining to take a look at some of those hints, which are lurking right there in second-semester intro physics.
Consider a uniformly charged wire alongside a particle of charge q:
(Apologies for the sloppy PowerPoint graphic, but it probably gets the gist across.) We know from freshman physics that (via Gauss’ law), the electric field generated by that charged wire is:
The force experienced by the charge q in the field is:
We’ll say the particle and the wire are both positively charged, so the force is repulsive and pointing radially outward from the while. For simplicity, we won’t bother with vector notation in this post, but do keep in mind that forces and fields are vectors and do have a direction that we have to pay attention to.
Now let’s start moving the wire and the particle to the right at a constant velocity v. Or equivalently, move ourselves to the left at a constant velocity v, leaving the wire and particle stationary in the lab frame. Physically, they are the same situation, and this ends up being a key part of relativity. A moving charged wire is a current carrying wire, since an electric current is just moving charge by definition. The current I is given by I = λv, since charge/time is the same thing as (charge/length)*(length/time). A current produces a magnetic field which wraps radially around the wire with a magnitude of
So now we have electric and magnetic fields, like so:
Now the force on a charged particle moving in a magnetic field is F = qvB, and in this case it’ll be directed radially inward, toward the wire.
Now hold on – when everything was standing still, the net force was qE, pointed away from the wire. When we changed nothing at all except sliding our own chair in the lab to the left at speed v, suddenly the net force is F = qE – qvB, which is something completely different. Substituting the expressions for E and B in, the net force is:
What the heck? The force is an objectively measurable thing which gives the particle a specific acceleration. It can’t possibly be different depending on whether we’re moving or not. If classical electromagnetism makes such a prediction, surely the theory is wrong. Right?
Right – if we assume that all these charges and fields and currents and lengths are all the same in both frames. And that simply isn’t the case. You need relativity.
But it’s 1860, and we haven’t got relativity yet. How might we go about groping in the dark toward an answer? Well, we might postulate that fields aren’t the same in each frame. In the frame where the system is stationary, we have (say) the electric field E, while in the moving frame we have some different electric field E’, given by some coefficient E = αE’, where α is a function of v. If we assume the force is the same in both frames (it isn’t, but we don’t know that in 1860), we can look for that coefficient by solving:
Which gives after a little algebra:
If this initial groping-in-the-dark attempt at fixing classical E&M to work like our Gallilean intuition says it should is right, the E field in the moving should be slightly bigger in the moving frame than in the rest frame. Maybe the motion with respect to the ether somehow magnifies it, I dunno. In any case the correction factor is very small. If we’re talking about laboratory speeds in the m/s range, the correction factor is on the order of parts-per-quadrillion.
But is it right experimentally, if we could measure it? As it would turn out, no – but it’s close. It turns out that α should be the Lorentz γ factor, but at small speeds his γ and our α have the same order of magnitude (though we’re still off by a factor of 2, it turns out).
In any case our first attempt at a relativity theory is wrong – but closer to right than we were without it. Don’t be tempted to think that even people like Einstein had their brilliant ideas spring into being fully formed. Even the seemingly sudden great advances represent a lot of hidden hard work, tentative steps, and false starts.