In Einstein's 1905 paper on electrodynamics, what he meant by energy of electromotive force? In his 1905 paper, Einstein says that when the magnet is in motion and conductor stationary, changing magnetic field in space develops electric field "of certain definite energy", and this starts current in the conductor. But in the reverse situation no electric field is produced in space but an electromotive force is produced in the conductor "to which there is no corresponding energy". What are these energies that he is referring to? Why there is no energy in the reverse case? Why asymmetry is troublesome? And how it indicates that Maxwell's laws are frame independent?
 A: Between Einstein's time and the present, there has been a lot of rounds of telephone tag and much context was lost. Part of this falls on Einstein's shoulders by not making explicit what he was referring to and part on his predecessors (Hertz, specifically) for providing the broken formulation of Maxwell's theory, which Einstein used in the 1905 paper, that further muddied the waters. So, I'll straighten everything out here.
Einstein returned to the general issue in 1908 and (with Laub) published a revised treatment of his 1905 paper, separately from and independently of Minkowski, who did the same. They each wrote down the equivalent of what we today call the Maxwell-Minkowski relations for the constitutive laws of the electromagnetic field.
This is our present situation today, particularly with respect to the Maxwell-Minkowski relations:
Maxwell's equations, in their present form - in contrast to the form Maxwell originally wrote them in - largely comes from this time and the following decades where Einstein returned to the matter once more. Our understanding today is that the equations can be separated into one group, which is completely independent of any consideration of metric and is the same in both the relativistic and non-relativistic form - and would have the same form even if time were a spatial dimension - and another group in which metrical properties and the distinction between different space-time structures has bearing.
The metric-independent subset of equations may be written as follows:
$$
 = ∇×, \hspace 1em  = -∇φ - \frac{∂}{∂t} \hspace 1em ⇒ \hspace 1em ∇· = 0, \hspace 1em ∇× + \frac{∂}{∂t} = , \\
∇· = ρ, \hspace 1em ∇× - \frac{∂}{∂t} =  \hspace 1em ⇒ \hspace 1em ∇· + \frac{∂ρ}{∂t} = 0,
$$
where:

*

*$$ is the magnetic potential (or "electromagnetic momentum" as Maxwell referred to it as) and $φ$ the electric potential,

*$$ the magnetic induction, $$ the electric field,

*$$ the electric displacement, $$ the magnetic field,

*$ρ$ the charge density and $$ the current density.

The fact that they are totally independent of metric (technically: *diffeomorphism* invariant) can be fully brought out by writing the fields as differential forms:

*

*$A = ·d - φ dt$ - the potential 1-form, where $d = (dx, dy, dz)$,

*$F = ·d + ·d∧dt$ - the field strength 2-form, where $d = (dy∧dz, dz∧dx, dx∧dy)$,

*$G = ·d - ·d∧dt$ - the response 2-form,

*$Q = ρ dV - ·d∧dt$ - the charge/current 3-form,

with the Maxwell equations being written as
$$
dA = F \hspace 1em ⇒ \hspace 1em dF = 0, \\
dG = Q \hspace 1em ⇒ \hspace 1em dQ = 0.
$$
It was actually Maxwell who wrote the fields as I just did above ... he used differential forms in most of his treatments, and vectors only sparingly (after appropriating quaternion algebra to serve this end). The only thing he didn't do was make full use of their Grassmann algebraic nature (i.e. $dx∧dy = -dy∧dx$) or combine everything with the use of $dt$. He only used, what we would today write as: $φ$ by itself, $·d$, $·d$, $·d$, $·d$, $·d$, $·d$ and $ρ dV$.
The other set of equations - which distinguish between the relativistic and non-relativistic versions - are the constitutive laws, which relate the response fields to the field strengths. The Maxwell-Minkowski relations assume that the constitutive laws are isotropic in at least one frame of reference. For all other frames of reference, there is the inclusion of a reference velocity $$ for this frame. The constitutive relations take on the following form:
$$ + \frac{1}{c^2} × = ε ( + ×), \hspace 1em  - \frac{1}{c^2} × = μ ( - ×),$$
where $ε$ is the permeability and $μ$ the permittivity.
One may, therefore, distinguish between the stationary frame, where $ = $, in which the constitutive laws reduce to isotropic form
$$ = ε , \hspace 1em  = μ ,$$
versus the moving frames where $ ≠ $. The respective forms of the equations are the "stationary form" and "moving form". Those terms were used in the 19th century, as well.
These equations generalize to the following form:
$$ + α × = ε ( + β ×), \hspace 1em  - α × = μ ( - β ×),$$
which are suitable for a combined space-time geometry in which the following are its invariants:
$$β dt^2 - α |d|^2, \hspace 1em dt \frac{∂}{∂t} + d·∇, \hspace 1em β ∇² - α \left(\frac{∂}{∂t}\right)^2.$$
The cases where $αβ > 0$ correspond to relativity, with a finite invariant speed $c$ given by $c = \sqrt{β/α}$. The cases $α ≠ 0$ and $β = 0$ would be the form suitable for a universe where the invariant speed is 0 (this is called the Carrollian universe, named after Lewis Carroll), while the cases where $α = 0$ and $β ≠ 0$ correspond to a universe where the invariant speed is infinite, where simultaneity is absolute; i.e. the universe of Newtonian physics, itself: non-relativistic physics. In non-relativistic form, normalizing $β = 1$, the Maxwell-Minkowski relations take the form
$$ = ε ( + ×), \hspace 1em  = μ ( - ×).$$
This is equivalent to the form which Lorentz derived in his theory. It is also the final form that Maxwell's constitutive laws took, after the $×$ term was added, as a correction by Thomson in the 1890's, I believe. Maxwell only ever wrote it as $ = μ $, which was a mistake (and muddied the picture). Also, he stuck the $×$ term in with $$, so his field-potential relation for $$ actually read: $ = -∇φ - ∂/∂t + ×$, so that he could write the other constitutive law as $ = ε $. But that's mostly just semantics
What Einstein showed is that in the relativistic case, where one has a finite invariant speed $c$, the constitutive laws are equivalent to the stationary form in all frames of reference ... provided that $εμ = 1/c^2$. If you actually work it out, you will find that the $$ vector will cancel out. Or, in Einstein's words: "$$ becomes superfluous". Under Einstein's assertion, the finite speed for electromagnetic wave propagation $V = 1/\sqrt{εμ}$, when in a vacuum, is one and the same as the finite invariant speed: $V = c$.
A proviso: the equivalence to stationary form in all frames is only true if $|| ≠ c$ and $|| ≠ V$. Ironically, that exception is the case (unaddressed) of what it would be like to "move along a light beam"; or more precisely: where the medium is at light speed.
More generally, the constitutive equations become $$-independent if $εμβ = α$. You can see the problem with the non-relativistic case where $α = 0$ ... or even the Carrollian case where $β = 0$. There's no way to get $$-independence for finite non-zero permittivity $ε$ and permeability $μ$.
This bothered Maxwell a lot. He made it one of his big issues, because he was always shilling for (Galilean) relativity of motion. And, yet he was staring directly at a set of equations that - in a "vacuum" (whatever it was) - showed different forms, dependent on some reference velocity $$; and flat-out contradicted his supposition of Galilean relativity. The vacuum was behaving like some kind of medium, by virtue of having this reference velocity sticking around.
He tried to talk his way out of this explicit breaking of Galilean relativity by surmising that maybe there was some kind of deeper theory that would describe a structure for the vacuum and give us a way to determine $$ (as well as $ε$ and $μ$). The question of what structure underlie the constitutive laws was one of the items on his end-of-treatise to-do lists. It was never resolved - as it is the same issue that underlies the matter of renormalization and regularization today in quantum field theory. It's what quantum field theory is pushing under the rug when it goes through its mental gymnastics.
In the later part of the 19th century, people made a distinction between the "stationary" versus "moving" form of Maxwell's equations. This is what they were referring to. Almost always, they used the stationary form; while they looked for some sign of $$. This was muddied, a little bit, by the fact that you're going to have a non-zero $$, even in Relativity because measurements are being done from within, or across, a medium other than a vacuum: the Earth's atmosphere. But taking that into account didn't seem to resolve the issue, when the power of resolution finally reached the point where people could take measurements and begin to get an estimate on $$.
What bothered Einstein is that you could have two entirely different situations which were not reflected in terms of anything tangible by which frame of reference you were in. The relations with $ ≠ $ would give you one set of results - including energies - while those with $ = $ would give you another set of results.
So, when he assumed that the wave speed in a vacuum is also an invariant speed, and that it's a finite invariant speed, he went from Galilean relativity, with $α = 0$ and $β ≠ 0$ to Lorentzian relativity with $αβ > 0$.
Even Lorentz couldn't do that, despite his stumbling on the Lorentz transforms. He missed the boat, because he only used the transforms as a way of staging Galilean transforms! That is: he wrote a Galilean transform $g$ as a Lorentz transform $l$ and a correction $l^{-1}g$ to get $g = l \left(l^{-1} g\right)$ ... which is why his constitutive laws ended up having non-relativistic form, rather than relativistic form. That discrepancy is something that Einstein, himself, noted rather late in the game. I think in 1920, when he finally got around to getting a proper understanding of Maxwell's actual theory and actual field-names, separate from Hertz' warped formulation.
It was Maxwell who gave the names those letters:

*

*$$ - for the "electromagnetic momentum" (the vector potential),

*$$ - for the magnetic induction,

*$$ - for the "total current":  + ∂/∂t,

*$$ - for the electric displacement,

*$$ - for the electric field,

*$$ - for the force ... he took different stabs at the force law, one being $ = ×$, and never got a handle on what the force law should be,

*$$ - for the magnetic force,

*$$ - for magnetization, proportional to the difference $ - μ_0 $, where $μ_0$ is the vacuum value of $μ$, but he had different normalizations for $$ and $μ$ than we have today, so that situation was muddied,

*$$ - the current density.

Haven't anyone ever wondered why there's a gap where $$ is supposed to be? Well, now you know. He used $$ and $$ interchangeably, and was never actually totally clear on what these velocities were being taken with reference to (thereby obscuring and undercutting his own advocacy of relativity of motion). You can't blame him. He was only in his 40's, died young and never got old (and experienced) - like that PBS drama on Einstein and Maxwell sneaked in a joke about.

Einstein's failing (he was only in his 20's) is in not explicitly calling out $$, and the other context surrounding the issue, oblivious to the fact that 100 years later nobody would know the context of these remarks. Research literature is supposed to dot every "i" and cross every "t", when it comes to references, so that you can (at least in principle) trace every talking point, fact, result and idea all the way back to its point of origin by following reference links. He didn't have a reference list. Referees are supposed to return papers like that and tell the author to put in a bibliography. (That's why we need to automate validation and get the human and referee out of the loop, and why I'm striving to do just that.)
And all of that is the context I just supplied here. Now you also know why the paper was called the electrodynamics of moving bodies. The word "moving" was in reference to $ ≠ $.
