Watts vs. volts amperes

I understand that a watt is a unit of power (change in energy per unit time) that describes the rate at which physical work can be done

Right.

The key thing to observe is that energy can move in both directions. It can move from the "supply" to the "load", but it can also move from the "load" to the "supply".

The power at any given instant is found by multiplying the current at that instant by the voltage at that instant. In a DC system our life is simple, voltage is a constant, current is a constant, so power is also a constant.

In an AC system however, voltage current and power all vary over time. So we attempt to use RMS voltage, RMS current and mean power to describe our AC systems.

There is a problem though, If voltage and current are sinusiodal and in-phase with each other then mean power is equal to RMS current times RMS voltage. However if they are 90 degrees out of phase then there is no net transfer or power. For two of the four quarter cycles energy flows from source to load. For the other two quarter cycles energy flows from load to source. Net energy transfer over the cycle is zero.

Is reactive power truly unusable? Suppose I had a magnet coil in a fictitious AC circuit, that is purely capacitive reactive, with no resistive component at all. It would exhibit purely reactive power flow. Wouldn't the coil produce a changing magnetic field,

Yes

that I can use to deflect a nearby magnet (converting electric energy into kinetic energy)?

What you would find is that when you introduced the magnet and placed a load on it that the current in the coil would no longer be purely reactive.

Are Watt and Volt Amps dimensionally equivalent, just e.g. like km/s and miles/hour?

Yes they are dimensionally equivilent, but by convention we use watts for "real" (mean) power and for instantationus power but not for "reactive" or "apparent" power.

If so, would using watts to denote apparent power be "technically correct" (and merely "wrong by convention")? If that's the case, then my understanding that VA is used instead of W, because it has a "documenting" purpose, by commenting exactly how these watts can be used. But this sounds strange, because then it seems like "not all watts are created equal"

Dimension is not a full specification of what a quantity means. For example both joules and newton-meters work out to kg⋅m²⋅s⁻² but noone would argue that torque and energy are the same thing.

What you understand is basically correct. Regarding DC circuits it is important to point out that 1 watt = 1 amp x 1 volt under steady state (long time) conditions when transients are gone. Under those conditions an ideal capacitor looks like an open circuit (no current flow) and an ideal inductor looks like a short-circuit (no voltage across the inductor). The only power is that dissipated in resistance.

Is reactive power truly unusable?

No. Reactive power results in energy being stored in the reactive circuit components. At any instant in time the energy stored in the electric field of a capacitor is

$$E_{C}(t)=\frac{CV(t)^2}{2}$$

And energy stored in the magnetic field of an inductor is

$$E_{L}(t)=\frac{LI(t)^2}{2}$$

That energy is available to either do work or generate heat. The energy of the magnetic field, for example, is essential to motor operation. You yourself pointed out that you could use a changing magnetic field to deflect a magnet (do work).

Are Watt and Volt Amps dimensionally equivalent, just e.g. like km/s and miles/hour

Yes. Both are units of Joules/sec. A volt is a Joule/Coulomb and an ampere is a Coulomb per second. The units of watts, whether they be electrical watts or watts of mechanical work, are Joules/sec.

If so, would using watts to denote apparent power be "technically correct" (and merely "wrong by convention")?

No it would not be technically correct. They both have units of Joules/sec but Volt-Amperes does not take into account any phase difference between voltage and current that would occur with reactive components (inductors, capacitors) in the circuit. On the other hand, instantaneous power in watts, is given by V(t)I(t)cosθ where θ is the angle between voltage and current. The angle is zero for purely resistive circuits so that the cos is 1.

But this sounds strange, because then it seems like "not all watts are created equal"

I think it's more accurate to say "not all volt-amperes are considered equal"

The product of voltage and current, without any knowledge of the circuit elements, is called volt-amperes and not watts. If the circuit is purely resistive, than volt-amps is the same as watts.

I guess you could say that all watts are volt-amperes but not all volt-amperes are watts.

Most of this makes sense. But what's troubling me is the idea that you use a different unit just because of context. "I guess you could say that all watts are volt-amperes but not all volt-amperes are watts." this makes sense, but it doesn't explain why we don't just universally use "VA", or universally use "W", and add on the stipulation of what we mean "Watts apparent" or "VA resistive".

I understand why this can be troublesome. First of all, the units are not different, but the same, i.e., Joules/sec. You don't have to think of the labels "watts" and "volt-amperes" as being electrical units. It is just that "watts" connotes in phase current and voltage and has traditionally been associated with resistive loads while "volt-amperes" leaves it open. I think you will find the term "volt-amperes" used when no detail is give on just what the voltage across and the current to particular circuit elements are. When in doubt, I always use the term volt-amperes since this is the most all inclusive term.

Hope this helps.

Suppose I had a magnet coil in a fictitious AC circuit, that is purely capacitive, with no resistive component at all. It would exhibit purely reactive power flow. Wouldn't the coil produce a changing magnetic field, that I can use to deflect a nearby magnet

The coil should have a purely inductive impedance (not capacitive), only it it were ideal, like an infinite solenoid with no resistance. A real solenoid could have a negligible resistance as well, but it would radiate, thus the impedance would deviate from the purely inductive. If you put a magnet that can move, the resistive part of the impedance will be even more. Basically, the energy transfer takes place only due to the resistive part of the impedance.

Are Watt and Volt Amps dimensionally equivalent, just e.g. like km/s and miles/hour?

Yes! The "mechanical" unit is the W (defined only in terms of kg, m and s). To get V and A you need the electrical charge, the C. But when you multiply V*A the C simplyfies.

If so, would using watts to denote apparent power be "technically correct"

Rigorously, it is correct. In principle, you can say that you have an ideal capacitor, and drive it with an AC current with 1 W of product between current and voltage, although no energy is transferred. It's just a unit of measurement. It is also necessary to clarify what are the reported currents and voltages: the peak values, the peak-to-peak, or the "power-average" values? (the last is the value which gives the average power). In any case, it is better to write what you are reporting and not to rely on the unit of measurement.