Science has the answer!
There are lots of fluxes in the world, of course, so I've simply chosen magnetic flux arbitrarily. Similar results will arise from electric flux (which might just as well be called a "flux inductor"!). If you try it with a mass flow, you'll get a regular capacitor, or something a lot like it (since after all, electrical current is a flux, which is 'capacitated' by a regular capacitor.)
Let us first reflect on what magnetic flux is:
When a current I flows, it creates a magnetomotive force NI = F which results in magnetic flux Φ. When talking about magnetic circuits, we are always talking about a system of permeable material, so the flux is carried primarily in the core pieces, and divided among them by Ohm's law (or as I like to say, Magnetohm's law). Instead of voltage, we have amperage; instead of current, we have flux; instead of resistance, we have reluctance. Curiously, rather than dissipating power, the system stores energy -- unlike an electric circuit containing resistances!
In this example, the left hand core segments, including the segment with the winding, represent a "source reluctance" R1. The center leg consists of two materials, one with permeability μ1 and resulting reluctance R2, and another of μ2 and reluctance R3. The right path is in parallel with the center leg, shunting some flux through R4.
So that serves as an introduction to magnetic circuits. You can see they are similar to simple electrical circuits, and they are analyzed the same way. Since reluctance is analogous to resistance, perhaps we can concieve of a component analogous to the capacitor. Replacing V with F, I with Φ, we get:
If you look at the voltage on the winding applying the MMF, you see EMF = -NdΦ/dt, good old Faraday's law. For a sinusoidial current on the left hand circuit, the resulting EMF is the derivative of the current, so it is phase shifted by 90° and no power is drawn on average. In the right hand circuit, however, the flux itself is phase shifted, so the EMF is the second derivative, and is always in phase with current. Evidently, a flux capacitor wouldn't be very useful for time travel, as it is simply a frequency-dependent resistor! You always suspected Doc Brown was a crackpot -- now you know.
What kind of materials posess this R′ quantity? Well, it turns out pretty much everything magnetic does, at some frequency or another. If we look at the magnetic circuit which combines reluctance and "flux capacitance" in parallel, and reduce it to material properties, we get a complex quantity for μ. The magnitude represents permeability, and the angle represents phase shift ('capacitivity', as it were). The real component is denoted μ′, and results in lossless energy storage (such as it is). The imaginary ("phase shifted") component is denoted μ′′ and represents lossiness. In fact, the angle of this quantity is often given in datasheets, for some reason as the relatively cryptic value "tan δ/μi". This compares with the tan δ of (regular) capacitors, except that because they divide it by initial permeability for some reason, the numbers generally end up mysteriously small. (Surely it would be a scandal if capacitor manufacturers began rating their multilayer ceramic chip capacitors in "tan δ/εi" -- Z5U is a pretty awful ceramic, but it typically has a permittivity over 20,000!)
In closing, if you'd like your own flux capacitor, simply make a really awful inductor. A ball of wire around a steel rod will do an excellent job. It may not look like the famous Flux Capacitor, but take confidence in your cheap object: Hollywood simply got the science wrong, as they so often do; it actually should've looked like your ball of wire!
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