Defining specific physics for a fictional element (or substance) used for airship lift

Question

On this Earth-like world with three moons and a human society with early-WWI technology and Victorian-like culture, a material known as "Suspendium" propels floating airships to war.

This Suspendium (based off of lore from Airships: Conquer The Skies) has a characteristic that make it interesting for making things float: Suspendium repels other Suspendium.

The element is omnipresent, from the dust in the air, to the bedrock in the ground. Airships use refined crystals or cheaper Suspendium dust to lift themselves up by repelling the ambient Suspendium within the ground. There is no anti-gravity magic; Suspendium repulsion behaves more like a fundamental force. Suspendium forms crystals in some unknown geological process. The larger the crystal, the rarer it is. Additionally, crystals do not repel Suspendium inside themselves, unless impurities are introduced.

Normally, Suspendium exerts next to no force while idle. However, they can be charged by running an electric current through crystals, which drastically increase their repulsion force, allowing them to hold up much more weight. Finer Suspendium dust produces a fair bit of lift for free, likely through some electrostatic process, though this hasn't been proved.

Suspendium's repulsive force also weakens with distance, constraining most airships to a service ceiling of up to 300 meters or so from ground level. Very light crafts can go higher, with diminishing returns.

While this is already plenty to support the universe with fictional yet consistent logic for explaining something that doesn't fit within real life physics, I would like to have a basic physics theory for modelling Suspendium forces in-universe, which could allow me to create some more esoteric mechanisms or weaponry while also explaining how they work, while staying consistent to the lore. How could "Suspendium" and its forces be defined using a simple supplemental physics theory?

Specifically, I am looking for a way to predict Suspendium-related interactions using mathematics, like using F=q(VxB) and such in calculating magnetic physics. The more that it fits within some criterions outlined below, the better.

Notes:

• Detail doesn't matter that much. This theory could be summarized in a single formula, which could (as an example) explain the relationship between the mass of two Suspendium objects, electric potential within said objects, and the distance between them. It could also be a lengthy paper which defines it as its own fundamental force, or anything in between. In any case, it should allow accurate predictions like other theories in physics, while respecting the existing properties of Suspendium.
• The existing lore doesn't provide a scale from which we can define some constants or units for use in formulas, so some educated guesses about physical properties of Suspendium are acceptable.
• Suspendium doesn't replace any existing parts of physics. All modern physics exist alongside Suspendium in this universe. Additionally, Suspendium shouldn't (but could when really necessary) violate existing laws in physics, such as conservation of energy.
• We don't know if Suspendium is an element, elementary particle, or some compound made from other things that aren't anomalous. Any interpretation will work as long as it is believable.
• Suspendium isn't a plot device that miraculously has many useful purposes. It's just a means to make things fly.
• Existing graphics draw Suspendium crystals as cyan octahedrons with a 2:1 height to width ratio. This could affect the molecular structure.
• It's not necessary to explain where Suspendium comes from, since there's not enough technological progression for that to matter yet.
• Suspendium has only been observed in a solid, crystalline state. This doesn't necessarily mean that it can't exist in other states though.

Answer 1

Modelling of the physics of Suspendium

(note: assuming classical physics ie no quantum or relativistic effects)

Modelling Suspendium as a material that has some quantity associated with it (which I'm calling $s$, i'm also thinking that $s$ is linearly dependant on the mass of the Suspendium (for example $1Kg$ of pure Suspendium have $s=1$, $2Kg$ has $s=2$, …)).

Following the form of the classical laws of gravity and electrostatics, you could model that the force between 2 pieces of Suspendium is proportional to the product of $s$, ie for a fixed distance the force is ($k$ is just a constant).

$$f=ks_1s_2$$

You could also have it so the Suspendium becomes 'energised' when exposed to an electric current, $I$, which results in an increase in $s$, where $I_0$ is a base line current, and $s_0$ is the of value of $s$ when the Suspendium is not exposed to a current. $$s=s_0(1+\frac{I}{I_0})$$

Now if you want to follow the more classical laws of physics , then you can have the force, $f$, between pieces (of $s1$ and $s_2$ respectively) seperated by a distance $r$ to be $$f=k\frac{s_1s_2}{r^2}=k\frac{{(s_1)}_0 {(s_2)}_0(1+\frac{I_1}{I_0})(1+\frac{I_2}{I_0})}{r^2}$$

Estimation of constants

If $s=5000$ for a cubic metre of Suspendium (which i'm estimating weighs $5000Kg$) and guesstimating that the a grain of dust has a volume of ~$10^{-15}$ $m^3$. Now for dust to fall the force of gravity has to be greater than force of repulsion, between the the dust and the Suspendium in the earths crust. Now assuming that the earth has $\frac{S}{M}=0.01$, where $M$ is the mass of the earth and $S$ is the total amount of $s$ in the earth. And the dust has $\frac{s}{m}=1$, then $k$ can be found.

$$f_{gravity}\geq f_{Suspendium}$$ $$\frac{GmM}{r^2}\geq \frac{ksS}{r^2}$$ $$G \geq k\frac{s}{m}\frac{S}{M}$$ $$G\geq0.01k$$ $$k\leq 100G= 6.67*10^{-9}$$

from this the force between repelling the dust is only just greater than the force of gravity, therefore the electromagnetic force would be greater over the short distances between the dust. so large crystals should be able hold themselves together.

Airship flight

The airships height can be adjusted by slightly adjusting the current through the crystal, which would increase the value of $s$, So to find the ratio of $\frac{I_1}{I_0}$, From the image, i'm estimating that the volume of the crystals are ~$7.46m^{3}$, and i'm guesstimating that the air ships weigh ~$10^5Kg$. For the air ships to float the force of gravity has to be greater than force of repulsion, between the the crystal and the Suspendium in the earths crust. So $m=10^5Kg$, $s_0=7.46*5000=37300$ and $\frac{S}{M}=0.01$,and am setting $k=10G$.

$$f_{gravity}\leq f_{Suspendium}$$

$$\frac{GmM}{r^2}\leq \frac{ks_0S}{r^2}$$

$$\frac{Gm}{ks_0}\frac{M}{S}\leq (1+\frac{I}{I_0})$$

$$(1+\frac{I}{I_0})\geq \frac{(G)(10^5)}{(10G)(37300)}(100)=26.81$$

$$\frac{I}{I_0}\geq25.81$$ When the ratio is exactly $25.81$ the air ship is neutrally buoyant (for this specific airship). If it is greater than $25.81$, then the air ship increases in altitude, similarly is less than $25.81$, it would sink.

To have a flight ceiling you could have $I_0$ to increase exponentially with altitude, so that for a fixed current the air ship would rise up to the altitude where $\frac{I}{I_0}=25.81$ (for $m=10^5$) then hold steady at that altitude.

For ships with lower mass they would require less current to maintain the same altitude. To lift larger ships to the same altitude you could, either increase current to the one crystal, use a larger crystal or use multiple crystals.

Energy conservation

Energy can also be conserved if the energy lost (into the current through the crystal) is equal to or greater than the increase in potential energy. This could also add to explaining the ceiling if the efficiency decreases with altitude.

hopefully that helps (i'll try to include more about energy)