What physical quantity has dimension [ML⁻³T⁻²] in electromagnetism?

[Kim Gi Hyuk]

TL;DR

Seeking a named physical quantity with dimension [ML⁻³T⁻²].
Two independent paths from Maxwell's equations (ρ²/ε₀ and μ₀J²)
both yield this dimension, suggesting it may be the apex of an
electromagnetic dimensional hierarchy. Is this quantity known in
established physics?

Background.
I have been building a systematic dimensional map of electromagnetic quantities, organized by their MLTIQNJ exponents. The map places mechanical quantities along a central vertical axis, with electric quantities to the left and magnetic quantities to the right — connected by structured dimensional steps (×d, ×E, ×H, etc.). The left-right symmetry reflects the duality between electric and magnetic sources.

The quantity χ.
When filling in all cells of the map using dimensional consistency with neighboring quantities, the top-center cell — the apex of the mechanical core axis — must have dimension:[χ] = M¹L⁻³T⁻²I⁰.

Two independent paths from Maxwell's equations both lead to this dimension:
- From Gauss's law (electric source): ρ²/ε₀ ~ [ML⁻³T⁻²]
- From Ampère's law (magnetic source): μ₀J² ~ [ML⁻³T⁻²]
where ρ is charge density [IL⁻³T] and J is current density [IL⁻²].

Note that (ρ, J) form the natural pair of electromagnetic source quantities — the four-current in relativistic notation.

My Questions
1. Does a named physical quantity with dimension [ML⁻³T⁻²] exist in established physics?
2. Is the structural relation ρ²/ε₀ ~ μ₀J² physically significant? For instance, does it appear in plasma physics, radiation reaction theory, or higher-dimensional theories such as Kaluza–Klein or Brane World models?
3. Both ρ and J — quantities with current index I≠0 — give rise, through ε₀ and μ₀, to a purely mechanical quantity χ with I⁰. Is there a known principle explaining why the apex of an electromagnetic dimensional hierarchy should be charge-free?

 

[Dale]

There isn’t any particularly deep significance for the dimensions of a particular quantity outside of the formulas where it appears and the system of units you use. Since you don’t specify either it isn’t much that can be said.

 

[Baluncore]

1. Does a named physical quantity with dimension [ML⁻³T⁻²] exist in established physics?

No.
That is the acceleration of the mass density.

 

[WernerQH]

There is some arbitrariness about dimensions. The Gaussian system has only three, but it is a mix of the electrostatic (esu) and electromagnetic (emu) systems, which have conflicting dimensions for the quantity of electric charge. By convention, charge was chosen to be measured in $\rm dyn^{1/2} cm$ (esu) in the Gaussian system, whereas $\rm dyn^{1/2} s$ (emu) was used in the practical system that evolved into the SI. It is rather unsettling to have different dimensions for a single physical quantity, depending on how it is measured! (As a force between two charges at rest, or between two electric currents.) Different units for charges at rest and charges in motion!

The dimensions differ by a velocity, and for the units we have the correspondence $$ \rm
1\ Biot \cdot second = 1\ dyn^{1/2} s \leftrightarrow 3 \times 10^{10} \ dyn^{1/2} cm = 3 \times 10^{10} \ Franklin
$$ which really means that 1 second is the same as 30 billion centimetres, if "charge" is to have only one reasonable dimension. Of course the speed of light is a fundamental constant, and the metre is now defined in terms of the second. Relativity has taught us that space and time are unified, and the distinction between dimensions of length and time is arbitrary. As for the "correct" dimension of charge, the existence of the fine-structure constant indicates that it should be dimensionless.

I found this an interesting read: Babel of Units. The Evolution of Units Systems in Classical Electromagnetism