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Most Planck units are a product of powers of all three of $\hbar$, $c$, and $G$, so we will not be able to fully understand their physical significance until we have a full theory of quantum gravity. But some of them are only powers of one or two of those three quantities, and we should be able to understand those units in a simpler physical regime. For example, the Planck charge $\sqrt{\hbar c}$ (under Lorentz-Heaviside conventions) sets the natural scale for the coupling constants of a relativistic quantum field theory in a non-dynamical $(3+1)$-dimensional spacetime (and indeed, at low energies the Standard Model's coupling constants are all within an order of magnitude or so of this charge scale).

Other examples are the Planck force $c^4/G$, the Planck power $c^5/G$, the Planck voltage $c^2/\sqrt{G}$, and the Planck current $c^3/\sqrt{G}$. These quantities are just powers of $c$ and $G$, so they should have an interpretation within completely classical general relativity. Indeed, as the Wikipedia page explains, the Planck force sets the scale of the gravitational force between any two Schwarzchild black holes (regardless of their masses) whose event horizons just touch, and also sets the scale for the effective gravitational forces that spacetime curvature induces on matter.

The physical significance of the Planck voltage and the Planck current in classical GR is less clear to me. In what sense do they set the natural scales for voltage and current in GR?

Two guesses, which seem to be in a similar spirit as the physical interpretation of the other Planck units, is that the effective current circulating around the equator of any Kerr-Newman black hole is on the order of the Planck current, and that the Planck voltage is the largest possible voltage difference before the electrostatic energy self-gravitation causes the system to collapse into a Reisser-Nordstrom black hole. (But those are basically just wild guesses.)

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    $\begingroup$ "These quantities are just powers of c and G so they should have an interpretation within completely classical general relativity" - While this is certainly possible (and even likely based on the magnitudes), there is no sufficient logic in this statement to guarantee that it is true. For example, the origin of G is unknown and may be related to quantum phenonena. $\endgroup$ – safesphere Dec 3 '17 at 3:15
  • $\begingroup$ I wonder if the interpretation of the black hole horizon as a membrane satisfying Ohm’s law with a surface resistivity of ρ = 4π ≈ 377 Ω is related. Not sure, since apparently it's non relativistic Ohm’s law. Interesting question, though. $\endgroup$ – Rexcirus Dec 3 '17 at 8:41
  • $\begingroup$ +1: for planck charge & voltage - I hadn't come across them before. $\endgroup$ – Mozibur Ullah Dec 3 '17 at 13:34
  • $\begingroup$ @Rexcirus That's not the surface resistivity of a BH, but rather the surface resistance. And it might be more relativistic than you suspect, because if you stop setting $c = 1$ then it's actually $R = 4 \pi / c = 377\, \Omega$ :-). As suggested by the lack of a $G$, that value is actually just the flat-spacetime impedance of free space and is not particular to black holes. $\endgroup$ – tparker Dec 3 '17 at 19:12

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