Why the length constant of passive current flow isn't depend on the membrane capacitance?

Question

I read that the equation for the length constant for passive conductance along a neuron depend on the resistance of the plasma membrane, the intracellular axoplasm and the extracellular medium. My question is why it's not dependent also on the membrane capacitance, since it can also affect how far the current can propagate (if for example the capacitance of the membrane is large, more negative charge will be on it's intracellular side, and will cause to the positive current to attract to the sides of the membrane). Thanks!

Answer 1

Length constants are defined in the 'steady state' regime of constant voltage; capacitance is important for time-dependent electrical functions but disappears in the steady state.

In a real neuron, where you have other time-dependent (dynamic) features like ion channel opening, time constants and length constants interact to determine how current flows in a length of neurite, as you intuit, but these passive properties by themselves are insufficient to describe that behavior: you need to also consider ion channel gating and conductances.

See also https://en.wikipedia.org/wiki/Cable_theory - note how the capacitance is always multiplied by a changing voltage dV/dt in the differential equations.