text: Goldmans equation seeks to determine the voltage E across a membrane.  A Cartesian coordinate system is used to describe the system, with the z direction being perpendicular to the membrane.  Assuming that the system is symmetrical in the x and y directions (around and along the axon, respectively), only the z direction need be considered; thus, the voltage E is the integral of the z' component of the electric field across the membrane.
  According to Goldmans model, only two factors influence the motion of ions across a permeable membrane: the average electric field and the difference in ionic concentration from one side of the membrane to the other.  The electric field is assumed to be constant across the membrane, so that it can be set equal to E/L, where L is the thickness of the membrane.  For a given ion denoted A with valence n, its flux j'—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula
  The first term corresponds to Ficks law of diffusion, which gives the flux due to diffusion down the concentration gradient, i.e., from high to low concentration.  The constant D is the diffusion constant of the ion A.  The second term reflects the flux due to the electric field, which increases linearly with the electric field; Formally, it is [A] multiplied by the drift velocity of the ions, with the drift velocity expressed using the Stokes–Einstein relation applied to electrophoretic mobility. The constants here are the charge valence n of the ion A (e.g., +1 for K, +2 for Ca and &minus;1 for Cl), the temperature T (in kelvins), the molar gas constant R, and the faraday F', which is the total charge of a mole of electrons.
  This is a first-order ODE of the form y = ay + b, with y = [A] and y = d[A]/dz; integrating both sides from z=0 to z=L with the boundary conditions [A](0) = [A] and [A](L) = [A], one gets the solution
  where μ is a dimensionless number
  and P is the ionic permeability, defined here as
  The electric current density J equals the charge q of the ion multiplied by the flux j
  Current density has units of (Amperes/m). Molar flux has units of (mol/(s m)). Thus, to get current density from molar flux one needs to multiply by Faraday's constant F (Coulombs/mol). F will then cancel from the equation below. Since the valence has already been accounted for above, the charge q of each ion in the equation above, therefore, should be interpreted as +1 or -1 depending on the polarity of the ion.
  There is such a current associated with every type of ion that can cross the membrane; this is because each type of ion would require a distinct membrane potential to balance diffusion, but there can only be one membrane potential.  By assumption, at the Goldman voltage E, the total current density is zero
  (Although the current for each ion type considered here is nonzero, there are other pumps in the membrane, e.g. Na/K-ATPase, not considered here which serve to balance each individual ions current, so that the ion concentrations on either side of the membrane do not change over time in equilibrium.) If all the ions are monovalent—that is, if all the n' equal either +1 or -1—this equation can be written
  whose solution is the Goldman equation
  where
  If divalent ions such as calcium are considered, terms such as e appear, which is the square of e; in this case, the formula for the Goldman equation can be solved using the quadratic formula.