Electromechanics and electrochemistry

From Coulomb to Maxwell to Nernst — and from there to mechanotransduction and the cochlear amplifier.

The cochlea is a mechanical device — fluid in a bony tube, membranes that vibrate, microscopic levers that gate ion channels — but it works because of electrochemical gradients. To follow how, we need a clean grounding in basic electromagnetism: the Coulomb field, the Maxwell equations, electric energy stored in capacitor-like membranes, then current and conductivity, and finally how chemical potentials acquire electrical contributions and the Nernst-equilibrium equation falls out. This chapter starts from scratch.

Coulomb’s law — the electric field of a point charge

A point charge $q$ produces an electric field

\mathbf{E}(\mathbf{r}) \;=\; \frac{1}{4\pi\varepsilon_0}\, \frac{q}{r^2}\, \hat{\mathbf{r}},

pointing radially outward (positive charge) or inward (negative). Two consequential properties:

Inverse-square scaling. Doubling distance quarters the field strength.
Linearity. The field of multiple charges is the vector sum of individual contributions (superposition).

q+1.00

r (probe)2.00

E = k_e q / r²0.250

V = k_e q / r0.500

Charge q = 1.00 Probe distance r = 2.00

Coulomb's law: a charge q at the origin produces an electric field E = k_e q / r² pointing radially outward (positive charge) or inward (negative). The field lines never cross; they spread radially with density falling as 1/r². The dashed circles are equipotentials — surfaces of constant V = k_e q/r — perpendicular to the field everywhere. Maxwell's first equation ∇·E = ρ/ε₀ is the integral statement of this field, applied to general charge distributions.

The field lines never cross; equipotential surfaces (constant $V = q/(4\pi\varepsilon_0 r)$ ) are perpendicular to the field everywhere. The electric potential and the field are related by $\mathbf{E} = -\nabla V$ .

For any charge distribution $\rho(\mathbf{r})$ , the field is given by the Coulomb superposition integral. But there is a much more powerful way to get the field directly when the distribution is symmetric.

Gauss’s law — Maxwell’s first equation

The integral statement, due to Carl Friedrich Gauss:

\oint_{\partial V} \mathbf{E}\cdot d\mathbf{A} \;=\; \frac{Q_\text{enc}}{\varepsilon_0}.

The flux of $\mathbf{E}$ through any closed surface equals the enclosed charge divided by $\varepsilon_0$ . The differential form is one of the four Maxwell equations:

\nabla \cdot \mathbf{E} \;=\; \frac{\rho}{\varepsilon_0}.

For symmetric charge distributions Gauss’s law gives the field without integration. The trick is to choose a surface over which $\mathbf{E}$ is constant in magnitude and either parallel or perpendicular to the area element.

Symmetry:

Gauss's law in integral form: ∮ E·dA = Q_enc/ε₀. The key is *choosing the right surface*: one over which E is constant in magnitude and parallel to the area element (or perpendicular to it on the side caps). For each symmetric charge distribution, this surface is the obvious one — spheres for point charges, pillboxes for planes, cylinders for lines. The resulting E-field formulas — 1/r², constant, 1/r — are completely fixed by symmetry, with no integration required.

Three canonical cases:

Spherical (point charge): $E = Q/(4\pi\varepsilon_0 r^2)$ — same as Coulomb’s law.
Planar (infinite charged sheet): $E = \sigma/(2\varepsilon_0)$ — constant, independent of distance.
Cylindrical (infinite charged line): $E = \lambda/(2\pi\varepsilon_0 r)$ — falls as $1/r$ .

The shape of $\mathbf{E}(\mathbf{r})$ — $1/r^2$ , constant, $1/r$ — is fixed by symmetry alone.

The full Maxwell equations

In differential form, with no sources except the fields themselves:

\nabla \cdot \mathbf{E} \;=\; \rho/\varepsilon_0, \qquad \nabla \cdot \mathbf{B} \;=\; 0,

\nabla \times \mathbf{E} \;=\; -\frac{\partial \mathbf{B}}{\partial t}, \qquad \nabla \times \mathbf{B} \;=\; \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}.

For our purposes the static-field versions (no time-derivatives) are sufficient: charges produce $\mathbf{E}$ via Gauss’s law, currents produce $\mathbf{B}$ via Ampère’s law, and the divergence of $\mathbf{B}$ is identically zero (no magnetic monopoles). The third equation — Faraday’s law — is what we need for any time-varying coupling.

Faraday’s law and electromagnetic induction

A changing magnetic flux through a closed loop induces an electromotive force:

\varepsilon \;=\; -\frac{d\Phi_B}{dt}, \qquad \Phi_B \;=\; \int \mathbf{B}\cdot d\mathbf{A}.

Mean B = 1.00 T Loop area A = 0.50 m² Tilt θ = 0°

Faraday's law: ε = −dΦ_B/dt. A changing magnetic flux through a loop induces an EMF equal to the time-derivative of the flux, with a minus sign (Lenz's law: the induced current opposes the change). This is the operative principle of electric generators, transformers, and induction stoves. In integral form, ∮ E·dℓ = −∂Φ_B/∂t over any closed loop — the third of Maxwell's equations.

The minus sign (Lenz’s law) ensures the induced current opposes the change in flux — energy conservation in disguise. Faraday’s law underlies generators, transformers, and inductive sensors, but for the cochlea it is mostly a background fact: the strong static magnetic susceptibility of biological tissue makes magnetic effects negligible.

Capacitance and stored electric energy

A capacitor stores charge $Q$ at voltage $V$ with proportionality $C$ :

Q \;=\; C V.

For a parallel-plate capacitor with area $A$ and gap $d$ , Gauss’s law gives the uniform field $E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)$ , and the voltage is $V = E d$ . Therefore $C = \varepsilon_0 A/d$ . The energy stored is

U \;=\; \tfrac12 C V^2 \;=\; \tfrac12 Q V \;=\; \tfrac12 \frac{Q^2}{C}.

Plate area A = 1000 cm² Plate separation d = 1.00 mm Voltage V = 10.0 V

The parallel-plate capacitor stores charge Q = CV on its plates and energy U = ½CV² in the field between them. C = ε₀A/d scales linearly with area and inversely with gap; E = V/d gives the operating field. Cell membranes are essentially nanoscale capacitors: with d ≈ 5 nm and dielectric ~5, the specific capacitance is ~1 μF/cm² — enough to support 100 mV potentials with manageable charge per cell.

A biological cell membrane is effectively a nanoscale capacitor: ~5 nm thick lipid bilayer with permittivity $\varepsilon_r \approx 5$ gives a specific capacitance $\sim 10\,\text{mF/m}^2 = 1\,\mu\text{F/cm}^2$ . A typical 100 mV membrane potential therefore stores $\sim 10^{-12}\,\text{C/cm}^2$ on each plate — a few thousand ions per square micron. The fact that this small surface charge sustains a 100 mV gradient is what makes membrane biophysics possible.

Current density and Ohm’s law

When charged particles drift through a medium under an electric field, they constitute a current. The current density $\mathbf{J}$ (charge crossing unit area per unit time) is

\mathbf{J} \;=\; n q \mathbf{v}_\text{drift} \;=\; \sigma \mathbf{E},

where $\sigma = nq\mu$ is the conductivity and $\mu = v_\text{drift}/E$ is the mobility — the drift velocity per unit applied field. This is the microscopic statement of Ohm’s law.

E100 V/m

v_drift = μE7.00 μm/s

σ = nqμ1.12 S/m

J = σE112.1 A/m²

Field E = 100 V/m Mobility μ = 7.00 × 10⁻⁸ m²/(V·s) Density n = 1.0 × 10²⁶ m⁻³

A charged species in a fluid drifts at terminal velocity v_drift = μE, where μ is the mobility (drift velocity per unit field). The current density is J = ρ_q v_drift = nqμE = σE, with σ = nqμ the *conductivity*. This is the microscopic form of Ohm's law. For K+ ions in physiological saline, μ ~ 7×10⁻⁸ m²/(V·s); typical fields in cell membranes (10⁷ V/m) give drift speeds of metres per second — consistent with ion channels' ms-scale gating times.

For ions in solution, the mobility comes from the same Stokes-Einstein machinery as the diffusion coefficient (viscosity & diffusion chapter): $\mu = q/\gamma_\text{drag}$ with $\gamma_\text{drag} = 6\pi\mu_\text{viscous} a$ . For a K⁺ ion of effective radius 0.14 nm in water, $\mu \sim 7\times 10^{-8}\,\text{m}^2/(\text{V·s})$ .

From the electric force to the electrochemical potential

A charged species in solution has not just the chemical potential $\mu_\text{chem}(c) = \mu_0 + k_B T \ln c$ from concentration but also an electrical contribution $zeV$ from the local potential. The total electrochemical potential is

\tilde\mu(\mathbf{r}) \;=\; \mu_0 \;+\; k_B T \ln c(\mathbf{r}) \;+\; z e\, V(\mathbf{r}).

At equilibrium, $\tilde\mu$ must be uniform across any compartment that allows the species to flow. For a membrane that selectively passes one ion species, this gives the Nernst potential:

▶ Nernst equation from electrochemical equilibrium

For an ion species $i$ at equilibrium across a permeable membrane, $\tilde\mu^\text{in} = \tilde\mu^\text{out}$ :

k_B T \ln c_i^\text{in} + z e V^\text{in} \;=\; k_B T \ln c_i^\text{out} + z e V^\text{out}.

Solving for $V_\text{Nernst} = V^\text{in} - V^\text{out}$ ,

V_\text{Nernst} \;=\; \frac{k_B T}{z e}\, \ln\!\frac{c_i^\text{out}}{c_i^\text{in}} \;=\; \frac{R T}{z F}\, \ln\!\frac{c_i^\text{out}}{c_i^\text{in}}.

At body temperature ( $T = 310\,\text{K}$ ), $RT/F = 26.7\,\text{mV}$ ; multiplying by $\ln 10$ gives the rule of thumb: a tenfold concentration ratio = ±61 mV per unit charge.

The Nernst potential is the equilibrium voltage at which electrical and chemical driving forces balance. If the actual membrane potential differs from $V_\text{Nernst}$ , the ion flows.

E_K (Nernst)-89.0 mV

E_Na60.6 mV

E_Cl-73.6 mV

V_rest (GHK)-69.8 mV

K⁺ [in] 140 mM [out] 5 mM P 1.00

Na⁺ [in] 15 mM [out] 145 mM P 0.04

Cl⁻ [in] 7 mM [out] 110 mM P 0.45

Presets:

Each ion's Nernst potential is the voltage at which its electrical and chemical gradients balance. The GHK resting potential is the weighted average — heavily weighted toward whichever species the membrane is most permeable to. In a resting neuron, P_K ≫ P_Na, so V_rest sits near E_K ≈ −90 mV. At the hair-cell apex bathed in endolymph (K⁺ ≈ 150 mM both sides), E_K ≈ 0 — and the +80 mV endocochlear potential becomes the driver of MET-channel current.

The Goldman–Hodgkin–Katz equation

A real membrane is permeable to multiple ion species. The steady-state resting potential is the voltage at which the total current vanishes:

V_\text{rest} \;=\; \frac{RT}{F}\, \ln\!\frac{P_K c_K^\text{out} + P_\text{Na} c_\text{Na}^\text{out} + P_\text{Cl} c_\text{Cl}^\text{in}}{P_K c_K^\text{in} + P_\text{Na} c_\text{Na}^\text{in} + P_\text{Cl} c_\text{Cl}^\text{out}}.

This is the Goldman–Hodgkin–Katz equation. The resting potential sits between the individual Nernst potentials, weighted by permeability.

P_K = 1.00 P_Na = 0.04 P_Cl = 0.45

The resting potential sits *between* the individual Nernst potentials, pulled toward the most-permeable ion. At a typical neuron's rest (P_K ≫ P_Na, P_Cl), V_rest is near E_K ≈ −90 mV. During an action potential, P_Na shoots up by ~100× and V_rest swings toward E_Na ≈ +60 mV — the depolarisation phase. The membrane is a *voltage-dependent permeability filter*, and the GHK equation is its statement.

At rest, $P_K \gg P_\text{Na}, P_\text{Cl}$ , so $V_\text{rest}$ is near $E_K \approx -90\,\text{mV}$ . During an action potential, $P_\text{Na}$ shoots up by ~100×, and $V_\text{rest}$ swings toward $E_\text{Na} \approx +60\,\text{mV}$ — the depolarisation. The membrane is a voltage-dependent permeability filter.

The endocochlear potential

The cochlear duct system has a uniquely complex electrochemistry. The scala media (containing endolymph) sits at +80 mV relative to surrounding perilymph — the endocochlear potential, maintained by active K⁺ pumping in the stria vascularis. The hair cells line the boundary between endolymph and perilymph; their apical (top) membranes face endolymph, their basolateral (side) membranes face perilymph.

The endocochlear potential is the unique feature of cochlear physiology. The stria vascularis actively pumps K+ from perilymph back to endolymph, maintaining the +80 mV endolymph-side voltage. Combined with the −60 mV resting voltage of the hair-cell soma, the *electrochemical* driving force across an open MET channel is +140 mV — a battery powerful enough to drive ~14 pA of K+ current per channel. The energy comes from ATP burned in the stria, not from the acoustic signal itself.

The driving voltage across an open apical MET channel is

\Delta V_\text{MET} \;=\; V_\text{endolymph} - V_\text{cytoplasm} \;=\; +80 - (-60) \;=\; +140\,\text{mV}.

The K⁺ Nernst potential across the apical membrane is near zero (both sides have $\sim 150\,\text{mM}$ K⁺), so this 140 mV is entirely electrical — the energy comes from ATP burned in the stria vascularis, not from the acoustic stimulus. A 100 pS MET channel passing 140 mV carries 14 pA of current — exactly the measured value.

Mechanically gated channels

The apical membrane of a hair cell carries ~50 MET channels per stereocilium. Each is gated by tip-link tension — a fine fibre attaching one stereocilium to the next-shorter one. Deflection $x$ of the hair bundle stretches the tip link with force $K_\text{TL} x$ , and the gating energy is

\Delta G(x) \;=\; \Delta G_0 \;-\; K_\text{TL}\, d\, x,

with $d$ the gate-swing of the channel. The open probability is the Fermi function from the free-energy chapter:

P_\text{open}(x) \;=\; \frac{1}{1 + e^{(\Delta G_0 - K_\text{TL}\,d\,x)/k_B T}}.

deflection x50 nm

P_open18.2%

x_½ (50% open)125 nm

Deflection x = 50 nm Zero-bias offset ΔG₀ = 2.5 k_BT K_TL · d (slope) = 1.00

Each stereocilium has a tip link — a fine fibre connecting it to the next-shorter stereocilium. Deflection of the hair bundle stretches the tip link, applying a gating force F = K_TL · x to the MET channel. The two-state Boltzmann probability of "open" is a Fermi function in F (or in x, with linear coupling). For a typical hair cell, P_open is 5-20% at rest and saturates near 1 at 100-nm deflection — matching the actual operating range during loud sounds. See [Hearing Ch 4.6](/hearing/cochlea/hair-cells).

The width of the sigmoid (in $x$ ) is $k_BT/(K_\text{TL}\,d) \approx 100\,\text{nm}$ — exactly the operating range over which hair-cell responses go from near-zero to saturation during loud sounds.

Piezoelectricity and prestin — the cochlear amplifier

A piezoelectric material couples mechanical strain to electric polarisation linearly:

\sigma_\text{stress} \;=\; c\,\varepsilon_\text{strain} \;-\; e\, E, \qquad D \;=\; e\,\varepsilon_\text{strain} \;+\; \epsilon\, E,

with cross-term $e$ tying strain to charge. Applying voltage changes length; applying strain changes charge. Quartz and certain ceramics are passive piezoelectrics.

The outer hair cells (OHCs) of the mammalian cochlea contain a membrane protein, prestin, that performs the same trick electrochemically: each prestin molecule changes conformation in response to membrane voltage, and there are $\sim 10^7$ copies per cell. The cell shortens and elongates at up to 80 kHz.

Membrane potential V_m = -60 mV

Prestin is the membrane protein of outer hair cells responsible for *electromotility* — voltage-driven length change. Each cell carries ~10⁷ prestin molecules, each undergoing a conformational change when the membrane voltage shifts. The collective response is a sigmoidal length-vs-voltage curve, with the cell shortening at depolarised V_m and lengthening at hyperpolarised V_m. Total length change is up to ~4% of the cell length, at frequencies up to 80 kHz. This is the active *cochlear amplifier*: prestin pumps energy into the basilar-membrane motion, undoing viscous damping. See [Hearing Ch 4.5](/hearing/cochlea/amplifier).

OHCs amplify the basilar-membrane motion by pumping energy back into the wave on each cycle — undoing the viscous damping that would otherwise dissipate the signal. This is the cochlear amplifier (Hearing Ch 4.5) and it is the closest thing biology has built to a fast piezoelectric actuator.

⏳ The history — From Nernst to Hodgkin-Huxley to prestin

Walther Nernst derived the equilibrium-membrane-potential formula in 1888 from thermodynamic equilibrium arguments — long before the molecular details of cell membranes were understood. The application to nerves was supplied by Julius Bernstein in 1902, who proposed (correctly) that resting nerve potentials originate from the K⁺ gradient maintained by the cell.

The GHK extension — multiple ions, finite permeabilities, constant-field assumption — was derived independently by David Goldman in 1943 and Alan Hodgkin and Bernard Katz in 1949. It is the workhorse formula for predicting resting potentials from measured concentrations.

Hodgkin and Andrew Huxley then spent the 1950s constructing the dynamical extension — voltage-gated channels with kinetics, the action-potential mechanism — using the squid giant axon. The Hodgkin–Huxley model won the 1963 Nobel Prize and remains the canonical mathematical framework for excitable membranes.

In the cochlea the story is more recent. The endocochlear potential was measured by Hallowell Davis and collaborators in 1958. The mechanism — pumping by the stria vascularis — was worked out over the following decades by Pierre Wangemann, Jochen Schacht, and others. Prestin, the electromotile motor of outer hair cells, was identified molecularly by Peter Dallos’s group in 2000.

For the cross-book applications — endocochlear power supply, MET-channel current, prestin amplifier, voltage-gated Ca²⁺ at ribbon synapse — see the key examples sub-page.