Introduction to Capacitors

Introduction

A capacitor has two metal plates separated by an insulator. Charging removes electrons from one plate and adds them to the other. The plates acquire equal and opposite net charges; electrons travel through the external wires, never across the insulating gap in this ideal model.[1]

Capacitance $C$, measured in farads (F), relates the upper plate’s positive charge $Q$, in coulombs (C), to the voltage difference $V_C$, in volts (V):

A battery at voltage $V_b$ charges the capacitor through resistance $R$, measured in ohms (Ω). Conventional current $I$, in amperes (A), changes the voltage over time $t$, in seconds (s):[2]

Charging slows as $V_C$ approaches $V_b$. Hold disconnects the external path; Discharge joins the plates through the resistor. Electron drift runs opposite to conventional current. The time constant $\tau$, in seconds, sets the charging and discharging timescale:

Choose a mode below, or adjust capacitance, resistance and battery voltage. Blue dots represent electrons. The dots exaggerate size and motion and represent groups of electrons. Changing a slider starts a fresh charge; switching modes preserves voltage. Restart returns to an uncharged capacitor. Pause freezes time. The model omits leakage and dielectric breakdown.

A biophysical neuron

A neuron’s membrane separates conducting fluids and acts as a capacitor. Here the moving charges are ions: sodium ($\mathrm{Na}^{+}$) entering the cell can raise its voltage, while potassium ($\mathrm{K}^{+}$) leaving can lower it. Voltage-dependent channels control these flows. The animation uses the classic Hodgkin–Huxley squid-axon membrane model.[3] To receive presynaptic spikes, we add an excitatory synapse; this addition supplies the input current $I_{\mathrm{syn}}$ in the voltage equation below.

The membrane voltage $V_m$ is the inside voltage relative to outside, in millivolts (mV). Membrane capacitance per area $C_m$ is in μF/cm²; inward synaptic current density $I_{\mathrm{syn}}$ and ionic current densities $I_{\mathrm{Na}}$, $I_{\mathrm{K}}$ and leak $I_L$ are in μA/cm². Time $t$ is in milliseconds (ms):

For each pathway $p\in\{\mathrm{Na},\mathrm{K},L\}$, conductance density $g_p$ is in mS/cm² and reversal potential $E_p$ is in mV:

These ionic currents are positive outwards; the current plot reverses their signs so upward means positive charge entering. The reversal potentials are $E_{\mathrm{Na}}=50$ mV, $E_{\mathrm{K}}=-77$ mV and $E_L=-54.3$ mV. Sodium activation $m$, sodium availability $h$ and potassium activation $n$ are dimensionless gates between zero and one:

The available maximum conductance densities are $\bar g_{\mathrm{Na}}=120f_{\mathrm{Na}}$ and $\bar g_{\mathrm{K}}=36f_{\mathrm{K}}$ mS/cm². The channel sliders set the fractions $f_{\mathrm{Na}}$ and $f_{\mathrm{K}}$ from zero to one, both initially one. Leak conductance is fixed at $g_L=0.3$ mS/cm². Each gate $x\in\{m,h,n\}$ evolves as:

The rates $\alpha_x$ and $\beta_x$, in ms⁻¹, drive the gate towards one and zero, respectively. With $u=V_m/(1\,\mathrm{mV})$ the dimensionless numerical voltage, the animation uses the classical squid-axon rates at 6.3 °C in the inside-minus-outside voltage convention:[3, 4]

Here $e$ is the base of the natural exponential. The apparent divisions by zero have finite limits: $\alpha_m=1$ ms⁻¹ at −40 mV and $\alpha_n=0.1$ ms⁻¹ at −55 mV. Each fresh demonstration starts at −65 mV with $x=\alpha_x/(\alpha_x+\beta_x)$, then runs for 200 ms without synaptic input before the displayed timeline begins. Membrane capacitance initially equals $C_m=1$ μF/cm².

In a biological synapse, an arriving presynaptic spike causes transmitter release, opening synaptic channels. We add a standard conductance-based synapse with instantaneous opening and exponential decay, as described in Neuronal Dynamics, Section 3.1, Equations (3.1)–(3.2). This is an extension to the original Hodgkin–Huxley membrane model.[5] Its conductance density $g_{\mathrm{syn}}$ is in mS/cm², and we write its current as inward-positive (the opposite sign convention to that reference):

The synaptic reversal potential is $E_{\mathrm{syn}}=0$ mV. Each arrival adds the selected weight $w$, in mS/cm², to this conductance; between arrivals it decays:

We choose a decay time $\tau_{\mathrm{syn}}=2$ ms and default weight $w=0.2$ mS/cm² for this demonstration. The Dirac impulse $\delta$ has unit area and units of ms⁻¹, so each arrival increases conductance by $w$; this is equivalent to the exponential conductance in the cited model. Arrival times are $t_j=(5+40j)$ ms for $j=0,1,2,\ldots$, following a schematic 4 ms journey from a source with prescribed spikes. These timings are also demonstration choices; transmitter release and the presynaptic neuron’s dynamics are not simulated.

Voltage and the three gates are integrated with fourth-order Runge–Kutta steps of 0.01 ms. Synaptic conductance decays exponentially, evaluated at each step’s start, midpoint and end. Action potentials emerge from the channel dynamics, without a firing threshold or voltage reset. Adjust capacitance, synaptic strength or the available channels to explore the response.