Capacitor Fundamentals, Behavior, and Energy Storage

Fundamental Capacitor Equation and Current

Starting with the fundamental relationship for a capacitor, which states that charge $(Q)$ is proportional to voltage $(V)$ and capacitance $(C)$: $(Q = C V)$. If we assume the capacitance (C) is static, meaning the plate distance $(d)$ and plate area $(A)$ are fixed, then $(C = \frac{\epsilon A}{d})$ remains constant. In this scenario, the only variable changing is the voltage $(V)$.

By definition, current $(I)$ is the time rate of change of charge $(Q)$ ($I = \frac{dQ}{dt}$). Substituting the capacitance relationship into this definition yields the fundamental equation for current in a static capacitor: $(I = C \frac{dV}{dt})$. This equation highlights that current flows when the voltage across the capacitor changes.

Squishy Capacitors: A Generalized Model

The concept of a "squishy capacitor" (one where the plate distance $(d)$ is not fixed) extends the fundamental equation. Imagine a dielectric like a sponge instead of rigid Teflon. If we apply a fixed voltage (e.g., $5$ volts) to this capacitor and then squish it, its capacitance becomes variable (a function of time, $C(t)$) because $(d)$ is changing. This concept is remarkably similar to how neurons work, functioning as 'squishy' capacitors whose membranes can be deformed by external stimuli.

To re-derive the current equation for a squishy capacitor with a fixed voltage, we start again with $(Q = C(t) V)$. Since $V$ is fixed and $C$ is a function of time, when we take the derivative for current $(I = \frac{dQ}{dt})$, we use the product rule:

$(I = \frac{d}{dt}(C(t)V))$
$(I = V \frac{dC(t)}{dt})$

This equation shows that current is generated even with a fixed voltage if the capacitance itself changes over time. In real-world applications, such as neurons in the ear, acoustic waveforms can squish cell membranes, changing their capacitance and generating a detectable charge, thus exciting the neuron. However, in this course, capacitors are generally considered