# Why doesn't a capacitor oscillate like a pendulum around equilibrium when discharging

Why doesn't a capacitor behave like a swinging pendulum when discharging? Please note that I'm not asking about solving the differential equation, but rather about how we are constructing it. I have a physical intuition about how things move towards equilibrium, like pendulums or a vibrating membrane (which I've seen as a model of a capacitor), and the solution for the capacitor is different.

When a capacitor discharges in a RC circuit, it reaches a voltage difference of 0 asymptotically. I understand that this is predicted from the differential equation:

0 + Q/C + dQ/dt = 0

dQ/dt = -Q/RC

I understand the approach of writing the equation by accounting for voltage drops around a loop. I understand that the solution to this equation is an exponential, and how to find the solution.

What I don't fully understand is why the voltage drop across the resistor is modeled using Ohm's Law as -RI or R dQ/dt. I know that Ohm's law is used to relate the voltage drop and current across the resistor, and I am comfortable with its use in situations where current is not changing. But here the current is changing, and I'm not certain that that the statistical justifications of Ohm's law that I've seen acknowledge the possibility of time-varying voltage or current.

What I'm wondering about is whether the moving charges have inertia that allow them to partially charge the capacitor in the reverse polarization. I asked a related question some time ago about AC circuits, but I feel that this scenario is actually simpler to describe. To that other question, someone referenced parasitic inductance as affecting the oscillation. I want to ask about this system of an RC capacitor to get at the issue of whether there are regimes (high voltage or current?) where the capacitor ever overshoots past 0 volts to reverse polarization.

The two possible scenarios that I see are: a) it never reverse polarization and does truly reach 0 asymptotically b) there is an acquired kinetic energy or inertia or parasitic inductance or some other reason why the capacitor overshoots and it oscillates asymptotically to 0 volts

Thank you

• Why do you want to know all this witbout solving a differential equation. This appears contradictory. – my2cts Mar 3 at 1:01
• Just to be sure, are you asking why the equations that come out of ideal circuit theory work well for physical circuits? – Alfred Centauri Mar 3 at 1:02
• The reason I ask is that, for example, the ideal RC circuit equations you allude to do not fully capture the behavior of a physical RC circuit but, nonetheless, are a very good approximation in the appropriate limit. – Alfred Centauri Mar 3 at 1:05

## 2 Answers

Modelling a circuit as a network of discrete components like capacitors and resistors, connected by perfect conductors, is valid so long as the rate of change of current and voltage in the circuit is slow compared with the speed at which changes can propagate around the circuit.

The propagation speed is usually a significant fraction (e.g. 0.1 to 0.5) of the speed of electromagnetic radiation in a vacuum.

For "small" electronic circuits (with physical dimensions of the order of say 100mm or less) the finite propagation speed starts to become important at frequencies of the order of MHz and is critically important at GHz frequencies (for example in computer circuits and cellphones).

In other situations the critical frequency can be much lower. For example for a power transmission line that is of the order of 1000km long, you can't ignore the finite propagation speed even at 50 or 60 Hz.

The analogy between electrical and mechanical dynamic systems is that capacitors correspond to springs, resistors correspond to mechanical dampers, and inductors correspond to masses (but note that the equivalent "spring stiffness" is inversely proportional the capacitance). Therefore, a circuit containing only capacitors and resistors can not self-oscillate, unless the propagation speed effects are significant.

At very high frequencies, any conductor has a significant inductance, which depends mostly on the its physical shape. Therefore, a real capacitor can have an oscillating response. Data sheets for capacitors will specify the "self resonant frequency" which is the oscillation frequency if the component is not connected to anything else. For small value capacitors, this is typically a few hundred MHz. For large capacitors, it may be a few orders of magnitude lower.

In practical circuits, you sometimes see several capacitors of different size connected in parallel (for example 100uF, 1uF, and 0.01uF). The total capacitance (101.01uF) is not significantly different from just the 100uF cap on its own, but the self-resonant frequency is higher because it depends on the smallest capacitor, not the biggest.

That depends on other elements of the circuit. Any "inertia" of the current is modeled as inductance. If you have an LC circuit (inductor plus capacitor, but no significant resistance) then the circuit will indeed oscillate or "ring".