# Gravity vs Electricity see-saw analogy and speed of light

Does electricity exhibit anything like the following steps:

1) A ball is rolled down from left side of a very long teeter-totter or see-saw

2) The see-saw rapidly (immediately) tilts once the ball is passed the half-way point so that the right side is higher

3) The ball will now continue moving to the right because of its accumulated kinetic energy or inertia, but also decelerates

4) The ball stops momentarily

5) The ball will change direction and roll back to the left side from where it was originally released

In Electricity, the analogy I know of is a square wave oscillating from +5 to -5 Volts across a resistor. I have a question about step 3

1) Electricity moves clockwise in a circuit driven by the +5 volt of a square wave

2) The square wave oscillates to -5 V

3) ??? Does the movement of the electron possess a mass inertia that keeps it moving in the original clockwise direction for a short period of time???

*** I am not an expert in this, but I suspect that I am not asking about the intrinsic material inductance or capacitance of the circuit. I understand that parasitic inductance can contribute to prolonging the original motion of the electrical current. BUT, I have a suspicion that mass inertia is a separate factor because

a) I've read derivations of the magnetic field by considering the strength of an electric field in different reference frames, and the fact that the invariance of the speed of light, as in Purcell's textbook

b) My intuition is that if inductance and/or capacitance were the ONLY reason for the continued transient motion of the electron after the voltage switches, then by analogy, one should be able to calculate the speed of light from the time that it takes the ball to stop moving after the see-saw has flipped. I don't believe that the speed of light factors factors into this, as it seems to be a purely Newtonian concept.

Thank you

??? Does the movement of the electron possess a mass inertia that keeps it moving in the original clockwise direction for a short period of time???

Only for an unimaginably short period of time. The typical drift velocity for electrons in a conductor is 10's of microns per second. Even the smallest electric field will stop them instantly:

$qE = F = ma$

$a = (q/m) E$

$q/m$ for an electron is $1.8×10^{11}$ C/kg so an electron in a 1V/m field will stop in

$\Delta t = v / a = 20 × 10^{-6} / 1.8×10^{11} \sim 10^{-16}$ seconds.

That's a very short time compared to any practical square wave.

The short answer is yes and the role of the "inertia" as you call, is played exactly by the electrical charge of the electron, $e$. If you study your situation at the level of single electrons instead of currents, then each of them is subject to a energy-potential, $U$ of, however it's kinetic energy is still includes its mass, (so there is actually a role played by $m$ too, however one expects $e$ to be more relevant due to EM strengt) $U=-eV$ so energy will oscillate between kinetic and potential as in the see-saw example. Assuming no loses, then conservation of energy gives you a simple equation that resembles a harmonic oscillator but with an straightened potential which is time-dependent rather than position. $$0= \frac{1}{2}mv^2 -e V(t)$$ where looks like (extending it periodically) $$V(t) = \left\{ \begin{array}{ll} +5 &|t| > 0\\ 0 & |t| < 0 \end{array}\right.$$

• Thanks, may I ask a followup about how this manifests in the behavior of electrical circuits? Ohm's Law seems to dictate that currents always moves towards lower potential, even in the treatment that I've seen for AC circuits. I am aware that material-based inductance can introduce a phase-shift. I'm curious if the analysis of the phase shift requires accounting for the mass of the electron if the frequency of voltage oscillation is high enough? In other words, are there formulas for the phase shift of V and I that explicitly account for the mass of the electron? – lamplamp May 14 '18 at 15:29
• I may not be able to answer in detail about circuits since it is not my area of expertise, however I can say it is a matter of scales. So If you are focused on currents, meaning at the scale of circuits, then Ohm law is the correct thing to use and any kind of oscillation of this $e$-inertia kind would probably occur too quick to be influential. Ohm's law is a collective behaviour, you might want to check the Drude model – ohneVal May 14 '18 at 15:34