Can the net motion of electrons in an AC circuit be considered an example of simple harmonic oscillation. Furthermore, how can the general formulae of SHM be adapted to suit a scenario of an AC current. More specifically how can formulae such as $a=-\frac{k \Delta x}{m}$ or $x=Acos(\omega t + \delta)$ be expressed in parameters more relevant to electricity (e.g. Voltage or net electron displacement).
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$\begingroup$ Using lumped circuit parameters (R, L, C) and their voltage and current formulas (easily found), a second-order or first-order (depending on charges or currents) differential equation whose solution mimics the differential equation common to SHO problems. $\endgroup$– K7PEHCommented Mar 14, 2016 at 15:29
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1$\begingroup$ Unsure about what the point of the question is. Yes, as K7PEH mentioned, of course one can model ac current flow as being due to an LCR circuit. But what would be the point? Pretty certain that the standard ac power frequency of 60 Hz doesn't relate to any sort of natural resonant frequency in ac power generation/transmission systems, so the L and C values of a model circuit wouldn't correspond to anything having physical meaning. $\endgroup$– user93237Commented Mar 14, 2016 at 17:52
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$\begingroup$ Imagine a weight bobbing up and down as it hangs from a spring. Now imagine ropes linking the weight to a dial with a pointer. As the weight bobs up and down, the pointer swings back and forth. Is the motion of the rope an oscillator? No! The oscillator is the weight/spring system. The motion of the rope merely communicates the state of the oscillator to the pointer. Likewise, there are electronic circuits called "oscillators", but the AC current that flows is not the oscillator: The oscillator is the assembly of components. $\endgroup$– Solomon SlowCommented Mar 14, 2016 at 19:05
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The actual motion of electrons in a conductor is not "simple" harmonic motion because of their interactions (scatter) with the material. But macroscopically (looking at current/voltage, rather than individual electrons), it is OK.
In that case, you can consider current = "velocity", driving force = voltage, and displacement = total amount of charge displaced.
For example if you have a simple parallel LC circuit, the charge $Q$ across the capacitor can be thought of as the displacement; the voltage $V$ is the "driving force" ($V=\frac{Q}{C}$), and the current $I=\frac{dQ}{dt}$ the "velocity".
We also know that a change in current in the inductor leads to a back emf: $V=-L\frac{dI}{dt} = -L\ddot Q$ - this is a bit like inertia (resisting a change in "velocity").
Putting $Q=A\cos\omega t$, $\dot Q = -\omega \sin\omega t$ and $\ddot Q = -\omega^2 Q$
If we now consider the sum of "driving voltage" and "inertial voltage" equal to zero, we find
$$-L\ddot Q + \frac{Q}{C}\\ LC\omega^2 Q = Q\\ \omega = \frac{1}{\sqrt{LC}}$$
This is of course all familiar SHM stuff...
