Simple harmonic motion (SHM) is periodic oscillation in which the restoring "force" is proportional to displacement. In AP Physics C: E&M (Topic 13.6), the charge on a capacitor in an LC circuit follows SHM, satisfying d²q/dt² = -(1/LC)q with angular frequency ω = 1/√(LC).
Simple harmonic motion is any oscillation where the thing pulling the system back toward equilibrium gets stronger in direct proportion to how far the system is displaced. Mathematically, that means the system obeys a differential equation of the form d²x/dt² = -ω²x, and the solution is always a sine or cosine.
In AP Physics C: E&M, SHM shows up in an unexpected place. Connect a charged capacitor to an inductor and the charge q on the capacitor obeys L(d²q/dt²) + q/C = 0, which rearranges to d²q/dt² = -(1/LC)q. That is exactly the SHM equation with charge playing the role of position. So q(t) = Q₀cos(ωt), the current oscillates as the derivative of charge, and energy sloshes back and forth between the capacitor's electric field and the inductor's magnetic field. Nothing is physically vibrating, but the math is identical to a mass on a spring.
SHM is the payoff of Topic 13.6 (Circuits with Capacitors and Inductors). The whole point of the LC circuit topic is recognizing that a circuit equation can have the same structure as a mechanics equation, and once you spot d²q/dt² = -(1/LC)q, you already know the answer from Mechanics. You can immediately write q(t) as a cosine, read off ω = 1/√(LC), and describe the energy exchange without solving anything new. This is also one of the cleanest examples of a skill Physics C rewards heavily, which is mapping a new physical situation onto a differential equation you have already solved.
Keep studying AP® Physics C: E&M Unit 13
LC Circuit (Unit 13)
The LC circuit is the system; SHM is its behavior. An inductor and capacitor trade energy back and forth, and the resulting charge oscillation q(t) = Q₀cos(ωt) is SHM in disguise. If an exam question gives you an LC circuit, your first move is writing Kirchhoff's loop rule and recognizing the SHM equation that falls out.
Angular Frequency (Unit 13)
Every SHM system has one number that sets how fast it oscillates. For an LC circuit, comparing d²q/dt² = -(1/LC)q to the standard form d²x/dt² = -ω²x tells you ω² = 1/LC, so ω = 1/√(LC). Bigger inductance or capacitance means slower oscillation, the same way a heavier mass or softer spring slows a mechanical oscillator.
Mass-Spring Oscillator (Physics C: Mechanics)
The LC circuit is a mass-spring system with the labels swapped. Inductance L plays the role of mass (it resists changes in current the way mass resists changes in velocity), 1/C plays the role of the spring constant, and charge plays the role of position. Compare ω = √(k/m) with ω = 1/√(LC) and you can see the analogy directly in the formulas.
Energy Conservation in Oscillators (Unit 13 / Mechanics)
In a mass-spring system, energy trades between kinetic and potential. In an ideal LC circuit, total energy is constant but trades between the capacitor (½q²/C, like spring PE) and the inductor (½LI², like kinetic energy). When the capacitor is fully charged, current is zero, and when current peaks, the capacitor is empty.
Multiple-choice questions on SHM in E&M almost always test one of three moves. First, deriving or identifying the correct differential equation for charge in an LC circuit, L(d²q/dt²) + q/C = 0. Second, reading the angular frequency off that equation, ω = 1/√(LC). Third, interpreting what the equation d²q/dt² = -(1/LC)q physically describes, which is oscillating charge, not exponential decay. On a free-response question, you may be asked to apply Kirchhoff's loop rule to an LC circuit, show that the result has SHM form, write q(t) with correct amplitude and phase from initial conditions, and describe the energy exchange between the capacitor and inductor. Showing the comparison to d²x/dt² = -ω²x explicitly is how you earn the justification points.
RC and RL circuits produce one-way exponential behavior (e^(-t/τ)) because a resistor dissipates energy, so the charge or current just dies off toward a final value. An ideal LC circuit has no resistance, so energy is conserved and the charge oscillates sinusoidally forever. Quick check: a first-order differential equation (one derivative) gives exponential decay, while a second-order equation with a minus sign, like d²q/dt² = -(1/LC)q, gives SHM. Add a resistor to an LC circuit and you get damped oscillation, which is no longer simple harmonic motion.
Simple harmonic motion happens whenever the restoring influence is proportional to displacement, which always produces the equation d²x/dt² = -ω²x with sinusoidal solutions.
In an LC circuit, Kirchhoff's loop rule gives L(d²q/dt²) + q/C = 0, so the capacitor's charge undergoes SHM exactly like a mass on a spring.
The angular frequency of an LC circuit is ω = 1/√(LC), which you read off by matching the circuit equation to the standard SHM form.
The analogy maps inductance to mass, 1/C to the spring constant, and charge to position, so every SHM result from Mechanics carries over to LC circuits.
Energy in an ideal LC circuit is conserved and oscillates between the capacitor's electric field (½q²/C) and the inductor's magnetic field (½LI²).
SHM means sinusoidal oscillation, not exponential decay; decay only appears when resistance is in the circuit.
It's periodic oscillation where the restoring force is proportional to displacement. In E&M it appears in Topic 13.6, where the charge on a capacitor in an LC circuit satisfies d²q/dt² = -(1/LC)q and oscillates as q(t) = Q₀cos(ωt).
No, not in an ideal LC circuit. With no resistance there is nothing to dissipate energy, so the charge oscillates sinusoidally forever instead of decaying. Exponential decay belongs to RC and RL circuits, which have a resistor.
Only the physical quantities change; the math is identical. Inductance L replaces mass, 1/C replaces the spring constant k, and charge q replaces position x, which turns ω = √(k/m) into ω = 1/√(LC).
ω = 1/√(LC). You get it by comparing the circuit's differential equation, d²q/dt² = -(1/LC)q, to the standard SHM form d²x/dt² = -ω²x and matching ω² = 1/LC.
Yes, through LC circuits in Topic 13.6. Expect questions asking you to identify the differential equation L(d²q/dt²) + q/C = 0, extract ω = 1/√(LC), and recognize that the equation describes oscillating charge rather than exponential decay.
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