LC circuit — oscillations (electrical analog of SHM): Foundations

Before you can enjoy that idea, you need to genuinely see every letter the parent note throws at you. Below, each symbol is built from nothing: plain words → a picture → why the topic can't live without it. They are ordered so each one leans on the ones above it — so a symbol is never used before its own section.

1. Charge $q$ and initial charge $q_{0}$ — the "amount of stuff" that moves

Picture two flat metal plates facing each other. One plate has a shortage of electrons — because electrons are negative, missing electrons leaves that plate net positive, so we call it the $+ q$ plate. The other plate has the matching surplus of electrons, making it net negative, the $- q$ plate. The number $q$ tells you how lopsided that pile is.

Figure — LC circuit — oscillations (electrical analog of SHM)

Read the figure: the lavender plate on the left is short of electrons, so it is net positive — marked with $+$ symbols, the $+ q$ plate. The coral plate on the right holds the surplus of electrons, so it is net negative — marked $-$ , the $- q$ plate. (Surplus electrons → negative plate; this matches the narrative exactly.) The mint arrows between them are the electric field — they always point from the $+ q$ plate toward the $- q$ plate, and there are more of them when $q$ is bigger.

Why the topic needs it: $q$ is the star of the whole story. In the parent note, $q$ plays the role of position $x$ in a spring, and $q_{0}$ is the "starting stretch" — how far you pulled the mass before letting go. Just as $x$ tells you how far the mass is from center, $q$ tells you how "pulled" the charge is from its balanced (empty) state.

2. Time $t$ and the idea of a function $q (t)$

Now the phrase "at the start" from §1 has a symbol: the starting instant is $t = 0$ , and at that instant the capacitor holds the initial charge $q_{0}$ , i.e. $q (0) = q_{0}$ . Half a millisecond later ( $t = 0.0005 s$ ) the plate holds a different amount, and $q (t)$ is the rule that hands you the right number for any $t$ you pick.

Why the topic needs it: oscillation is change over time. Without the idea of a quantity that depends on $t$ , "sloshing back and forth" has no meaning. (The exact recipe for the movie is $q (t) = q_{0} cos (ω t)$ , but every symbol in it — including $ω$ — is only used once it has been built in §12; we are not using it yet.)

3. Current $i$ and the two starting conditions

Back to the dam: charge $q$ is the water stored; current $i$ is the flow rate through the pipe. A big lake draining slowly can have small current; a small puddle rushing out can have big current. Stored amount and flow rate are different things — this distinction is the whole secret of the 90° phase later.

Why the topic needs it: $i$ is the analog of velocity $v$ . Same relationship: velocity is how fast position changes; current is how fast charge changes. And just as velocity has a sign (left/right), current has a sign (which way it circulates).

4. The derivative $\frac{d q}{d t}$ — "rate of change" written in symbols

The topic writes $i = \frac{d q}{d t}$ . That fraction-looking thing is called a derivative, and here is what it means before it means anything scary.

WHY this tool and not just "change over time"? Because ordinary "change ÷ time" ( $Δ q /Δ t$ ) only gives an average over some stretch. We want the value at a single instant — the exact steepness at one point on the curve. The derivative is the tool built precisely to answer "how fast, right at this instant?". No other tool answers that.

Look at the figure: the straight red line just kissing the curve is the slope at that instant. Its steepness is the current at that moment.

Why the topic needs it: the sentence "current is the rate of change of charge" only becomes an equation through the derivative. Everything downstream ( $i (t)$ , the 90° phase, the energy) is built on it.

5. The second derivative $\frac{d ^{2} q}{d t ^{2}}$ — "the rate of change of the rate of change"

Picture a car: position is where you are, first derivative is your speed, second derivative is your acceleration (how hard you're pressing the pedal). For charge: $q$ is the pile, $\overset{q}{˙}$ is the flow, $\overset{q}{¨}$ is how quickly the flow is speeding up or slowing.

Why the topic needs it: the master equation $\overset{q}{¨} = - \frac{1}{L C} q$ is a statement about the second derivative. It says "the acceleration of the charge points back toward zero" — which is the mathematical fingerprint of every oscillation. Because it is a second-order derivative equation, it needs the two starting facts from §3. See Simple Harmonic Motion for the same fingerprint in a spring.

6. Voltage $V$ — the "push" that drives charge

Before we touch $C$ or $L$ we must name the quantity both of them produce: voltage.

Picture walking along the loop in the chosen (clockwise) direction. As you step across a component from its $-$ mark to its $+$ mark, you rise by $V$ ; the other way you drop by $V$ . Fixing these little $+ / -$ arrows once is what lets us add voltages consistently in §11.

Why the topic needs it: every component in the loop speaks the language of voltage. To add them up around the loop (Kirchhoff) we first need $V$ defined with a sign, not just a magnitude.

7. Capacitance $C$ — how easily a capacitor stores charge

The defining relationship (built in full in Capacitance and Energy in Capacitors) is:

Why the topic needs it: $V_{C}$ is the "restoring push." Just as a stretched spring pushes back harder the more you stretch it, a charged capacitor pushes charge back harder the more charge sits on it. That's why $1/ C$ plays the role of the spring constant $k$ : a small $C$ (stiff, cramped container) gives a big $1/ C$ (strong pushback).

8. Energy in the capacitor $U_{E}$

Picture the stretched spring again: energy stored is $\frac{1}{2} k x^{2}$ . Swap $x \to q$ and $k \to 1/ C$ and you land exactly on $\frac{1}{2} \cdot \frac{1}{C} \cdot q^{2} = \frac{q ^{2}}{2 C}$ . Same picture, same math.

Why the topic needs it: this is one of the two "buckets" the energy sloshes between.

9. Inductance $L$ — the electrical "heaviness"

Detailed origin lives in Inductance and Self-Induction. The key rule:

Read the figure — two panels, same idea:

Left (lavender block on wheels): a heavy mass $m$ . The coral arrow is a push; the caption reminds you the mass is slow to speed up and slow to stop — that reluctance is inertia.
Right (mint coil): an inductor $L$ . The coral arrow is the current $i$ flowing through it; the caption shows the back-EMF $V_{L} = L d i / d t$ pointing so as to oppose the change in that current.
The pairing to notice: the coral push/current sits in the same spot on both panels, and both captions say "resists change." That side-by-side is the whole point — $L$ is to current what $m$ is to motion.

Why the topic needs it: $L$ is the analog of mass $m$ . It supplies the "inertia" that makes the circuit overshoot instead of quietly settling.

10. Energy in the inductor $U_{B}$

Compare a mass's kinetic energy $\frac{1}{2} m v^{2}$ : swap $m \to L$ and $v \to i$ and you get $\frac{1}{2} L i^{2}$ exactly. This is the second bucket. Energy trades back and forth between $U_{E}$ (§8) and $U_{B}$ .

Why the topic needs it: the whole oscillation is $U_{E} \leftrightarrow U_{B}$ , and total $U_{E} + U_{B}$ staying fixed is the proof it runs forever (in the ideal, resistor-free case).

11. Kirchhoff's Voltage Law — the loop rule (and why the signs add)

Picture walking around a circular hiking trail: however much you climb, you must descend exactly as much to arrive back at the same spot. Voltage works the same way around a loop.

Intuition Why

$V_{C} + V_{L} = 0$ — read it off the loop diagram The figure shows the whole loop with a single clockwise walking arrow and the $+ / -$ reference marks drawn on each component. Follow the walk:

The capacitor (lavender) has its $+$ mark on the $+ q$ plate. Walking clockwise you cross it from $+$ to $-$ , so you record a drop of $V_{C} = q / C$ .
The inductor (mint) has its $+ / -$ marks set the same way relative to the walk, so crossing it you also record a drop of $V_{L} = L \frac{d i}{d t}$ .
Both are drops in the same walking direction, so KVL (rises minus drops $= 0$ , i.e. the drops sum to zero) gives: $V_{C} + V_{L} = 0 \Rightarrow \frac{q}{C} + L \frac{d i}{d t} = 0.$ They add rather than subtract precisely because both voltages were measured with the same orientation as the walk — this is exactly what the diagram's matched $+ / -$ marks show. If you had drawn one component's polarity backwards, you'd write a minus and get the identical physics after fixing the sign.

Now substitute the sign-consistent $i = \overset{q}{˙}$ (from §3): $L \overset{q}{¨} + \frac{q}{C} = 0 \Rightarrow \overset{q}{¨} = - \frac{1}{L C} q .$ The minus sign — the restoring nature — is a direct consequence of these consistent sign choices: when $q$ is positive, $\overset{q}{¨}$ is negative, pulling it back toward zero.

Why the topic needs it: this single equation is the doorway from the physical circuit to the differential equation. Full treatment in Kirchhoff's Voltage Law.

12. Angular frequency $ω$ , period $T$ , frequency $f$

These three describe how fast the sloshing happens.

For this circuit the answer turns out to be: $ω = \frac{1}{L C}, T = 2 π L C, f = \frac{1}{2 π L C}$

Why the topic needs it: these are the payoff — the actual rhythm of the oscillation. Bigger $L$ (more inertia) or bigger $C$ (softer spring) → slower rhythm, exactly as intuition demands.

13. $cos$ and $sin$ — the shape of oscillation, and the $9 0^{\circ}$ between them

The two are the same wave shifted by a quarter turn ( $9 0^{\circ}$ ). That quarter-turn shift is the entire reason charge and current are "90° out of phase":

$q (t) = q_{0} cos (ω t)$ — starts at maximum $q_{0}$ (matching $q (0) = q_{0}$ ).
$i (t) = - q_{0} ω sin (ω t)$ — starts at zero (matching $i (0) = 0$ ).

When one is at its peak (flat slope → zero speed), the other is crossing zero (steepest slope → max speed). We'll live this out fully in the next deep dive.

Why the topic needs it: $cos$ describes the charge movie; $sin$ (its derivative's cousin) describes the current movie; their built-in quarter-turn offset is the physics of "current is max when charge is zero."

Prerequisite map

Equipment checklist

Test yourself — cover the right side and see if you can answer each before revealing.

What does

q

physically represent, and what mechanical quantity is it the analog of?

The charge piled on a capacitor plate; it is the analog of position

x

Which plate is the

+ q

plate — the one with surplus or shortage of electrons?

The one with a shortage of electrons (missing negatives leaves it net positive); the surplus-electron plate is

- q

What is

q_{0}

, and at what instant is it measured?

The initial charge on the capacitor at time

t = 0

(switch just closed); it is the maximum the charge reaches.

What are the TWO starting conditions this circuit needs, and why two?

q (0) = q_{0}

and

i (0) = 0

; a second-order (second-derivative) equation needs two pieces of starting information.

In plain words, what is

\frac{d q}{d t}

, and what does its sign mean?

The instantaneous slope of the charge-vs-time graph — the current

i

; positive means charge is piling onto the

+ q

plate, negative means it is draining.

Why do we need a derivative instead of just "change over time"?

A plain ratio gives only an average over an interval; the derivative gives the exact rate at a single instant.

What does the second derivative

\overset{q}{¨}

mean here?

How fast the current itself is changing — the "acceleration" of the charge.

What is voltage

V

, and why does it need a reference polarity?

The electrical push (energy per charge) across two points; it needs

+ / -

marks so its sign is meaningful when we add voltages around a loop.

State the capacitor rule

V_{C} = ?

and say why

1/ C

acts like spring constant

k

V_{C} = q / C

; a stiff (small

C

) capacitor pushes back harder per unit charge, giving a large

1/ C = k

State the inductor rule

V_{L} = ?

and why

L

acts like mass

m

V_{L} = L d i / d t

; it resists changes in current the way inertia resists changes in motion.

Write the two energy buckets.

U_{E} = \frac{q ^{2}}{2 C}

(electric) and

U_{B} = \frac{1}{2} L i^{2}

(magnetic).

Why do the two voltages add in KVL as

V_{C} + V_{L} = 0

Because both are measured with the same walking direction around the single loop, so both count as drops in that direction and sum to zero.

Give

ω

T

f

in terms of

L

and

C

ω = \frac{1}{L C}

T = 2 π L C

f = \frac{1}{2 π L C}

Why are

cos

and

sin

the right shapes, and what is the phase between them?

They are the shadows of a dot circling steadily — the natural oscillation shape — and they are shifted by

9 0^{\circ}

, matching charge vs current.