LC circuit — oscillations (electrical analog of SHM)

Intuition The big picture

An LC circuit is a capacitor and an inductor connected in a loop, with no battery and no resistor. Charge the capacitor, then close the switch — the charge does NOT just drain away. It sloshes back and forth forever, exactly like a mass bouncing on a spring.

WHY oscillation? Two "stores" of energy fight over the same energy: the capacitor stores energy in its electric field ( $\tfrac{1}{2}\tfrac{q^2}{C}$ ), the inductor stores energy in its magnetic field ( $\tfrac{1}{2}Li^2$ ). Neither can hold all of it at rest because the inductor resists changes in current — so energy keeps overshooting from one to the other.
The analogy: $q \leftrightarrow x$ (position), $i = \dot q \leftrightarrow v$ (velocity), $L \leftrightarrow m$ (inertia), $1/C \leftrightarrow k$ (springiness).

Setting it up from scratch

The solution and the energy

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

Formula Energy conservation — derive the constant

Electric (capacitor) energy: $U_E=\dfrac{q^2}{2C}=\dfrac{q_0^2}{2C}\cos^2(\omega t)$ . Magnetic (inductor) energy: $U_B=\dfrac12 Li^2=\dfrac12 L q_0^2\omega^2\sin^2(\omega t)$ .

Use $\omega^2=1/LC$ so $L\omega^2=1/C$ : $U_B=\frac{q_0^2}{2C}\sin^2(\omega t)$ Add them — Why? to check total energy is constant: $U_{\text{total}}=\frac{q_0^2}{2C}\big[\cos^2\omega t+\sin^2\omega t\big]=\frac{q_0^2}{2C}=\text{const}$ Energy is traded $U_E\leftrightarrow U_B$ but the total stays fixed (no resistor).

Common mistake Steel-manned common errors

(1) "Current is max when charge is max." Why it feels right: more charge ⟹ more push, surely more flow. The fix: current is $\dot q$ , the slope of $q(t)$ . At the peak of a cosine the slope is zero. Max current happens when $q=0$ (steepest slope). They are 90° apart.

(2) Using $T=2\pi/\sqrt{LC}$ (forgetting the square root is in the denominator differently). Why it feels right: $\omega=1/\sqrt{LC}$ and $T=2\pi/\omega$ get rushed together. The fix: $T=2\pi/\omega=2\pi\sqrt{LC}$ — the $\sqrt{LC}$ ends up in the numerator of $T$ .

(3) "Energy decreases over time." Why it feels right: real circuits die out. The fix: only because real wires have resistance $R$ . The ideal LC circuit has $R=0$ , so total energy is exactly constant — perpetual oscillation.

Recall Feynman: explain to a 12-year-old

Imagine a swing. A capacitor full of charge is like a swing pulled all the way up — lots of "stored push," not moving. Let go: it rushes down (current grows) and energy turns into motion. At the bottom it's moving fastest (max current) but the "height" (charge) is zero. It overshoots and climbs the other side, refilling the capacitor backwards. With no friction it swings forever. The inductor is the swing's "heaviness" that keeps it going past the bottom; the capacitor is the "height" that pulls it back.

Flashcards

What is the differential equation of an ideal LC circuit?

L\ddot q + q/C = 0

, i.e.

\ddot q = -\dfrac{1}{LC}q

What is the angular frequency of LC oscillations?

\omega = \dfrac{1}{\sqrt{LC}}

What is the period of an LC circuit?

T = 2\pi\sqrt{LC}

In the mechanical analogy, what plays the role of mass

m

The inductance

L

In the mechanical analogy, what plays the role of spring constant

k

1/C

What is the analog of position

x

in the LC circuit?

The charge

q

on the capacitor.

What is the analog of velocity

v

The current

i = \dot q

By what phase do charge and current differ?

90^\circ

(current is max when charge is zero).

Why does the ideal LC circuit oscillate forever?

It has no resistance, so total energy

q_0^2/2C

is conserved (no dissipation).

Write the energy stored in the capacitor and inductor.

U_E=\dfrac{q^2}{2C}

and

U_B=\dfrac12 Li^2

At what fraction of the period is energy shared equally?

T/8

(and odd multiples), where

\omega t=\pi/4

How is peak current related to peak charge?

i_0 = q_0\,\omega = q_0/\sqrt{LC}

Connections

Simple Harmonic Motion — same equation $\ddot x=-\omega^2 x$ ; everything maps over.
Capacitance and Energy in Capacitors — source of $U_E=q^2/2C$ .
Inductance and Self-Induction — origin of back-EMF $L\,di/dt$ .
Kirchhoff's Voltage Law — used to write the loop equation.
Damped Oscillations / LCR Circuit — what happens when $R\neq 0$ .
Resonance and AC Circuits — $\omega=1/\sqrt{LC}$ is the resonant frequency.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Socho ek capacitor ko charge karke ek inductor ke saath loop mein joed diya, na battery na resistor. Charge seedha khatam nahi hota — woh aage-peeche oscillate karta hai, bilkul spring pe lage mass ki tarah. Isiliye isko SHM ka electrical analog kehte hain. Yahan capacitor ka charge $q$ position $x$ jaisa hai, current $i=\dot q$ velocity jaisa hai, inductor $L$ mass (inertia) jaisa, aur $1/C$ spring constant $k$ jaisa.

Kirchhoff's voltage law lagao: $q/C + L\,di/dt = 0$ . Ise solve karo to milta hai $\ddot q = -\tfrac{1}{LC}q$ — yeh exactly SHM equation hai. Isliye $\omega = 1/\sqrt{LC}$ aur $T=2\pi\sqrt{LC}$ . Bada $L$ ya bada $C$ matlab slow oscillation, kyunki zyada inertia ya softer spring.

Sabse important baat: charge aur current 90 degree out of phase hain. Jab charge maximum hota hai, current zero hota hai (cosine ka peak pe slope zero). Jab charge zero hota hai, current maximum (sabse steep slope). Yeh wahi galti hai jo bachche karte hain — "zyada charge to zyada current" — galat! Current to charge ka rate of change hai.

Energy ki baat karein to capacitor mein electric energy $q^2/2C$ aur inductor mein magnetic energy $\tfrac12 Li^2$ . Yeh dono aapas mein energy exchange karte rehte hain, lekin total energy $q_0^2/2C$ constant rehta hai kyunki resistor nahi hai. Real circuit mein resistance hota hai isliye dheere-dheere energy khatam ho jaati hai (damping), par ideal LC mein hamesha chalta rehta hai. Exam mein $\omega=1/\sqrt{LC}$ , peak current $i_0=q_0\omega$ , aur $T/8$ pe energy equally share — yeh teen cheezein pakki yaad rakho.

LC circuit — oscillations (electrical analog of SHM)

Setting it up from scratch

The solution and the energy

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections