LC circuit — oscillations (electrical analog of SHM): Question bank

1.8.30 · D5 · Physics › Electromagnetism › LC circuit — oscillations (electrical analog of SHM)

Machinery ke reminders jo tumhe chahiye honge (sab parent note mein earn kiye gaye hain):

$q$ = capacitor par charge (yeh "position" hai), $i = \overset{q}{˙}$ = current (yeh "velocity" hai).
$q (t) = q_{0} cos (ω t)$ , $i (t) = - q_{0} ω sin (ω t)$ , jahan $ω = \frac{1}{L C}$ .
$ω$ = angular frequency (cycle ke radians jo har second sweep hote hain). Period $T$ woh time hai jisme ek full cycle complete hoti hai: kyunki ek full cycle $2 π$ radians ki hoti hai, $T = \frac{2 π}{ω} = 2 π L C$ . To " $t = T /4$ " ka matlab sirf itna hai ki "ek cycle ka quarter raasta."
Energies: $U_{E} = \frac{q ^{2}}{2 C}$ (capacitor, electric field), $U_{B} = \frac{1}{2} L i^{2}$ (inductor, magnetic field), total fixed rehti hai $\frac{q _{0}^{2}}{2 C}$ par.

Traps se pehle, apne dimag mein pictures pakki kar lo.

Figure 1 — $9 0^{\circ}$ wala dance. Cyan curve charge $q (t)$ hai aur amber curve current $i (t)$ hai, dono ek hi time axis par drawn hain. Ek cheez dhundho: jahan cyan curve flat ho (peak ya trough par), wahan amber curve zero cross karti hai, aur vice-versa. Double-headed white arrow dono peaks ke beech quarter-cycle gap ko mark karta hai — wahi $9 0^{\circ}$ phase difference visible form mein hai.

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

Figure 2 — do bucket. Same time axis, ab energy mein. Cyan capacitor ki energy $U_{E}$ hai, amber inductor ki energy $U_{B}$ hai, aur upar dashed white line dono ka sum hai. Do cheezein dhundho: ek bucket ki crest hamesha doosre ki trough ke upar baiThe hoti hai (energy aage-peechhe hand ho rahi hai), aur dashed total kabhi nahi hilta — woh ek flat line hai, jo energy conservation hai tumhari aankhon ke saamne. White dots woh equal-sharing instants mark karte hain jahan dono curves half-height par cross karti hain.

Figure 3 — resistance ki ek phusphusahat. Faint cyan wave ideal $R = 0$ oscillation hai; bold amber wave wahi circuit hai jisme ek chhota resistor add kiya gaya hai. Do changes dhundho: amber wave dashed white envelope ke andar shrink hoti hai (energy ab leak ho rahi hai), aur iske peaks cyan peaks ke thoda right mein drift karte hain — frequency $ω$ se $ω^{'}$ par aa gayi hai. Dono effects neeche ke edge-case questions ka subject hain.

True or false — justify karo

LC circuit mein charge aur current dono ek hi instant mein peak karte hain.

False. Current,

q (t)

ka slope hai; cosine ke top par slope zero hota hai, to jab

q

maximum ho tab current zero hoti hai. Woh

9 0^{\circ}

apart hain — Figure 1 mein offset peaks dekho.

Ek ideal LC circuit mein total energy sirf average par constant rehti hai, har instant par nahi.

False. Woh har instant par constant hai:

\frac{q _{0}^{2}}{2 C} (cos^{2} ω t + sin^{2} ω t) = \frac{q _{0}^{2}}{2 C}

hamesha. Figure 2 mein dono energy humps hamesha ek hi flat line mein sum hote hain. Dissipate karne ke liye kuch nahi hai kyunki

R = 0

hai.

Capacitance ko double karne se oscillation faster ho jaati hai.

False.

ω = 1/ L C

, to zyada

C

ka matlab chhota

ω

hai — slower. Ek bada

C

ek "softer spring" hai (

1/ C

shrink hota hai), to wापas swing karne mein zyada time lagta hai.

Agar tum circuit ko capacitor uncharged aur koi current nahi lekar shuru karo, to bhi woh oscillate karega.

False.

q_{0} = 0

aur

i_{0} = 0

ke saath system mein zero energy hai, to

q (t) = 0

for all time. Pehle energy inject karni hogi (cap ko charge karo ya current chalao).

Inductor ki magnetic energy exactly tab sabse badi hoti hai jab capacitor fully charged ho.

False. Ulta hai:

U_{B}

tab peak karti hai jab

q = 0

ho (current maximum), aur

U_{E}

tab peak karti hai jab

i = 0

ho (charge maximum). Figure 2 mein ek bucket ki crest doosre ki trough ke upar baiThe hai.

Jis moment

q = 0

ho, circuit mein koi energy nahi hoti.

False.

q = 0

par electric energy zero hai, lekin saari energy inductor ke magnetic field mein baiThe hai (

U_{B}

apne maximum par hai). Energy ne sirf stores change kiye hain.

Current har period mein do baar direction reverse karti hai.

True.

i = - q_{0} ω sin (ω t)

zero se pass hoti hai aur sign flip karti hai

t = T /4

aur

t = 3 T /4

par (jahan sine zero cross karta hai). Woh har cycle mein do turnarounds hain: charge ek taraf slosh karta hai, momentarily ruk jaata hai, phir doosri taraf slosh karta hai.

Analogy mein, inductance

L

spring constant ka role play karta hai.

False.

L

mass

m

ka role play karta hai (inertia — yeh current mein changes resist karta hai, bilkul jaise mass velocity mein changes resist karti hai). Spring constant

1/ C

se map hota hai.

Ek real circuit se resistor remove karna hi perpetual oscillation allow karta hai.

True (idealised).

R = 0

ke saath koi dissipation term nahi hai, to damping vanish ho jaati hai aur amplitude kabhi decay nahi karta. Real wires mein hamesha kuch

R

hota hai, to true perpetual motion ek limit hai, lab fact nahi.

Error dhundho

"Kyunki

ω = 1/ L C

, to period

T = 2 π / L C

hai."

Wrong.

T = 2 π / ω = 2 π L C

ω

ko invert karne se

L C

numerator mein aa jaata hai, to

T

L

aur

C

ke saath grow karta hai, shrink nahi.

"Energy

t = T /4

par

L

aur

C

ke beech equally share hoti hai."

Wrong.

T /4

par (

ω t = π /2

) charge zero hai aur saari energy magnetic hai. Equal sharing ke liye

cos^{2} = sin^{2}

chahiye, yaani

ω t = π /4

, jo

t = T /8

hai.

"Capacitor ka voltage aur inductor ka voltage milke battery voltage ke barabar hote hain."

Wrong. LC circuit mein koi battery nahi hoti. Kirchhoff's loop deta hai

V_{C} + V_{L} = 0

; dono voltages har instant par equal aur opposite hain.

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

, to magnetic energy amplitude

\frac{1}{2} L q_{0}^{2}

hai."

Wrong. Current amplitude

i_{0} = q_{0} ω

hai, to

U_{B, m a x} = \frac{1}{2} L (q_{0} ω)^{2} = \frac{1}{2} L q_{0}^{2} ω^{2}

hai.

ω^{2} = 1/ L C

use karne par yeh

q_{0}^{2} /2 C

ke barabar hota hai — wahi jo

U_{E, m a x}

hai, jaise energy conservation demand karta hai.

"Back-EMF

V_{L} = L i

hai."

Wrong. Ek inductor current ki change ko oppose karta hai, to

V_{L} = L \frac{d i}{d t}

hai — current ke rate of change ke proportional, current ke nahi. Inductance and Self-Induction dekho.

"Kyunki current charge se

9 0^{\circ}

lag karta hai, current

q_{0} ω cos (ω t)

hai."

Wrong sign/function.

q_{0} cos (ω t)

ko differentiate karne par

- q_{0} ω sin (ω t)

milta hai, ek negative sine, cosine nahi. Lag ko concretely dekhne ke liye, peaks compare karo: charge

q_{0} cos (ω t)

t = 0

par peak karta hai, jabki current

- q_{0} ω sin (ω t)

apna positive peak baad mein,

ω t = 3 π /2

par (yaani

t = 3 T /4

) reach karta hai. Kyunki current ka matching peak charge ke peak ke ek quarter-cycle baad aata hai, hum kahte hain current charge se

9 0^{\circ}

lag karta hai. (Equivalently

- sin θ = cos (θ + 9 0^{\circ})

, lekin peaks se read karna lead/lag seedha rakhne ka sabse safe tarika hai.)

Why questions

Charge zero ko overshoot kyun karta hai, wahan bas rukta kyun nahi?

Kyunki

q = 0

par current (energy) maximum hai, aur inductor us current mein kisi bhi sudden change ko resist karta hai. Pendulum ko uske swing ke bottom par imagine karo: woh fastest move kar raha hai, aur uska inertia use rokne nahi deta. Yahan inductor wahi inertia hai — woh current ko

q = 0

ke past flow karta rehta hai, aur woh continuing current hai charge jo doosri taraf plates par pile ho raha hai. Overshoot wahi inertia hai jo instantly brake karne se mana karta hai.

Hum

1/ C

(na ki

C

) ko spring constant

k

ka analog kyun kehte hain?

Ek spring ki restoring force

F = - k x

hoti hai: displacement

x

ek push produce karta hai

k

ke proportional. Yahan "displacement" charge

q

hai, aur restoring "push" capacitor voltage

V_{C} = q / C

hai. To displacement aur restoring push ke beech proportionality constant

1/ C

hai, yahi

k

ka matlab hai. Ek chhota

C

isliye ek stiff spring ki tarah behave karta hai: thoda sa charge bhi ek bada voltage build karta hai jo current ko wापas slam karta hai, ek fast, high-

ω

oscillation deta hai.

Kirchhoff's Voltage Law yahan sahi starting tool kyun hai, energy conservation kyun nahi?

Kirchhoff's Voltage Law har instant par voltages ke baare mein ek statement hai, to yeh

q (t)

mein ek differential equation produce karta hai — ek rule jo

q

, uske slope, aur uske curvature ko sabhi times par together tie karta hai. Wahi full waveform

q_{0} cos (ω t)

ko pin down karta hai. Energy conservation ek single number hai jo poori motion mein true hai; yeh peak charge se peak current batata hai, lekin yeh ek cosine ko kisi bhi doosre curve se distinguish nahi kar sakta jiske paas same energy ho, to yeh kabhi khud time-dependence generate nahi kar sakta.

Charge aur current exactly

9 0^{\circ}

apart kyun honni chahiye, koi aur phase kyun nahi?

Current defined hai charge ke derivative (slope) ke roop mein, aur differentiation ek cosine ko

- sin

mein shift kar deta hai, jo same shape ko quarter-cycle over move karta hai. To phase gap ek physical coincidence nahi hai jise tum tune kar sako — yeh "rate of change" ke meaning mein baked in hai. Jahan bhi

q

flat ho (peak ya trough par), uska slope zero hai, to

i = 0

; jahan

q

zero se race kare, uska slope steepest hai, to

∣ i ∣

peak karta hai. Figure 1 is slope-to-height correspondence ko visible banata hai.

Ek bada inductor oscillation ko slow kyun karta hai?

L

electrical inertia hai: current ko change karne ke liye jo voltage woh demand karta hai woh

L d i / d t

hai, to ek bada

L

matlab hai ki current mein thodi si change bhi ek bada voltage maangti hai. Current isliye sluggishly build aur drain hoti hai, aur poora slosh zyada time leta hai. Mechanical analogy mein yeh ek heavier mass hai — chalana mushkil, rokna mushkil — aur

ω = 1/ L C

theek se shrink hota hai jab

L

grow karta hai, period

T = 2 π L C

ko stretch karta hai.

ω = 1/ L C

ko resonant frequency kyun kehte hain?

Akela chhodne par, yeh LC pair exactly

ω = 1/ L C

par oscillate karta hai — uski "natural note," sirf itne se set hoti hai ki uske paas kitna inertia (

L

) aur springiness (

1/ C

) hai. Agar ab tum ise ek AC source se drive karo, to drive ko is natural note se match karne par har push motion ke saath step mein aata hai jo use reinforce karta hai, to amplitude ek peak par climb karti hai. Wahi resonant peak exactly woh hai jiske liye radios mein

1/ L C

ko tune kiya jaata hai — Resonance and AC Circuits ka subject.

Edge cases

Jab

C \to \infty

(ek huge capacitor) ho tab

ω

aur

T

ka kya hoga?

ω = 1/ L C \to 0

aur

T = 2 π L C \to \infty

. Ek infinitely "soft spring" charge ko wापas kheenchne ke liye kabhi itna voltage nahi build kar paata, to period infinity tak stretch ho jaati hai — effectively koi oscillation nahi.

Jab

C \to 0

(vanishing capacitance) ho tab

ω

aur

T

ka kya hoga?

Opposite singular limit:

ω = 1/ L C \to \infty

aur

T = 2 π L C \to 0

. Ek infinitely "stiff spring" (

1/ C \to \infty

) charge ko instantly wापas slam karta hai, to oscillation infinitely fast ho jaati hai. Physically tiny capacitor almost koi charge hold nahi karta, to trade karne ke liye barely koi energy hoti hai aur idealised LC model is extreme par break down ho jaata hai.

Jab

L \to 0

(ek ideal wire, koi inductor nahi) ho tab kya hoga?

ω \to \infty

aur

T \to 0

: infinitely fast,

C \to 0

case ko mirror karta hai. Physically magnetic energy store karne ke liye koi inertia nahi hai, to circuit degenerate ho jaata hai — capacitor wire ke through instantly discharge ho jaata, aur LC model break down ho jaata hai.

Agar tum ek small resistance

R

introduce karo, to kya oscillation immediately harmonic rehna band ho jaati hai?

Nahi, lekin woh ab strictly harmonic nahi rahi. Amplitude exponentially decay karti hai, aur frequency shift ho kar

ω^{'} = \frac{1}{L C} - (\frac{R}{2 L})^{2}

ho jaati hai, jo ideal

ω = 1/ L C

se thodi neeche hoti hai. To waveform ek shrinking envelope ke neeche

ω^{'}

par oscillate karti hai (Figure 3 dekho). Yeh LCR Circuit aur Damped Oscillations ka underdamped regime hai.

R

badhate raho — kya koi aisi value hai jahan oscillation completely band ho jaaye?

Haan. Jab

R = 2 L / C

ho tab square root ke andar ki term,

\frac{1}{L C} - (\frac{R}{2 L})^{2}

, exactly zero hit karti hai, to

ω^{'} = 0

. Yeh critically damped case hai: charge jitni jaldi ho sake without overshooting zero par wापas aata hai — bilkul koi back-and-forth nahi. Yeh "abhi bhi wiggle karta hai" aur "kabhi wiggle nahi karta" ke beech sharp boundary hai.

Aur zyada resistance,

R > 2 L / C

, ka kya?

\frac{1}{L C} < (\frac{R}{2 L})^{2}

hai, to root ke neeche ki quantity negative hai aur

ω^{'}

ab koi real oscillation frequency nahi rahi. Yeh overdamped regime hai: charge slowly do decaying exponentials ke sum ke roop mein zero par wापas oozta hai, koi oscillation nahi.

R \to \infty

ki limit mein current completely choke off ho jaati hai aur charge essentially frozen ho jaata hai — swing molasses mein dubi hoti hai.

Capacitor fully discharged shuru hota hai lekin current

i_{0}

already flow kar rahi hai. Kya yeh phir bhi oscillate karega?

Haan. Energy ab completely inductor mein shuru hoti hai (

U_{B}

maximum), aur

q (t) = \frac{i _{0}}{ω} sin (ω t)

— ek sine solution. Charge aur current phir bhi oscillate karte hain, bas ek shifted starting phase ke saath.

Kya koi aisa instant hai jab dono

U_{E} = 0

aur

U_{B} = 0

simultaneously ho?

Nahi. Kyunki

U_{E} + U_{B} = q_{0}^{2} /2 C

ek fixed positive constant hai, woh dono ek saath vanish nahi ho sakte. Jab ek zero hoti hai tab doosri poori energy hold karti hai.

t = T /2

par,

q

aur

i

t = 0

se compare karke kaisa hoga?

q (T /2) = q_{0} cos (π) = - q_{0}

(fully charged opposite taraf) aur

i (T /2) = - q_{0} ω sin (π) = 0

(momentarily at rest). Yeh start ka mirror image hai — swing doosri side ke far top par.

Recall Ek-line self-test

Upar har answer ko cover karo aur reveal karne se pehle apna reasoning aloud bolo. Agar koi bhi answer tumhare dimaag mein sirf "true" ya "false" hai bina kisi kyunki ke, to tumne use abhi seekha nahi hai.

Connections

Parent topic
Simple Harmonic Motion — woh analogy jo in mein se aadhe traps resolve karti hai.
Capacitance and Energy in Capacitors · Inductance and Self-Induction — jahan se $U_{E}$ aur $V_{L}$ aate hain.
Kirchhoff's Voltage Law — loop rule jo equation of motion ke peeche hai.
Damped Oscillations / LCR Circuit — $R \neq = 0$ edge cases (underdamped, critical, overdamped).
Resonance and AC Circuits — kyun $ω = 1/ L C$ resonant frequency hai.