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

1.8.30 · D1 · Physics › Electromagnetism › LC circuit — oscillations (electrical analog of SHM)

Iss idea ko enjoy karne se pehle, tumhe genuinely har woh letter dekhna hoga jo parent note tumhare saamne rakhta hai. Neeche, har symbol ko zero se banaya gaya hai: saadhe words → ek picture → kyun yeh topic uske bina exist nahi kar sakta. Inhe is tarah order kiya gaya hai ki har ek unse upar wale pe lean karta hai — toh koi symbol apne khud ke section se pehle kabhi use nahi hota.

1. Charge $q$ aur initial charge $q_{0}$ — "cheez ki miqdar" jo move karti hai

Do flat metal plates ko ek doosre ke saamne picture karo. Ek plate mein electrons ki kami hai — kyunki electrons negative hote hain, missing electrons us plate ko net positive chodti hain, isliye hum use $+ q$ plate kehte hain. Doosri plate mein electrons ka matching surplus hai, jisse woh net negative ho jaati hai, woh $- q$ plate. Number $q$ batata hai ki woh pile kitni lopsided hai.

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

Figure padhiye: baayein lavender plate electrons ki kami mein hai, isliye woh net positive hai — $+$ symbols se marked, yeh $+ q$ plate hai. Daayein coral plate electrons ka surplus rakhti hai, isliye woh net negative hai — $-$ marked, yeh $- q$ plate. (Surplus electrons → negative plate; yeh narrative se bilkul match karta hai.) Unke beech mint arrows electric field hain — woh hamesha $+ q$ plate se $- q$ plate ki taraf point karte hain, aur jab $q$ bada hota hai toh inki sankhya bhi zyada hoti hai.

Topic ko iske liye kyun zaroori hai: $q$ poori kahaani ka star hai. Parent note mein, $q$ wahi role karta hai jo spring mein position $x$ karti hai, aur $q_{0}$ woh "starting stretch" hai — tune mass ko chhorne se pehle kitna kheencha tha. Jaise $x$ batata hai mass center se kitna door hai, $q$ batata hai charge apni balanced (empty) state se kitni "pulled" hai.

2. Time $t$ aur function $q (t)$ ka idea

Ab §1 ka "shuru mein" wala phrase ek symbol paata hai: starting instant $t = 0$ hai, aur us instant pe capacitor initial charge $q_{0}$ rakhta hai, yaani $q (0) = q_{0}$ . Aadhe millisecond baad ( $t = 0.0005 s$ ) plate alag miqdar rakhti hai, aur $q (t)$ woh rule hai jo tumhare choose kiye kisi bhi $t$ ke liye sahi number deta hai.

Topic ko iske liye kyun zaroori hai: oscillation time ke saath badlav hai. Kisi aisi quantity ke idea ke bina jo $t$ pe depend kare, "aage-peeche sloshing" ka koi matlab nahi. (Movie ki exact recipe $q (t) = q_{0} cos (ω t)$ hai, lekin isme har symbol — including $ω$ — tab tak use nahi hota jab tak woh §12 mein build nahi ho jaata; hum ise abhi use nahi kar rahe.)

3. Current $i$ aur do starting conditions

Dam wapas: charge $q$ stored paani hai; current $i$ pipe mein flow rate hai. Ek bada lake dheere drain ho sakta hai aur small current ho sakti hai; ek chhota puddle tezi se nikle toh badi current ho sakti hai. Stored amount aur flow rate alag cheezein hain — yahi distinction baad mein 90° phase ka poora secret hai.

Topic ko iske liye kyun zaroori hai: $i$ velocity $v$ ka analog hai. Wahi relationship: velocity rate hai jis par position change hoti hai; current rate hai jis par charge change hota hai. Aur jaise velocity ka sign hota hai (left/right), current ka bhi sign hota hai (kis taraf circulate karta hai).

4. Derivative $\frac{d q}{d t}$ — "rate of change" symbols mein likha

Topic likhta hai $i = \frac{d q}{d t}$ . Woh fraction-jaisi cheez derivative kahlati hai, aur yahan woh matlab kya hai kuch bhi scary hone se pehle.

YEH tool kyun, sirf "time mein badlav" nahi? Kyunki ordinary "change ÷ time" ( $Δ q /Δ t$ ) sirf kuch stretch pe average deta hai. Hum ek single instant ki value chahte hain — curve pe ek point pe exact steepness. Derivative woh tool hai jo precisely "kitni tezi se, bilkul is instant?" ka jawaab dene ke liye bana hai. Koi doosra tool woh jawaab nahi deta.

Figure dekho: curve ko sirf kiss karne wali seedhi laal line us instant ki slope hai. Uski steepness us moment ki current hai.

Topic ko iske liye kyun zaroori hai: "current charge ki rate of change hai" wali sentence sirf derivative ke through ek equation banti hai. Sab kuch downstream ( $i (t)$ , 90° phase, energy) isi pe build hai.

5. Second derivative $\frac{d ^{2} q}{d t ^{2}}$ — "rate of change ki rate of change"

Ek car picture karo: position wahan ho tum, first derivative tumhari speed hai, second derivative tumhari acceleration (kitna pedal press kar rahe ho). Charge ke liye: $q$ pile hai, $\overset{q}{˙}$ flow hai, $\overset{q}{¨}$ hai kitni tezi se flow speed up ya slow ho raha hai.

Topic ko iske liye kyun zaroori hai: master equation $\overset{q}{¨} = - \frac{1}{L C} q$ second derivative ke baare mein ek statement hai. Yeh kehta hai "charge ki acceleration zero ki taraf wapas point karti hai" — jo har oscillation ka mathematical fingerprint hai. Kyunki yeh ek second-order derivative equation hai, ise §3 ke do starting facts chahiye. Wahi fingerprint spring mein dekhne ke liye Simple Harmonic Motion dekho.

6. Voltage $V$ — woh "push" jo charge drive karta hai

$C$ ya $L$ ko touch karne se pehle humein woh quantity naam deni chahiye jo dono produce karte hain: voltage.

Loop mein chosen (clockwise) direction mein chalte hue imagine karo. Jab tum kisi component ke $-$ mark se $+$ mark ki taraf step karte ho, tum $V$ se rise karte ho; doosri taraf tum $V$ se drop karte ho. Yeh chhote $+ / -$ arrows ek baar fix karna hi hai jo §11 mein voltages consistently add karne deta hai.

Topic ko iske liye kyun zaroori hai: loop mein har component voltage ki language bolta hai. Unhe loop ke around add karne ke liye (Kirchhoff) humein pehle $V$ sign ke saath define karna hoga, sirf magnitude nahi.

7. Capacitance $C$ — capacitor kitni aasaani se charge store karta hai

Defining relationship (Capacitance and Energy in Capacitors mein fully built) hai:

Topic ko iske liye kyun zaroori hai: $V_{C}$ "restoring push" hai. Jaise stretched spring jitna zyada stretch karo utna harder pushback karta hai, waise ek charged capacitor jitna zyada charge ho utna harder charge pushback karta hai. Isliye $1/ C$ spring constant $k$ ki role play karta hai: chhota $C$ (stiff, cramped container) bada $1/ C$ (strong pushback) deta hai.

8. Capacitor mein energy $U_{E}$

Stretched spring phir picture karo: stored energy $\frac{1}{2} k x^{2}$ hai. $x \to q$ aur $k \to 1/ C$ swap karo aur tum exactly $\frac{1}{2} \cdot \frac{1}{C} \cdot q^{2} = \frac{q ^{2}}{2 C}$ pe land karo. Same picture, same math.

Topic ko iske liye kyun zaroori hai: yeh un do "buckets" mein se ek hai jinke beech energy slosh karti hai.

9. Inductance $L$ — electrical "heaviness"

Detailed origin Inductance and Self-Induction mein hai. Key rule:

Figure padhiye — do panels, same idea:

Left (lavender block on wheels): ek heavy mass $m$ . Coral arrow ek push hai; caption remind karta hai mass speed up hone mein slow aur rukne mein slow hai — woh reluctance inertia hai.
Right (mint coil): ek inductor $L$ . Coral arrow current $i$ hai jo iske through flow kar rahi hai; caption dikhata hai back-EMF $V_{L} = L d i / d t$ us current mein change oppose karne ke liye point karta hai.
Notice karne wali pairing: coral push/current dono panels pe same jagah baithe hain, aur dono captions kehte hain "resists change." Yeh side-by-side poora point hai — $L$ current ke liye wahi hai jo $m$ motion ke liye hai.

Topic ko iske liye kyun zaroori hai: $L$ mass $m$ ka analog hai. Yoh "inertia" supply karta hai jo circuit ko quietly settle hone ki jagah overshoot karwa deti hai.

10. Inductor mein energy $U_{B}$

Mass ki kinetic energy $\frac{1}{2} m v^{2}$ se compare karo: $m \to L$ aur $v \to i$ swap karo aur exactly $\frac{1}{2} L i^{2}$ milta hai. Yeh doosra bucket hai. Energy $U_{E}$ (§8) aur $U_{B}$ ke beech trade back and forth karti hai.

Topic ko iske liye kyun zaroori hai: poori oscillation $U_{E} \leftrightarrow U_{B}$ hai, aur total $U_{E} + U_{B}$ fixed rehna is baat ka proof hai ki yeh hamesha chalta rahega (ideal, resistor-free case mein).

11. Kirchhoff's Voltage Law — loop rule (aur signs kyun add hote hain)

Ek circular hiking trail pe chalte hue imagine karo: jitna bhi chadhe, utna hi utarna hoga wapas usi jagah pahunchne ke liye. Voltage ek loop ke around same tarah kaam karta hai.

Intuition

$V_{C} + V_{L} = 0$ kyun — loop diagram se padho Figure poora loop dikhata hai ek single clockwise walking arrow aur har component pe $+ / -$ reference marks drawn ke saath. Walk follow karo:

Capacitor (lavender) ka $+$ mark $+ q$ plate pe hai. Clockwise chalte hue tum ise $+$ se $-$ cross karte ho, isliye tum ek drop of $V_{C} = q / C$ record karte ho.
Inductor (mint) ke $+ / -$ marks walk ke relative same tarah set hain, isliye ise cross karte hue tum ek drop of $V_{L} = L \frac{d i}{d t}$ bhi record karte ho.
Dono same walking direction mein drops hain, isliye KVL (rises minus drops $= 0$ , yaani drops zero mein sum hote hain) deta hai: $V_{C} + V_{L} = 0 \Rightarrow \frac{q}{C} + L \frac{d i}{d t} = 0.$ Yeh add hote hain subtract ki jagah precisely kyunki dono voltages same walk orientation ke saath measure kiye gaye — yeh exactly wahi hai jo diagram ke matched $+ / -$ marks dikhate hain. Agar tum ne ek component ki polarity ulti draw ki hoti, tum minus likhte aur sign fix karne ke baad identical physics paate.

Ab sign-consistent $i = \overset{q}{˙}$ (§3 se) substitute karo: $L \overset{q}{¨} + \frac{q}{C} = 0 \Rightarrow \overset{q}{¨} = - \frac{1}{L C} q .$ Minus sign — restoring nature — in consistent sign choices ka direct consequence hai: jab $q$ positive hota hai, $\overset{q}{¨}$ negative hota hai, use zero ki taraf wapas kheeenchta hai.

Topic ko iske liye kyun zaroori hai: yeh single equation physical circuit se differential equation tak ka doorway hai. Full treatment Kirchhoff's Voltage Law mein.

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

Yeh teen describe karte hain ki sloshing kitni tezi se hoti hai.

Is circuit ke liye answer nikalta hai: $ω = \frac{1}{L C}, T = 2 π L C, f = \frac{1}{2 π L C}$

Topic ko iske liye kyun zaroori hai: yeh payoff hain — oscillation ki actual rhythm. Bada $L$ (zyada inertia) ya bada $C$ (softer spring) → dhheema rhythm, exactly jaise intuition demand karta hai.

13. $cos$ aur $sin$ — oscillation ki shape, aur unke beech $9 0^{\circ}$

Dono same wave hain jo ek quarter turn ( $9 0^{\circ}$ ) shift hai. Woh quarter-turn shift hi poori wajah hai ki charge aur current "90° out of phase" hain:

$q (t) = q_{0} cos (ω t)$ — maximum $q_{0}$ pe start karta hai ( $q (0) = q_{0}$ match karta hai).
$i (t) = - q_{0} ω sin (ω t)$ — zero pe start karta hai ( $i (0) = 0$ match karta hai).

Jab ek apne peak pe hota hai (flat slope → zero speed), doosra zero cross kar raha hota hai (steepest slope → max speed). Hum yeh next deep dive mein fully live karenge.

Topic ko iske liye kyun zaroori hai: $cos$ charge movie describe karta hai; $sin$ (uske derivative ka cousin) current movie describe karta hai; unka built-in quarter-turn offset hi "current max hai jab charge zero hai" ki physics hai.

Prerequisite map

Equipment checklist

Khud ko test karo — right side cover karo aur reveal se pehle dekho ki jawaab de sakte ho ya nahi.

q

physically kya represent karta hai, aur yeh kisi mechanical quantity ka analog hai?

Capacitor plate pe pila charge; yeh position

x

ka analog hai.

Kaun si plate

+ q

plate hai — electrons ke surplus wali ya shortage wali?

Shortage wali electrons ki (missing negatives use net positive chodti hain); surplus-electron plate

- q

hai.

q_{0}

kya hai, aur kis instant pe measure kiya jaata hai?

t = 0

pe capacitor pe initial charge (switch abhi closed); yeh maximum hai jo charge kabhi reach karta hai.

Is circuit ko DO starting conditions kyun chahiye, aur kyun do?

q (0) = q_{0}

aur

i (0) = 0

; ek second-order (second-derivative) equation ko unique movie pin down karne ke liye do pieces of starting information chahiye.

Saadhe shabdon mein

\frac{d q}{d t}

kya hai, aur uske sign ka kya matlab hai?

Charge-vs-time graph ki instantaneous slope — current

i

; positive matlab charge

+ q

plate pe pile ho raha hai, negative matlab drain ho raha hai.

Sirf "time mein change" ki jagah derivative kyun chahiye?

Saadha ratio sirf ek interval pe average deta hai; derivative ek single instant pe exact rate deta hai.

Yahan second derivative

\overset{q}{¨}

ka kya matlab hai?

Current khud kitni tezi se change ho rahi hai — charge ki "acceleration."

Voltage

V

kya hai, aur ise reference polarity kyun chahiye?

Do points ke across electrical push (energy per charge); ise

+ / -

marks chahiye taaki iska sign meaningful ho jab hum loop ke around voltages add karein.

Capacitor rule

V_{C} = ?

state karo aur batao ki

1/ C

spring constant

k

jaisa kyun kaam karta hai.

V_{C} = q / C

; ek stiff (chhota

C

) capacitor har unit charge pe harder pushback karta hai, bada

1/ C = k

deta hai.

Inductor rule

V_{L} = ?

state karo aur

L

mass

m

jaisa kyun kaam karta hai.

V_{L} = L d i / d t

; yeh current mein changes ka resist karta hai jaise inertia motion mein changes ka resist karti hai.

Do energy buckets likhiye.

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

(electric) aur

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

(magnetic).

KVL mein do voltages add kyun hote hain

V_{C} + V_{L} = 0

mein?

Kyunki dono single loop ke around same walking direction se measure kiye gaye hain, isliye dono us direction mein drops count hote hain aur zero mein sum ho jaate hain.

ω

T

f

L

aur

C

ke terms mein do.

ω = \frac{1}{L C}

T = 2 π L C

f = \frac{1}{2 π L C}

cos

aur

sin

sahi shapes kyun hain, aur unke beech phase kya hai?

Yeh steadily circling dot ki shadows hain — natural oscillation shape — aur yeh

9 0^{\circ}

shift hain, charge vs current match karta hai.