1.8.30
Ek LC circuit mein ek capacitor aur ek inductor ek loop mein connected hote hain, koi battery nahi, koi resistor nahi. Capacitor ko charge karo, phir switch band karo — charge bas drain nahi hota. Yeh hamesha aage-peechhe slosh karta rehta hai , bilkul waisi hi tarah jaise ek mass spring pe bounce karta hai.
Oscillation KYU hoti hai? Do "energy stores" ek hi energy ke liye ladte hain: capacitor energy apne electric field mein store karta hai (2 1 C q 2 magnetic field mein store karta hai (2 1 L i 2 changes ka virodh karta hai — isliye energy ek se doosre mein overshoot karti rehti hai.Analogy: q ↔ x i = q ˙ ↔ v L ↔ m 1/ C ↔ k
Capacitance C V C = q / C Inductance L V L = L d t d i Koi resistor nahi ⟹ no energy loss ⟹ hamesha ke liye undamped oscillation.
ω = 1/ L C L q ¨ + C 1 q = 0 m x ¨ + k x = 0 ω = k / m m → L k → 1/ C ω = ( 1/ C ) / L = 1/ L C L C slower oscillation. Samajh mein aata hai.
Worked example Worked example 1 — frequency nikalnaL = 2 mH C = 5 μ F f T
ω = L C 1 = ( 2 × 1 0 − 3 ) ( 5 × 1 0 − 6 ) 1 = 1 0 − 8 1 = 1 0 4 rad/s Kyun? Matched SHM formula mein plug in karo.f = ω /2 π ≈ 1592 Hz Kyun? ω = 2 π f T = 1/ f ≈ 0.628 ms Kyun? Period, frequency ka reciprocal hota hai.
Worked example Worked example 2 — energy se peak current nikalnaCapacitor ko usi L , C q 0 = 2 × 1 0 − 4 i 0
Energy method: swap point par saari electric energy saari magnetic energy ban jaati hai.
2 C q 0 2 = 2 1 L i 0 2 ⇒ i 0 = q 0 L C 1 = q 0 ω Kyun? Conservation: peak U E U B
i 0 = ( 2 × 1 0 − 4 ) ( 1 0 4 ) = 2
Common mistake Common galtiyan (steel-manned)(1) "Current max hota hai jab charge max hota hai."
Kyun sahi lagta hai: zyada charge ⟹ zyada push, zaroor zyada flow. Fix: current q ˙ q ( t ) slope . Cosine ki peak par slope zero hota hai. Max current tab hota hai jab q = 0
(2) T = 2 π / L C
Kyun sahi lagta hai: ω = 1/ L C T = 2 π / ω Fix: T = 2 π / ω = 2 π L C L C T numerator mein aata hai.
(3) "Energy time ke saath decrease hoti hai."
Kyun sahi lagta hai: real circuits die out karte hain. Fix: sirf isliye kyunki real wires mein resistance R Ideal LC circuit mein R = 0
Recall Feynman: ek 12-saal ke bacche ko explain karoEk swing imagine karo. Charge se bhara capacitor ek swing ki tarah hai jo poori tarah upar khichi hui ho — bahut "stored push," move nahi kar rahi. Chodo: yeh neechhe rush karta hai (current badhta hai) aur energy motion mein badal jaati hai. Sabse neechhe yeh sabse fast move kar raha hota hai (max current) lekin "height" (charge) zero hai. Yeh overshoot karta hai aur doosri taraf chadhta hai, capacitor ko ulta refill karta hai. Koi friction nahi toh hamesha ke liye swing karta rehta hai. Inductor swing ki "heaviness" hai jo ise neechhe se aage le jaata hai; capacitor woh "height" hai jo ise wapas kheenchti hai.
Mnemonic Yaad rakhne ka tarika"CLOCK" ⟷ C harge aur L 's current dance karte hain, O ut-of-phase 90° se, C onserved energy, K eeps time T = 2 π L C "Mass is L, spring is one-over-C." (m → L , k → 1/ C
Ideal LC circuit ki differential equation kya hoti hai? L q ¨ + q / C = 0 q ¨ = − L C 1 q
LC oscillations ki angular frequency kya hoti hai? LC circuit ki period kya hoti hai? Mechanical analogy mein mass m Inductance L
Mechanical analogy mein spring constant k 1/ C
LC circuit mein position x Capacitor par charge q
Velocity v Current i = q ˙
Charge aur current mein kitne phase ka fark hota hai? 9 0 ∘
Ideal LC circuit hamesha kyun oscillate karta hai? Ismein koi resistance nahi hoti, isliye total energy q 0 2 /2 C
Capacitor aur inductor mein stored energy likhiye. U E = 2 C q 2 U B = 2 1 L i 2
Period ke kitne fraction par energy equally share hoti hai? T /8 ω t = π /4
Peak current ka peak charge se kya relation hai? Kirchhoff loop V_C + V_L = 0
i = -i0 sin(omega t) 90 deg phase
