Class 12 Physics · Chapter 7
Short answer:
Is chapter me total 16 exercise questions cover kiye gaye hain — RMS/peak value calculations, pure R/L/C circuits me current-voltage relation, LCR series circuit ka impedance aur resonant frequency, power factor aur power consumption, aur transformer ke turns-ratio numericals. Derivations trim karke seedha final formula se numerical solve kiya gaya hai — yehi is chapter ka board-exam pattern hai. AC voltage applied to resistor/inductor/capacitor, LCR series circuit (qualitative), resonance (brief), power in AC circuit, aur transformer (basic) — yehi scope hai.
AC circuit me current aur voltage same time pe peak/zero nahi hote — beech me ek phase difference hota hai, jo depend karta hai circuit me resistor, inductor, capacitor kaunsa element hai. DC circuit me sirf resistance current rokta hai, par AC circuit me inductor aur capacitor bhi ek 'effective resistance' (reactance) dikhate hain jo frequency pe depend karta hai. Pure resistor me current-voltage same phase me hote hain, pure inductor me current 90° peeche rehta hai, pure capacitor me current 90° aage rehta hai. Jab teeno LCR series me lagte hain, overall phase difference in dono extremes ke beech kahin hota hai — depend karta hai XL aur XC me se kaun bada hai. Yehi is chapter ka core concept hai, aur isi par saare numericals bane hain.
Chapter 7 Summary — 5 Minute Revision
1. AC voltage — basic definitions
Alternating voltage/current time ke saath sinusoidally badalta hai:
- v = Vm sin ωt , i = Im sin ωt
- Vm, Im = peak (amplitude) values
- ω = 2πf = angular frequency, f = frequency (Hz)
RMS (root mean square) value woh DC value hai jo same heating effect deta hai. Ammeter/voltmeter RMS hi measure karte hain — jab tak "peak" na likha ho, koi bhi AC voltage/current value (jaise ghar ka 220V) RMS hi hoti hai.
- Vrms = Vm/√2 ≈ 0.707 Vm
- Irms = Im/√2
2. AC applied to resistor, inductor, capacitor — akele akele
| Element | Opposition | Phase (i vs v) | Avg power/cycle |
|---|---|---|---|
| Resistor R | R (same for AC/DC) | Same phase (φ = 0) | P = VrmsIrms |
| Inductor L | XL = ωL = 2πfL | Current 90° peeche (lags) | Zero |
| Capacitor C | XC = 1/ωC = 1/2πfC | Current 90° aage (leads) | Zero |
↔ Table ko side me swipe karein
Yaad rakhne ka tareeka: "ELI the ICE man" — inductor (L) me EMF (voltage) current se pehle aata hai (ELI), capacitor (C) me current voltage se pehle aata hai (ICE).
Pure L ya pure C circuit me average power zero hoti hai kyunki poore cycle me energy source se load me jaati hai aur wapas source me aa jaati hai — koi net consumption nahi (isliye inhe "wattless current" bhi kehte hain).
3. AC applied to series LCR circuit
Jab R, L, C teeno series me ek AC source se jude hote hain, current sabme same hoti hai lekin voltage phasors alag phase pe hote hain. Total impedance (Z) — resistance ka AC version:
Phase angle φ jisse current voltage se peeche/aage rehta hai:
- Agar XL > XC → circuit inductive hai, current voltage se peeche (lag) rehta hai
- Agar XC > XL → circuit capacitive hai, current voltage se aage (lead) rehta hai
- Agar XL = XC → circuit purely resistive jaisa behave karta hai, φ = 0 (yehi resonance hai)
4. Resonance (brief)
Jis frequency pe XL = XC hota hai usse resonant frequency (f0) kehte hain. Us par impedance minimum (Z = R) hoti hai aur current maximum:
Resonance pe circuit purely resistive ban jaata hai — current aur voltage same phase me aa jaate hain. (Sharpness of resonance / Q-factor ki detail curriculum se hata di gayi hai — sirf resonant frequency ka concept aur formula chahiye.)
5. Power in AC circuit — power factor
AC circuit me average power sirf Vrms × Irms nahi hoti — phase difference ka factor lagta hai:
cos φ ko power factor kehte hain — batata hai kitna fraction power actually consume ho raha hai:
- Pure R circuit: cos φ = 1 (max power)
- Pure L ya pure C circuit: cos φ = 0 (zero power — wattless)
- Resonance pe: cos φ = 1 (kyunki Z = R)
6. Transformer — basic principle
Transformer mutual induction (EM induction) ke principle par kaam karta hai — sirf AC pe chalता hai, DC pe nahi. Do coils (primary Np turns, secondary Ns turns) ek common iron core par lipti hoti hain. Ideal transformer (100% efficient, koi energy loss nahi) me:
- Step-up: Ns > Np → voltage badhta hai, current ghat jaata hai (power transmission me use hota hai)
- Step-down: Ns < Np → voltage ghat jaata hai, current badh jaata hai (ghar ke appliances ke liye)
Real transformer me kuch power loss (copper loss, iron/eddy-current loss) hota hai isliye efficiency 100% se kam hoti hai — numerical me agar efficiency di ho to Output power = η × Input power use karo.
Poore Class 12 Physics ke handwritten colour notes
IITian & district toppers ke banaye short notes — revision-ready, diagram ke saath. Board se pehle poora syllabus 3 din me revise.
Exercise Questions — Solutions (Q1–Q16)
Q1. Ek 100Ω resistor 220V, 50Hz AC supply se connect hai. (a) Circuit me rms current nikalo. (b) Ek poore cycle me resistor ki net power consumption nikalo.
Diya hai: R = 100Ω, Vrms = 220V
Irms = Vrms/R = 220/100 = 2.2 A
Pure resistor circuit me power seedha VrmsIrms ya Irms²R se milta hai (cos φ = 1 kyunki phase difference zero hai):
P = Irms² R = (2.2)² × 100 = 484 W
Q2. (a) Ghar ki AC supply '220V, 50Hz' likhi hoti hai — yeh peak value hai ya rms value? (b) Is supply ki peak voltage kitni hai?
(a) Jab tak explicitly 'peak' na likha ho, AC voltage/current values hamesha rms hoti hain — kyunki voltmeter/ammeter rms hi measure karte hain aur yehi practical significance rakhta hai (heating effect ke through). Toh 220V yeh Vrms hai.
(b)
Vm = Vrms × √2 = 220 × 1.414 = 311.1 V
Q3. Ek 44 mH pure inductor 220V, 50Hz AC supply se joda gaya hai. Circuit me rms current nikalo.
Diya hai: L = 44 mH = 0.044 H, f = 50 Hz, Vrms = 220V
XL = 2πfL = 2 × 3.14 × 50 × 0.044 = 13.82 Ω
Irms = Vrms/XL = 220/13.82 = 15.92 A
Q4. Ek 60 μF pure capacitor 110V, 60Hz AC supply se joda gaya hai. Circuit me rms current nikalo.
Diya hai: C = 60 μF = 60×10⁻⁶ F, f = 60 Hz, Vrms = 110V
XC = 1/(2πfC) = 1/(2 × 3.14 × 60 × 60×10⁻⁶) = 44.21 Ω
Irms = Vrms/XC = 110/44.21 = 2.49 A
Q5. Q3 aur Q4 wale circuits (pure L aur pure C) me ek poore cycle me source se absorb hui net power kitni hai? Reason bhi do.
Dono circuits (pure inductor aur pure capacitor) me net power zero hoti hai.
P = Vrms Irms cos φ
Pure L me current voltage se 90° peeche rehta hai aur pure C me 90° aage — dono cases me phase difference φ = 90°, isliye cos φ = cos 90° = 0. Poore cycle me energy source se element me jaati hai aur wapas source me laut aati hai (energy store hoti hai magnetic/electric field me, consume nahi hoti) — isliye is current ko 'wattless current' kehte hain.
Q6. Ek series LCR circuit me R = 20Ω, L = 1.5 H, C = 35 μF hai, jo 200V (rms) variable-frequency AC source se joda gaya hai. (a) Resonant frequency nikalo. (b) Resonance par impedance aur rms current nikalo.
Diya hai: R = 20Ω, L = 1.5H, C = 35×10⁻⁶ F, Vrms = 200V
ω0 = 1/√(LC) = 1/√(1.5 × 35×10⁻⁶) = 1/√(5.25×10⁻⁵) = 138.0 rad/s
f0 = ω0/2π = 138.0/6.28 = 21.97 Hz ≈ 22 Hz
Resonance par XL = XC, isliye impedance sirf resistance ke barabar rehta hai:
Z = R = 20 Ω
Irms = Vrms/Z = 200/20 = 10 A
Q7. Ek series LCR circuit me R = 30Ω, L = 0.3H, C = 100 μF hai, 200V (rms), 50Hz supply se joda gaya. Impedance, rms current, power factor aur average power consumed nikalo.
Diya hai: R = 30Ω, L = 0.3H, C = 100×10⁻⁶ F, f = 50Hz, Vrms = 200V
XL = 2πfL = 2 × 3.14 × 50 × 0.3 = 94.25 Ω
XC = 1/(2πfC) = 1/(2 × 3.14 × 50 × 100×10⁻⁶) = 31.83 Ω
Yahan XL > XC, toh circuit inductive hai.
Z = √(R² + (XL−XC)²) = √(30² + (94.25−31.83)²) = √(900 + 3897) = 69.25 Ω
Irms = Vrms/Z = 200/69.25 = 2.89 A
cos φ = R/Z = 30/69.25 = 0.433 (φ ≈ 64.3°, current voltage se peeche)
P = Vrms Irms cos φ = 200 × 2.89 × 0.433 = 250.2 W
Q8. Ek series RL circuit me R = 8Ω, XL = 6Ω hai, 100V (rms) supply se joda gaya. Impedance, current, power factor aur average power nikalo.
Diya hai: R = 8Ω, XL = 6Ω (koi capacitor nahi, XC = 0), Vrms = 100V
Z = √(R² + XL²) = √(64 + 36) = √100 = 10 Ω
Irms = 100/10 = 10 A
cos φ = R/Z = 8/10 = 0.8
P = Vrms Irms cos φ = 100 × 10 × 0.8 = 800 W
Q9. Ek series LCR circuit me R = 15Ω, L = 0.2H, C = 200 μF hai, 50Hz supply se joda gaya. XL, XC aur Z nikalo, aur batao circuit inductive hai ya capacitive.
Diya hai: R = 15Ω, L = 0.2H, C = 200×10⁻⁶ F, f = 50Hz
XL = 2πfL = 2 × 3.14 × 50 × 0.2 = 62.83 Ω
XC = 1/(2πfC) = 1/(2 × 3.14 × 50 × 200×10⁻⁶) = 15.92 Ω
Z = √(R² + (XL−XC)²) = √(15² + (62.83−15.92)²) = √(225 + 2201) = 49.26 Ω
XL > XC hai, isliye circuit inductive hai — current voltage se peeche rahega.
Q10. Ek LC circuit me L = 2H aur C = 32 μF hai. Iska resonant (natural) frequency nikalo.
Diya hai: L = 2H, C = 32×10⁻⁶ F
ω0 = 1/√(LC) = 1/√(2 × 32×10⁻⁶) = 1/√(6.4×10⁻⁵) = 125 rad/s
f0 = ω0/2π = 125/6.28 = 19.9 Hz
Q11. Ek step-down transformer ka primary coil 2000 turns ka hai aur secondary 100 turns ka. Primary ko 220V AC diya jaata hai. Secondary voltage (ideal transformer, koi loss nahi) nikalo.
Diya hai: Np = 2000, Ns = 100, Vp = 220V
Vs/Vp = Ns/Np
Vs = Vp × (Ns/Np) = 220 × (100/2000) = 11 V
Kyunki Ns < Np, yeh step-down transformer hai — voltage kam kar diya.
Q12. Ek step-up transformer ka primary coil 500 turns ka hai aur 220V AC input hai. Agar secondary output 4400V chahiye, secondary turns kitne honge?
Diya hai: Np = 500, Vp = 220V, Vs = 4400V
Ns = Np × (Vs/Vp) = 500 × (4400/220) = 500 × 20 = 10,000 turns
Kyunki Ns > Np aur V badha, yeh step-up transformer hai.
Q13. Ek ideal transformer ka Np = 1000, Ns = 50 hai. Primary current Ip = 2A hai. Secondary current Is kitna hoga?
Diya hai: Np = 1000, Ns = 50, Ip = 2A. Ideal transformer me power conserve hoti hai (VpIp = VsIs), isliye current turns ratio ke ulta chalta hai:
Ip/Is = Ns/Np ⇒ Is = Ip × (Np/Ns)
Is = 2 × (1000/50) = 2 × 20 = 40 A
Note: yeh step-down transformer hai (Ns < Np) isliye voltage kam, current zyada — power constant rehta hai.
Q14. Ek transformer ka primary 220V, 5A input le raha hai, secondary output 11,000V hai, efficiency 90% hai. Secondary current Is nikalo.
Diya hai: Vp = 220V, Ip = 5A, Vs = 11,000V, η = 90% = 0.9
Input power Pin = Vp Ip = 220 × 5 = 1100 W
Output power Pout = η × Pin = 0.9 × 1100 = 990 W
Is = Pout/Vs = 990/11000 = 0.09 A
Real transformer me efficiency <100% hoti hai (copper loss + iron loss ki wajah se) isliye output power seedha input power ke barabar nahi hota — η factor lagana padta hai.
Q15. Ek AC source ka instantaneous voltage v = 311 sin(314t) volts diya hai. Peak voltage, rms voltage aur frequency nikalo.
Standard form v = Vm sin ωt se compare karke:
Vm = 311 V
Vrms = Vm/√2 = 311/1.414 = 220 V
ω = 314 rad/s ⇒ f = ω/2π = 314/6.28 = 50 Hz
Yehi Indian household AC supply ka standard equation hai.
Q16. Pure inductor aur pure capacitor circuit me current-voltage ka phase relationship samjhao — kaunsa aage, kaunsa peeche rehta hai, aur kyun.
Pure inductor circuit me current voltage se 90° peeche (lag) rehta hai — kyunki inductor changing current ka virodh karta hai (self-induced EMF, Lenz's law), toh current build hone me time lagta hai.
Pure capacitor circuit me current voltage se 90° aage (lead) rehta hai — kyunki capacitor pehle charge hona shuru karta hai (current turant flow hoti hai) aur voltage capacitor par charge accumulate hone ke saath dheere badhta hai.
LCR series circuit me overall phase difference in dono extremes (+90° aur −90°) ke beech kahin hota hai — depend karta hai XL aur XC me se kaun bada hai: XL > XC ho to circuit inductive (current lags), XC > XL ho to capacitive (current leads).
Important Equations — Ek Nazar Me
| Quantity | Formula |
|---|---|
| Instantaneous voltage/current | v = Vm sin ωt, i = Im sin ωt |
| RMS value | Vrms = Vm/√2, Irms = Im/√2 |
| Inductive reactance | XL = ωL = 2πfL |
| Capacitive reactance | XC = 1/ωC = 1/(2πfC) |
| Impedance (series LCR) | Z = √(R² + (XL − XC)²) |
| Current (series LCR) | Irms = Vrms/Z |
| Phase angle | tan φ = (XL − XC)/R |
| Resonant angular frequency | ω0 = 1/√(LC) |
| Resonant frequency | f0 = 1/(2π√(LC)) |
| Impedance at resonance | Z = R (minimum) |
| Average power | P = Vrms Irms cos φ |
| Power factor | cos φ = R/Z |
| Transformer relation (ideal) | Vs/Vp = Ns/Np = Ip/Is |
| Transformer efficiency | η = Pout/Pin |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Peak value ko power formula me daal dena. Power hamesha P = VrmsIrmscos φ se nikalti hai, na ki peak values se. Agar sawaal me Vm diya ho, pehle Vrms = Vm/√2 nikalo, phir power formula lagao — nahi to answer √2 factor se galat aayega.
- Impedance me sign error jab XC > XL ho. Z = √(R² + (XL−XC)²) me hamesha (XL−XC) ka square hota hai, isliye chahe XC bada ho ya XL, Z hamesha positive aata hai. Galti tab hoti hai jab students XC > XL ke case me impedance negative ya undefined maan lete hain.
- AC power calculation me power factor bhool jaana. P = VI (jaise DC me) likh dena galat hai — AC me hamesha cos φ ka factor lagta hai. Sirf pure resistive circuit (φ=0, cos φ=1) me P = VI chalega, LCR circuit me nahi.
- Pure L aur pure C ka phase relationship ulta karna. Yaad rakho: Inductor me current peeche (lag) rehta hai, Capacitor me current aage (lead) rehta hai. 'ELI the ICE man' trick use karo — confuse mat ho jaana especially LCR circuit ke net phase decide karte waqt.
- Step-up aur step-down transformer ratio ulta lagana. Step-up transformer me Ns > Np hota hai (voltage badhta hai), step-down me Ns < Np (voltage ghatta hai). Current hamesha voltage ke ulta chalta hai — step-up me current ghatta hai, step-down me current badhta hai. In dono ko mix mat karo.
- Resonance ko 'maximum voltage' samajh lena instead of 'maximum current'. Resonance par XL=XC hone se impedance minimum (=R) hoti hai, isliye current maximum hoti hai — voltage to source ne fix kiya hua hai, woh change nahi hota. Kai students resonance ko voltage-related maan lete hain jo galat hai.
Board-Style Important Questions
- 1 mark: Pure inductor se AC circuit me current, voltage se kitne phase se aage ya peeche rehta hai?
- 1 mark: Resonant angular frequency ω₀ ka formula likho L aur C ke terms me.
- 2 marks: Ek 200Ω resistor ko 220V, 50Hz AC supply se joda gaya hai. Rms current aur average power consumption nikalo.
- 2 marks: Power factor kya hota hai? Pure resistive circuit aur pure inductive circuit ke liye power factor ki value batao.
- 3 marks: Ek series LCR circuit me R, L, C diye gaye hain. Impedance ka formula derive kiye bina, dikhao ki circuit resonance par purely resistive kaise behave karta hai.
- 3 marks: Transformer ka basic principle samjhao aur ideal transformer ke liye Vs/Vp = Ns/Np = Ip/Is relation likho. Step-up aur step-down transformer me farak batao.
Aksar Poochhe Jaane Wale Sawaal
AC voltage/current ki value hamesha RMS hoti hai ya peak?
Jab tak explicitly 'peak' ya 'amplitude' na likha ho, koi bhi quoted AC value (jaise ghar ka '220V supply') hamesha RMS value hoti hai — kyunki voltmeter/ammeter RMS hi measure karte hain.
Resonance kya hota hai aur kab hota hai?
Series LCR circuit me jab X_L = X_C ho jaata hai, us frequency ko resonant frequency kehte hain. Us par impedance minimum (Z=R) hoti hai, current maximum hoti hai, aur circuit purely resistive jaisa behave karta hai (phase difference zero).
Pure L ya pure C circuit me power zero kyun hoti hai?
Kyunki in dono me current-voltage ke beech 90° ka phase difference hota hai, isliye cos φ = cos 90° = 0. Energy source se element me jaati hai aur wapas laut aati hai — net consumption zero. Isliye ise 'wattless current' kehte hain.
Transformer DC pe kaam kyun nahi karta?
Transformer mutual induction (changing magnetic flux) ke principle par kaam karta hai. DC me current constant hoti hai isliye flux change nahi hota, aur secondary me koi EMF induce nahi hoti. Isiliye transformer sirf AC ke saath kaam karta hai.
Power factor low hone ka kya matlab hai?
Low power factor (cos φ chhota) ka matlab hai circuit me actual power consumption kam hai jabki current zyada bah rahi hai — yeh transmission lines me energy loss badhata hai. Isliye industries me power factor correction (capacitor banks lagana) common practice hai.
Impedance aur resistance me kya farak hai?
Resistance (R) sirf resistor ki AC/DC dono me current rokne ki property hai — frequency independent. Impedance (Z) poore AC circuit (R, L, C sabka combined effect) ka opposition hai — frequency-dependent hoti hai kyunki X_L aur X_C dono frequency par depend karte hain.
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