Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL)

1.8.18 · Physics › Electromagnetism

YE LAWS EXIST KYUN KARTI HAIN?

KCL KYUN? Charge create ya destroy nahi ho sakta, aur steady state mein yeh kisi ek point par accumulate nahi ho sakta (ek node mein charge store karne ki capacity nahi hoti). Isliye jo bhi current andar aati hai woh bahar nikal ni chahiye.

KVL KYUN? Electric potential ek point ki property hai. Agar tum koi bhi round-trip lo aur same point par wapas aao, toh potential wahi hogi jab tum gaye the — isliye loop mein total potential change zero hai. Yeh sirf yeh keh raha hai ki electrostatic field conservative hai ( $\oint E \cdot d l = 0$ steady fields ke liye).

KCL — Kirchhoff's Current Law

First principles se derivation. Charge conservation kehti hai ki ek tiny region mein charge change ki rate net current flowing in ke barabar hai: $\frac{d Q _{node}}{d t} = \sum I_{in} - \sum I_{out} .$ Steady state mein (ya ek idealized node ke liye jisme charge storage nahi hai), $\frac{d Q _{node}}{d t} = 0$ . Isliye $\sum I_{in} = \sum I_{out} ⟹ \sum I_{k} = 0.$ Yeh step kyun? $d Q / d t = 0$ set karna hi KCL ka poora content hai — ek node battery ya capacitor nahi hai; yeh charge hoard nahi kar sakta.

KVL — Kirchhoff's Voltage Law

First principles se derivation. Potential har point par single-valued hoti hai. Point $A$ se shuru karo jahan potential $V_{A}$ hai, elements se guzro gain/lose karte potential, aur $A$ par wapas aao: $V_{A} + (Δ V_{1}) + (Δ V_{2}) + \dots + (Δ V_{n}) = V_{A} .$ $V_{A}$ subtract karo: $\sum_{k} Δ V_{k} = 0.$ Yeh step kyun? Tum same point par wapas aate ho, isliye uski potential change nahi hui hai — trip ka net zero hona chahiye.

Sign conventions (woh part jo sabko confuse karta hai)

Apni chosen direction mein loop mein ghoomte hue:

Resistor se current ki direction mein guzarna → potential drop hoti hai $I R$ se (likho $- I R$ ).
Resistor se current ke against guzarna → potential rise hoti hai ( $+ I R$ ).
Battery mein − se + terminal ki taraf guzarna → potential rise hoti hai ( $+ ε$ ).
Battery mein + se − ki taraf guzarna → potential drop hoti hai ( $- ε$ ).

Figure — Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL)

Worked Example 1 — Single loop

Ek loop mein EMF $ε = 12 V$ aur do resistors $R_{1} = 2 Ω$ , $R_{2} = 4 Ω$ series mein hain. Current $I$ find karo.

Current direction assume karo, maano clockwise. Kyun? KVL ko ek assumed direction chahiye; galat guess sirf negative answer dega.
KVL apply karo battery ke − terminal se clockwise jaate hue: $+ ε - I R_{1} - I R_{2} = 0.$ Yeh step kyun? Battery − se + rise hai; current ki direction mein resistors drops hain.
Solve karo: $I = \frac{ε}{R _{1} + R _{2}} = \frac{12}{6} = 2 A .$ Yeh step kyun? Algebra — aur yeh series-resistance result reproduce karta hai, ek accha sanity check.

Worked Example 2 — Teen branches wala Node (KCL)

Currents $I_{1} = 3 A$ aur $I_{2} = 2 A$ ek node mein andar flow karti hain; $I_{3}$ bahar flow karti hai. $I_{3}$ find karo.

KCL: $I_{1} + I_{2} - I_{3} = 0$ . Kyun? In-currents positive, out-current negative.
$I_{3} = 3 + 2 = 5 A$ . Yeh step kyun? Andar aaya charge bahar jaane wale charge ke barabar hona chahiye — kuch gayab nahi ho sakta.

Worked Example 3 — Two-loop circuit (dono laws saath mein)

Battery $ε = 10 V$ ek node ko feed karti hai jo $R_{1} = 5 Ω$ aur $R_{2} = 20 Ω$ mein parallel mein split ho jaata hai, phir wapas aata hai. Branch currents find karo.

Top node par KCL: $I = I_{1} + I_{2}$ jahan $I_{1}$ $R_{1}$ se, $I_{2}$ $R_{2}$ se. Kyun? Total current branches ke beech split hoti hai.
KVL, $R_{1}$ wala loop: $ε - I_{1} R_{1} = 0 \Rightarrow I_{1} = 10/5 = 2 A$ .
KVL, $R_{2}$ wala loop: $ε - I_{2} R_{2} = 0 \Rightarrow I_{2} = 10/20 = 0.5 A$ . Kyun? Dono branches par same voltage $ε$ hoti hai (parallel).
KCL par wapas: $I = 2 + 0.5 = 2.5 A$ . Check: $R_{eq} = (5 \cdot 20) / (25) = 4 Ω$ , isliye $I = 10/4 = 2.5 A$ . ✓

Common mistake Classic errors ko steel-man karte hain

Mistake 1: "Current resistor mein use ho jaati hai, isliye kam nikaali hoti hai." Kyun sahi lagta hai: Energy use hoti hai, isliye lagta hai current bhi honi chahiye. Fix: Ek resistor energy consume karta hai, charge nahi. KCL: same current ek series resistor mein enter aur exit karti hai. Voltage drop hoti hai, current nahi.

Mistake 2: KVL mein sign flips bhool jaana. Kyun sahi lagta hai: "Saari voltages add karna" natural lagta hai. Fix: Tum signed changes sum kar rahe ho. Ek loop direction choose karo aur strictly apply karo: rise $+$ , drop $-$ . Jo drop tum ulti direction mein traverse karo woh rise ban jaati hai.

Mistake 3: Negative current aane par panic karna. Kyun sahi lagta hai: Negative current "galat lagti hai." Fix: Iska matlab bas yeh hai ki tumhari assumed direction ulti thi. Magnitude sahi hai — arrow ulta kar do.

Recall Feynman: 12-saal ke bache ko explain karo

Wires ko paani ke pipes samjho. KCL: T-junction par, andar aane wala saara paani bahar nikalna chahiye — paani gayab nahi hota ya pile up nahi karta. KVL: socho voltage height hai ek pahadi path par. Battery ek pump hai jo tumhe upar uthata hai; resistors slides hain jo tumhe niche laate hain. Pahadi ke around ek pura loop ghoomke wahi aao jahan se shuru kiya — isliye saare ups aur downs bilkul zero ho jaane chahiye. Bas itna hai: paani conserved hai, aur pahadi par round trip karke same height par wapas aate hain.

Flashcards

KCL ke peeche kaun sa physical conservation law hai?

Electric charge ka conservation (koi charge node par accumulate nahi hota).

KVL ke peeche kaun sa physical conservation law hai?

Energy ka conservation (potential single-valued hoti hai; round trip same potential par wapas laata hai).

KCL mathematically state karo.

\sum_{k} I_{k} = 0

kisi bhi node par (in-currents positive, out negative); equivalently

\sum I_{in} = \sum I_{o u t}

KVL mathematically state karo.

\sum_{k} V_{k} = 0

kisi bhi closed loop mein (rises positive, drops negative).

Current ki direction mein resistor se guzarne par potential rise hoti hai ya fall?

Yeh

I R

se fall hoti hai (ek drop,

- I R

Battery mein − se + terminal ki taraf guzarne par rise ya fall?

Potential

ε

se rise hoti hai (

+ ε

Agar KVL se negative current aaye, toh iska kya matlab hai?

Tumhari assumed current direction ulti thi; magnitude sahi hai, bas arrow reverse karo.

Kya current resistor mein "use ho jaati hai"?

Nahi — KCL current ko series mein same rakhti hai; sirf energy (voltage) dissipate hoti hai.

EMF ke across parallel mein do resistors ke liye, kya same hota hai aur kya differ karta hai?

Dono par same voltage; currents resistance ke inversely differ karti hain.

Connections

Ohm's Law — $V = I R$ provide karta hai, KCL/KVL ke saath combine karke koi bhi DC network solve karta hai.
Series and Parallel Resistors — directly KCL + KVL se derive hote hain.
Conservation of Charge — KCL ka basis.
Conservation of Energy — KVL ka basis.
Electric Potential — single-valuedness KVL ko justify karti hai.
Wheatstone Bridge — dono laws simultaneously require karne wala application.
Mesh and Nodal Analysis — KVL aur KCL par built systematic algorithms.