Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL): Question bank

1.8.18 · D5 · Physics › Electromagnetism › Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL)

Shuru karne se pehle, is page par use kiye gaye har symbol ki definition, zero se:

$Q$ = electric charge = "electrical stuff" ki total matra (coulombs, C mein). Ise paani ki tarah socho. Dekho Conservation of Charge.
$I$ = current = har second ek point se guzarne wale charge $Q$ ki matra, isliye $I = \frac{d Q}{d t}$ (amperes, A mein). Socho paani ek pipe par bane nishaan se guzar raha hai.
$V$ = potential difference (voltage) = neeche diye hilly-landscape picture mein aap kitna girte ya chadhte ho (volts, V mein). Dekho Electric Potential.
$R$ = resistance = ek component current ko kitna rokta hai, Ohm's Law $V = I R$ se juda hua (ohms, $Ω$ mein).
$ε$ = EMF (electromotive force) = battery ka fixed voltage "push" jo wo apne $-$ terminal se $+$ terminal tak deti hai (volts, V mein). Picture mein ye pump ki height hai.
$r$ = internal resistance = ek real battery ke andar chhupa hua chhota resistance (ohms, $Ω$ mein).

Worked mini-derivations — har sign choice dekho

Traps se pehle, do complete step-by-step derivations taaki tumne dekha ho ki ek real circuit mein har sign aur step kyun hota hai.

Worked example Ek single loop ke around KVL — poori working

Setup:

ε = 12 V

R_{1} = 2 Ω

R_{2} = 4 Ω

ek loop mein.

I

find karo.

Ek direction assume karo — maano clockwise. Kyun: KVL ko ek reference direction chahiye; galat guess sirf ek negative number deta hai, jise hum tab reinterpret karte hain.
Loop clockwise walk karo battery ke $-$ terminal se shuru karke, har signed change add karte hue:
- Battery ke through $- \to +$ : tum pump par chadhte ho → $+ ε$ .
- Current direction mein $R_{1}$ ke through: tum slide se neeche jaate ho → $- I R_{1}$ .
- Current direction mein $R_{2}$ ke through: doosri slide → $- I R_{2}$ . $+ ε - I R_{1} - I R_{2} = 0$ Ye zero kyun sum karta hai: tum usi point par khatam hue jahan se chale the — same height (KVL). Har rise aur drop dekhne ke liye figure mein arrows follow karo.
Solve karo. $I = \frac{ε}{R _{1} + R _{2}} = \frac{12}{6} = 2 A$ . Positive hai, isliye hamara clockwise guess sahi tha.

True or false — justify karo

A resistor current use kar leta hai, isliye andar se jitna jaata hai usse kam bahar aata hai.

False. Ek resistor energy dissipate karta hai, charge nahi. KCL ke hisaab se ek series resistor mein same current andar jaata hai aur bahar aata hai; sirf voltage uske across girta hai.

KCL sirf steady state (DC) mein kaam karta hai.

Generally False. KCL as

\sum I = 0

ko node par

\frac{d Q}{d t} = 0

chahiye (

Q

= stored charge); agar node charge store kar sakta hai (ek capacitor plate) toh tumhe

\frac{d Q}{d t}

include karna hoga. Ordinary wire junctions ke liye ye hold karta hai kyunki ek bare junction charge hoard nahi kar sakta.

KVL kehta hai ek loop mein saare voltages ki magnitudes ka sum zero hota hai.

False. Ye kehta hai signed changes ka sum zero hota hai. Rises

+

hain, drops

-

hain (pump-and-slide figure dekho); magnitudes kabhi zero sum nahi karte jab tak loop trivial na ho.

Agar tumhare equations se current negative nikle, toh tumne algebra mein galti ki.

False. Negative answer sirf matlab hai tumhara assumed direction ulta tha; magnitude sahi hai, isliye arrow palat do.

Battery ke across parallel mein do resistors same current carry karte hain.

False. Unka same voltage share hota hai (wo same do nodes se connected hain), lekin currents alag hote hain:

I = V / R

, isliye chhota resistor zyada current carry karta hai.

KVL charge conservation ka statement hai.

False. KVL energy conservation se aata hai (potential single-valued hai, ek round trip same height par wapas aata hai). Charge conservation woh hai jo tumhe KCL deta hai.

Ek ideal battery ke liye, terminal voltage hamesha uske EMF

ε

ke barabar hota hai.

Ideal battery ke liye True (zero internal resistance

r

). Ek real battery

I r

andar drop karta hai, isliye terminal voltage

= ε - I r

hota hai aur current badhne par girta hai.

Tum KVL ek aisi loop par apply kar sakte ho jahan se ek gap hai jahan koi wire nahi hai.

Circuit terms mein True sirf tab agar tum gap ke across potential difference account karo (jaise ek open switch ya capacitor). KVL ke liye ek "loop" koi bhi closed potential path hai; ek open air gap simply uske across poora voltage difference carry karta hai.

Spot the error

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

"Main node mein aane wale currents add karunga aur zero ke barabar set karunga:

I_{1} + I_{2} + I_{3} = 0

, teeno andar flow kar rahe hain."

Agar teeno genuinely andar flow kar rahe hain, toh teeno positive nahi ho sakte aur zero sum nahi kar sakte (jab tak sab zero na hon). Real junctions mein kam se kam ek current bahar jaana chahiye; galti ye hai ki har arrow ko andar assign kiya (figure mein left node dekho).

"Apna loop walk karte hue mujhe current ke against ek resistor mila, isliye main

- I R

likhta hoon."

Galat sign. Current ke against tum slide par ulta chadhte ho → ye ek rise hai:

+ I R

likho. Current direction mein ye drop

- I R

hai.

"Battery

ε

ka rise deti hai chahe main kisi bhi direction mein uske through chalu."

Sign tumhari travel direction par depend karta hai.

-

+

tak ye

+ ε

(rise) hai;

+

-

tak ye

- ε

(drop) hai. Direction poore loop ke liye ek baar choose ki jaati hai, phir rigidly apply ki jaati hai.

"KCL fail ho gayi — mere node mein currents balance nahi kar rahe, isliye charge create ho raha hai."

Zyada sambhav hai tumne ek direction galat label ki ya ek branch miss kar li. KCL exact hai; apparent imbalance ek bookkeeping error signal karta hai, physics violation nahi.

"Ye do resistors parallel mein hain, isliye main unke resistances add kar deta hoon."

Parallel resistors reciprocals ki tarah add hote hain:

1/ R_{e q} = 1/ R_{1} + 1/ R_{2}

. Tum resistances directly sirf series mein add karte ho. Dekho Series and Parallel Resistors.

"Maine loop ke liye clockwise aur branch current ke liye clockwise choose kiya, isliye current definitely clockwise hai."

Loop direction sirf ek bookkeeping convention hai; ye physical current fix nahi karta. Sirf equations solve karna (aur result ka sign) tumhe sahi direction batata hai.

"Ek wire mein koi resistor nahi hai, isliye main usse KVL mein ignore kar sakta hoon."

Sirf ek ideal wire ke liye theek hai (

V = I R = 0

kyunki

R = 0

). Ye phir bhi nodes ko equal potential par connect karta hai — wire ko ignore karna sahi hai, lekin us wire ke nodes ko kabhi ignore mat karo.

Why questions

Charge ek ordinary node par pile up kyun nahi kar sakta?

Ek bare junction mein charge

Q

store karne ki koi capacity nahi hoti; koi bhi imbalance turant ek huge electric field banata jo charge ko wapas bahar dhakelta. Isliye effectively

\frac{d Q}{d t} = 0

hota hai aur andar = bahar.

KVL equivalent kyun hai ye kehne ke ki electrostatic field conservative hai?

Ek conservative field matlab

\oint E \cdot d l = 0

— kisi bhi closed loop mein charge per unit work zero hota hai, jo exactly potential changes ka sum zero hona hai. Dekho Electric Potential.

Hum current direction assume karne ke liye kyun keh sakte hain jab hum jaante bhi nahi?

Equations linear aur honest hain: galat guess same magnitude ka ek negative number produce karta hai. Assume karna humein consistently signs likhne deta hai; algebra khud ko correct kar leta hai.

Parallel branches ek hi voltage kyun share karte hain?

Unke endpoints same do nodes hain. Potential kisi point ki property hai, isliye un do points ke beech ka difference ek number hai jo unke beech har branch ko feel karna hoga.

Series-resistance formula

R_{e q} = R_{1} + R_{2}

KVL se kyun nikalta hai?

Ek current

I

dono se guzarta hai; KVL deta hai

ε = I R_{1} + I R_{2} = I (R_{1} + R_{2})

, isliye pair ek single resistor

R_{1} + R_{2}

ki tarah behave karta hai. Dekho Series and Parallel Resistors.

Hume ek two-loop circuit solve karne ke liye dono laws kyun chahiye, sirf ek nahi?

KCL akela unknown currents relate karta hai lekin unki values pin nahi karta; KVL akela nahi jaanta ki nodes par currents kaise split hote hain. Saath milkar ye saare unknowns ke liye enough independent equations dete hain — ye Mesh and Nodal Analysis ki basis hai.

Ek balanced Wheatstone Bridge mein middle branch mein zero current kyun hota hai?

Jab bridge ke midpoints par do node potentials equal hote hain, KVL middle branch ke across zero voltage deta hai, isliye Ohm's law se zero current — charge ko push karne ke liye koi round-trip height difference nahi.

Edge cases

Do connections ke saath ek node par KCL kya kehta hai jo series mein hain?

Current andar = current bahar — ek series path mein everywhere same

I

. Do elements wala ek "node" actually sirf ek wire par ek point hai.

KVL kya hoga agar ek loop mein ek capacitor ho jisme koi current flow nahi ho raha (steady state)?

Capacitor ek voltage carry karta hai lekin koi current nahi; KVL phir bhi un signed terms mein se ek ke roop mein us capacitor voltage ka use karke hold karta hai. Resistor drops zero hain kyunki

I = 0

Ek branch mein zero resistance — KVL kya force karta hai?

Us branch ke across voltage

I R = 0

hai, isliye uske do nodes equal potential par hain. Koi bhi current flow kar sakta hai; branch ek short hai.

Ek branch mein infinite resistance (open circuit) — KCL kya force karta hai?

Usse koi current flow nahi kar sakta, isliye us branch ki current zero hai. Poora loop voltage open gap ke across appear hota hai (KVL), kuch bhi nahi push karta.

Ek ideal wire se short-circuited ideal battery — laws kya predict karte hain?

KVL deta hai

ε = I R = I \cdot 0 = 0

, ek contradiction jab tak

ε = 0

na ho. Ye idealisation break down hai: real batteries mein internal resistance

r

hota hai, jo finite

I = ε / r

deta hai.

Ek node jahan saare currents momentarily zero hain — kya KCL satisfy hai?

Haan, trivially:

\sum 0 = 0

. KCL ek balance hai, aur khali balance bhi balance hai.

Pure ideal wire ka ek loop (koi source nahi, koi resistor nahi) — kitna current flow karta hai?

KVL

0 = 0

sum karta hai, koi constraint nahi deta jo current force kare. Koi EMF drive karne ke liye nahi, physical current zero hai; kuch pump nahi karta.

Agar tum same loop ko opposite direction mein traverse karo, kya KVL badalta hai?

Nahi. Har rise ek drop ban jaata hai aur har drop ek rise, isliye poora equation

- 1

se multiply hota hai:

0 = 0

phir bhi. Physics direction-independent hai.

Edge cases — jahan laws khud break hoti hain

KVL fail kab hona shuru hota hai?

Jab circuit signal wavelength

λ

ke comparable ho (high frequency / fast edges). Tab

\oint E \cdot d l = - \frac{d Φ _{B}}{d t} \neq = 0

(ek changing magnetic flux

Φ_{B}

EMF induce karta hai), isliye loop voltages ab zero sum nahi karte — tumhe instead Faraday's law use karni hogi.

KCL fail kab hona shuru hota hai, aur ise kya fix karta hai?

Fast, non-quasi-static regimes mein charge momentarily build up kar sakta hai (jaise capacitor plates ke beech). Maxwell ka fix hai displacement current

I_{d} = ε_{0} \frac{d Φ _{E}}{d t}

(ek changing electric flux

Φ_{E}

se); ise real current mein add karne se

\sum I = 0

full Ampère–Maxwell sense mein restore ho jaata hai.

Ek capacitor par displacement current kyun chahiye?

Plates ke beech gap ko physically koi charge cross nahi karta, phir bhi wires mein current flow karta hai. Plates ke beech changing electric field ek displacement current

I_{d} = ε_{0} \frac{d Φ _{E}}{d t}

carry karta hai jo exactly conduction current ko continue karta hai, KCL-style balance intact rakhta hai.

Ek slow (DC ya low-frequency) circuit ke liye hum ye sab safely ignore kyun kar sakte hain?

Quasi-static condition hold karta hai: loops ke through magnetic flux aur displacement current negligible hain, isliye plain KVL (

\sum V = 0

) aur KCL (

\sum I = 0

) excellent approximations hain — exactly is chapter ka regime.

Connections

Ohm's Law — upar har "why" quietly $V = I R$ use karta hai.
Conservation of Charge — reason ki koi current resistor mein "gayab" nahi hoti.
Conservation of Energy — kyun ek round trip net zero voltage deta hai.
Electric Potential — single-valuedness har KVL trap ke neeche hai.
Series and Parallel Resistors — parallel/series confusion items.
Wheatstone Bridge — balanced-bridge edge case.
Mesh and Nodal Analysis — kyun dono laws saath chahiye.