Apply Kirchhoff's Voltage Law (KVL)

WHAT is KVL?

WHY is it true? (Derivation from first principles)

Voltage between two points is defined as the work per unit charge to move a charge between them: $V_{AB} = \frac{W_{AB}}{q} = -\int_A^B \vec{E}\cdot d\vec{l}$

This means voltage (electric potential) is a conservative quantity — it depends only on the endpoints, not the path.

Now take a closed loop: start at point $A$ and return to point $A$ . The endpoints are the same point, so the total work per charge to go around must be zero (a conservative field does zero net work on a closed path): $\oint \vec{E}\cdot d\vec{l} = 0 \;\Longrightarrow\; \sum_k V_k = 0$

(Caveat: this holds when there is no changing magnetic flux threading the loop. In the presence of a time-varying $B$ -field, Faraday's law gives $\oint \vec E \cdot d\vec l = -d\Phi_B/dt \ne 0$ . For standard lumped DC/low-frequency circuits, KVL holds.)

HOW to apply it — the sign convention (the part everyone gets wrong)

Figure — Apply Kirchhoff's Voltage Law (KVL)

Worked Example 1 — Single loop, one source

A $12\,\text{V}$ battery drives current through two series resistors $R_1 = 4\,\Omega$ and $R_2 = 2\,\Omega$ . Find the current $I$ .

Worked example Solution

Traverse clockwise starting at the battery's $-$ terminal.

Cross battery ( $-$ to $+$ ): rise $\Rightarrow +12$ . Why: we exit the $+$ terminal, so potential goes up.
Cross $R_1$ (current enters $+$ side, we go with current): drop $\Rightarrow -IR_1$ . Why: resistors drop voltage in the direction of current.
Cross $R_2$ : drop $\Rightarrow -IR_2$ .

KVL: $+12 - IR_1 - IR_2 = 0$ $12 = I(4+2) = 6I \;\Rightarrow\; I = 2\,\text{A}$

Check with Feynman logic: voltage divides — $V_{R_1}=8\,$ V, $V_{R_2}=4\,$ V, and $8+4=12\,$ V ✓. The battery's push exactly equals the total drop.

Worked Example 2 — Two sources opposing

Loop with a $10\,\text{V}$ source, a $4\,\text{V}$ source opposing it, and one resistor $R = 3\,\Omega$ . Find $I$ .

Worked Example 3 — Verify with a known answer (Forecast-then-Verify)

Series: $9\,$ V source, $R_1=3\,\Omega$ , $R_2=6\,\Omega$ . Forecast: total $R = 9\,\Omega$ , so $I = 1\,$ A; drops $3\,$ V and $6\,$ V.

Steel-manning the classic mistakes

Active Recall

Recall Explain KVL to a 12-year-old (Feynman)

Voltage is like how high up you are. A battery is an escalator that lifts you up; a resistor is a slide that brings you down. If you walk in a full circle and come back to where you started, you must be at the same height — so all the ups and downs cancel out to zero. That's KVL!

What does KVL state about a closed loop?

The algebraic sum of all voltages around any closed loop equals zero.

What physical principle makes KVL true?

Electric potential is conservative — a closed path returns to the same potential, so net voltage change is zero.

When can KVL fail?

When a time-varying magnetic flux threads the loop (Faraday's law), giving a nonzero loop EMF.

If your assumed current comes out negative, what does it mean?

The actual current flows in the opposite direction; the magnitude is still correct.

Sign rule when traversing an element and exiting its

+

terminal?

Write it as a voltage rise,

+V

For a resistor, in which direction is the voltage drop?

In the direction of the current (current enters the

+

terminal).

In Example 1 (12 V, 4 Ω, 2 Ω series), what is the current?

I = 2

Does changing traversal direction change the answer?

No — it just multiplies the whole equation by

-1

; the solution is identical.

Connections

Kirchhoff's Current Law (KCL) — the companion law (charge conservation at nodes).
Ohm's Law — used to turn $V$ into $IR$ inside KVL equations.
Voltage Divider — a direct consequence of KVL in a series loop.
Conservation of Energy — KVL is energy conservation per unit charge.
Mesh Analysis — systematic application of KVL to multiple loops.
Faraday's Law — the reason for the "no changing flux" caveat.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, KVL ka core idea bilkul simple hai: kisi bhi closed loop mein saare voltages ka algebraic sum zero hota hai. Socho tum ek gol pahaadi trail pe chal rahe ho — jitna upar chadhoge utna hi neeche utarna padega, aur jab wapas starting point pe aaoge toh tumhari height ka net change zero hoga. Circuit mein voltage hi "height" (electric potential) hai. Battery tumhe upar le jaati hai (rise), resistor tumhe neeche laata hai (drop). Full loop poora karo, sab cancel out — total zero.

Sabse important cheez hai sign convention. Yahi pe zyaadatar log galti karte hain. Rule yaad rakho: jis terminal se tum exit karte ho, wahi tumhara sign hai. Agar $+$ terminal se bahar nikle toh $+V$ likho (rise), agar $-$ terminal se nikle toh $-V$ likho (drop). Resistor mein current jis direction mein jaata hai usi direction mein drop hota hai. Bas isko consistently follow karo poore loop mein.

Ek aur mast baat — agar tumne current ki direction galat guess kar li, tension mat lo! KVL self-correcting hai. Answer negative aayega, iska matlab actual current opposite direction mein bah raha hai, par magnitude bilkul sahi rahega. Isliye bindaas koi bhi direction pick karo aur solve kar do.

Yeh law kyun matter karta hai? Kyunki yeh basically energy conservation hai — per unit charge. Ohm's law ke saath combine karke tum series circuits, voltage dividers, aur mesh analysis sab solve kar sakte ho. Real hardware — power supplies, PCB traces, sab jagah engineer KVL laga ke voltages nikaalte hain. Isko strongly pakad lo, aage ka poora circuit analysis isi pe khada hai.

WHAT is KVL?

WHY is it true? (Derivation from first principles)

HOW to apply it — the sign convention (the part everyone gets wrong)

Worked Example 1 — Single loop, one source

Worked Example 2 — Two sources opposing

Worked Example 3 — Verify with a known answer (Forecast-then-Verify)

Steel-manning the classic mistakes

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Circuit Analysis Fundamentals

Connections