Kirchhoff's Voltage Law

Equivalent forms

\mathcal{E} = \sum_k I R_k

\oint \vec{E}\cdot d\vec{l} = 0

A statement about energy conservation that turns a tangle of wires into a few solvable linear equations.

Symbol	Name	SI unit	Dimension	Range
$\mathcal{E}$	EMF Source voltage driving the loop	V	M·L²·T⁻³·I⁻¹	1 – 24
$R_1$	Resistance 1 First series resistor	Ω	M·L²·T⁻³·I⁻²	1 – 20
$R_2$	Resistance 2 Second series resistor	Ω	M·L²·T⁻³·I⁻²	1 – 20
$I$	Loop currentoutput Current circulating in the loop	A	I	0 – 10

Unit systems

Symbol	SI
$\mathcal{E}$	V
$R_1$	Ω
$R_2$	Ω
$I$	A

Where it holds

Exact for lumped circuits where the loop is small compared to the signal wavelength (quasi-static regime). Fails when changing magnetic flux threads the loop (then

\oint E\cdot dl = -d\Phi /dt

, a finite EMF) or at frequencies where distributed inductance/capacitance matter.

Dimensional analysis

[\mathcal ${E}]/([R_1]+[R_2]) = V/\Omega = A \checkmark$ (each term $in \Sigma V_k$ carries units of volts)

Discovery

Gustav Kirchhoff · 1845

While still a 21-year-old student, Kirchhoff generalized Ohm's law to networks of arbitrary loops and junctions, publishing two rules that still anchor every circuit-analysis course. They are direct consequences of energy and charge conservation.

Try this

Walk all the way around any circuit loop and you end up where you started — at the same voltage. What does that simple fact let you calculate?

A 12 V battery drives a series loop with a 4 Ω and an 8 Ω resistor. Use KVL to find the current and the voltage across each resistor.

Research status: stable

Real-world applications

Solving multi-loop networks
Designing voltage dividers
Battery-pack balancing
SPICE and every circuit simulator

Common misconceptions

KVL always holds — it breaks when a time-varying magnetic flux links the loop (transformers, inductive pickups)
Voltage 'gets used up' in wires — ideal wires have zero drop; drops live in the resistive elements

Experimental verification

Place a voltmeter across each element of a series loop and sum the readings with sign — the total is zero to meter precision. Demonstrated daily in every undergraduate lab.

Derivation

The electrostatic field is conservative, so the line integral of E around any closed path is zero:

\oint E\cdot dl = 0

.

Splitting the loop into circuit elements, the integral of E across each element is its voltage rise or drop, so the signed sum of all element voltages around the loop is zero.

Limiting cases

R_1 + R_2 \to 0

⟶

I \to \infty

A short across the source draws unbounded current (limited only by internal resistance).

R_2 \to \infty

⟶

I \to 0

An open branch stops the loop current; the full EMF appears across the gap.

What if…

What if a magnetic field through the loop changes in time?

KVL in the simple form fails; the loop gains an induced $EMF -d\Phi /dt$ and you must use Faraday's law.

What if you add a third resistor in series?

Same rule — the current drops to \mathcal{E}/(R_1+R_2+R_3) and the drops still sum to the EMF.

1

12 V across 4 Ω + 8 Ω

Given ·

\mathcal{E}:: 12
R 1:: 4
R 2:: 8

Find · I, V_1, V_2

Steps

KVL: \mathcal ${E} - I R_1 - I R_2 = 0$
$I =$ \mathcal ${E}/(R_1 + R_2) = 12/12 = 1$ A
$V_1 = I R_1 = 4\,\mathrm{V}$ , $V_2 = I R_2 = 8\,\mathrm{V}$ ; their sum equals the source $\checkmark$

Result ·

I = 1

A; drops of 4 V and 8 V sum to the 12 V source.

kirchhoff current lawOhm's Law→resistors series parallel