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

WHY do these laws exist?

WHY KCL? Charge cannot be created or destroyed, and in steady state it cannot accumulate at a single point (a node has no capacity to store charge). So whatever current flows in must flow out.

WHY KVL? Electric potential is a property of a point. If you take any round-trip and come back to the same point, the potential is the same as when you left — so the total potential change around the loop is zero. This is just saying the electrostatic field is conservative ($\oint \vec{E}\cdot d\vec{l}=0$ for steady fields).

KCL — Kirchhoff's Current Law

Derivation from first principles. Charge conservation says the rate of charge change in a tiny region equals net current flowing in: $\frac{dQ_{\text{node}}}{dt} = \sum I_{\text{in}} - \sum I_{\text{out}}.$ In steady state (or for an idealized node with no charge storage), $\dfrac{dQ_{\text{node}}}{dt}=0$. Therefore $\sum I_{\text{in}} = \sum I_{\text{out}} \quad\Longrightarrow\quad \sum I_k = 0.$ Why this step? Setting $dQ/dt=0$ is the whole content of KCL — a node is not a battery or capacitor; it can't hoard charge.

KVL — Kirchhoff's Voltage Law

Derivation from first principles. Potential is single-valued at each point. Start at point $A$ with potential $V_A$, walk through elements gaining/losing potential, and return to $A$: $V_A + (\Delta V_1) + (\Delta V_2) + \dots + (\Delta V_n) = V_A.$ Subtract $V_A$: $\sum_k \Delta V_k = 0.$ Why this step? You return to the same point, so its potential hasn't changed — the trip must net to zero.

Sign conventions (the part that trips everyone)

Travel around the loop in a chosen direction:

• Through a resistor in the direction of current → potential drops by $IR$ (write $-IR$).
• Through a resistor against the current → potential rises ($+IR$).
• Through a battery from − to + terminal → potential rises ($+\varepsilon$).
• Through a battery from + to − → potential drops ($-\varepsilon$).

Worked Example 1 — Single loop

A loop with EMF $\varepsilon = 12\text{ V}$ and two resistors $R_1 = 2\ \Omega$, $R_2 = 4\ \Omega$ in series. Find the current $I$.

1. Assume a current direction, say clockwise. Why? KVL needs an assumed direction; a wrong guess just gives a negative answer.
2. Apply KVL going clockwise starting from the battery's − terminal: $+\varepsilon - IR_1 - IR_2 = 0.$ Why this step? Battery − to + is a rise; resistors in the current direction are drops.
3. Solve: $I = \frac{\varepsilon}{R_1+R_2} = \frac{12}{6} = 2\text{ A}.$ Why this step? Algebra — and it reproduces the series-resistance result, a good sanity check.

Worked Example 2 — Node with three branches (KCL)

Currents $I_1 = 3\text{ A}$ and $I_2 = 2\text{ A}$ flow into a node; $I_3$ flows out. Find $I_3$.

1. KCL: $I_1 + I_2 - I_3 = 0$. Why? In-currents positive, out-current negative.
2. $I_3 = 3 + 2 = 5\text{ A}$. Why this step? Charge in must equal charge out — none can vanish.

Worked Example 3 — Two-loop circuit (both laws together)

Battery $\varepsilon = 10\text{ V}$ feeds a node that splits into $R_1 = 5\ \Omega$ and $R_2 = 20\ \Omega$ in parallel, then returns. Find branch currents.

1. KCL at top node: $I = I_1 + I_2$ where $I_1$ through $R_1$, $I_2$ through $R_2$. Why? Total current splits between the branches.
2. KVL, loop through $R_1$: $\varepsilon - I_1 R_1 = 0 \Rightarrow I_1 = 10/5 = 2\text{ A}$.
3. KVL, loop through $R_2$: $\varepsilon - I_2 R_2 = 0 \Rightarrow I_2 = 10/20 = 0.5\text{ A}$. Why? Both branches see the same voltage $\varepsilon$ across them (parallel).
4. Back to KCL: $I = 2 + 0.5 = 2.5\text{ A}$. Check: $R_{\text{eq}} = (5\cdot20)/(25) = 4\ \Omega$, so $I = 10/4 = 2.5\text{ A}$. ✓

Flashcards

Connections

• Ohm's Law — supplies $V=IR$, combined with KCL/KVL solves any DC network.
• Series and Parallel Resistors — derived directly from KCL + KVL.
• Conservation of Charge — the basis of KCL.
• Conservation of Energy — the basis of KVL.
• Electric Potential — single-valuedness justifies KVL.
• Wheatstone Bridge — application requiring both laws simultaneously.
• Mesh and Nodal Analysis — systematic algorithms built on KVL and KCL.

Concept Map

Hinglish (regional understanding)