Foundations — Apply Kirchhoff's Voltage Law (KVL)

Before you can apply KVL, you have to be able to read every mark on the page. This note builds each symbol from nothing — a smart 12-year-old who has never seen a circuit should finish able to read the parent note line by line.

1. Charge — the stuff that flows

Why the topic needs it: every voltage in KVL is defined as work per unit charge. Without $q$ there is no voltage to sum.

2. Current — the rate of flow

The slash in $I=q/t$ is just division: "charge shared out over the time it took." We use division (not subtraction) because we want a rate — an amount per second — and "per" always means divide.

Why the topic needs it: in KVL we write resistor voltages as $IR$, so $I$ must be understood first. An arrow on the wire shows current's direction — the way we assume positive charge flows.

3. Voltage — electrical height

Here $W$ is work (energy spent, in joules, J) and $q$ is the charge from step 1. Dividing work by charge answers: "how much energy per unit of charge?" — the "steepness of the electrical hill."

Why the topic needs it: KVL sums voltages. This "height" picture is the whole intuition — the parent note's hiking-trail story is exactly this landscape.

4. The two element types: source and resistor

The little $+$ and $-$ marks are terminal signs: they tell you which end is the "high ground." You will read them constantly when walking a loop.

5. Ohm's Law — turning current into voltage

Why the topic needs it: KVL gives an equation full of resistor voltages. Ohm's Law lets us replace each unknown resistor voltage $V$ by the known expression $IR$, so we can solve for the current $I$.

6. The closed loop and the traversal arrow

Why the topic needs it: KVL is a statement about a closed loop. "Return to the same point ⇒ same height ⇒ voltages sum to zero" is the entire proof.

7. The summation symbol $\sum$

The little $k=1$ underneath is a counter that ticks through each element; the $n$ on top is how many elements there are. It is only a compact way to say "$V_1+V_2+\cdots +V_n$."

8. The conservation idea (the deepest root)

Why the topic needs it: because voltage is conservative, a closed loop (start = end) must have zero total voltage change. That single fact is KVL.

9. Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Apply Kirchhoff's Voltage Law (KVL) — the topic these foundations serve.
• Ohm's Law — converts each resistor voltage into $IR$.
• Conservation of Energy — why voltage is conservative and the loop sum vanishes.
• Kirchhoff's Current Law (KCL) — the companion law built on charge (step 1).