Electronics · Math shelf · Optional — for the curious · build 2026.09.23-1644

P = IV

The law

P = I × V. Power equals current times voltage drop — measured in watts (W), the rate energy is being delivered. Every part in a circuit is converting energy at some number of watts, right now.

What power is

Energy is the stuff (measured in joules); power is how fast it moves (joules per second = watts). A part with current flowing through it and voltage dropping across it is taking energy out of the circuit at P = I × V watts — the LED turns its share into light, the resistor turns its share into heat. (Why the units multiply out so neatly is on the What's a Coulomb? sheet — it's arithmetic, not coincidence.)

Three ways to hold it

You knowYou wantUse
current and voltagethe powerP = I × V
power and voltagethe currentI = P ÷ V
power and currentthe voltageV = P ÷ I
P I V
Same triangle trick as V = IR: cover the one you want. Units shortcut, too: mA × V = mW (milliwatts).

Worked: the energy books of one LED loop

The classic loop — 5 V supply, red LED (2.0 V drop), 220 Ω resistor (3.0 V drop), 14 mA everywhere:

PartP = I × VBecomes
LED14 mA × 2.0 V = 28 mWlight (and a little warmth)
Resistor14 mA × 3.0 V = 42 mWheat, on purpose
Supply delivers14 mA × 5.0 V = 70 mW28 + 42 ✓ — the books balance

Notice the quiet scandal: the resistor throws away more power than the LED uses. That's the price of the simple circuit — the resistor's job is to burn off the extra push as heat, and 42 mW of heat is so little you'll never feel it.

Why kit resistors have a rating

Our resistors are rated ¼ W = 250 mW — the most heat they can shed before they cook. The classic loop's 42 mW isn't close. But watch what happens as resistance falls across the full 5 V:

Straight across 5 VCurrent (I = V ÷ R)Power (P = I × V)Verdict
1 kΩ5 mA25 mWcool
220 Ω≈ 23 mA≈ 114 mWwarm, fine
100 Ω50 mA250 mWat the limit — noticeably hot
a bare wire (≈ 0 Ω)everything the supply hasall of it, in the wirethis is a short — this is why shorts burn

Smaller resistance means more current and the same voltage — power climbs fast. A short circuit is the limit of that story: huge current, all the supply's power converted to heat in a wire that was never meant to shed any.

For scale

Two shortcuts, for the very curious

Substitute Ohm's law (V = IR) into P = I × V and two more forms fall out: P = I² × R (swap V for IR) and P = V² ÷ R (swap I for V÷R). Handy when you only know two things about a resistor — engineers reach for all three without thinking.

Practice

Answers at the bottom. The currents come from V = IR — that sheet first if these feel unfair.

  1. The blue-LED loop: 3.2 V drop, 8 mA. What power does the blue LED take?
  2. Same loop: the 220 Ω resistor has the other 1.8 V. Its power?
  3. A 1 kΩ resistor sits straight across the 5 V rails. Power? Is ¼ W enough?
  4. What resistance straight across 5 V would hit the ¼ W limit exactly? (Two steps: I = P ÷ V first, then R = V ÷ I.)

Answers

1. 8 mA × 3.2 V ≈ 26 mW. 2. 8 mA × 1.8 V ≈ 14 mW — this loop wastes less than the red one; the blue LED claims more of the 5 V. 3. I = 5 mA, P = 25 mW — a tenth of the rating; fine. 4. I = 0.25 ÷ 5 = 50 mA, then R = 5 ÷ 0.05 = 100 Ω — which is why nothing smaller than 220 Ω belongs across the rails.