Why follow it all the way to dollars?
Because electricity is the one thing in the house you pay for without ever seeing it. A litre of milk has a size; a kilowatt-hour does not, and the numbers on an appliance — 120 V, 12.5 A, 1500 W — mean nothing until you can turn them into energy and cost.
The chain does exactly that. The label gives volts and amps, or watts. Multiply by the time it runs and you have the energy; multiply by the price of a kilowatt-hour and you have what it costs to run. It also explains things that otherwise look random: why a kettle and a toaster on one circuit trip the breaker, why an LED bulb pays for itself, and why the fridge with the bigger price tag can cost less to own.
Where it turns up
- The EnerGuide label — major appliances sold in Canada, such as fridges and washers, show their yearly energy use in kWh, so models can be compared before you buy
- The breaker panel — a 15 A breaker on a 120 V circuit allows at most 15 × 120 = 1800 W on that circuit at once
- A phone charger — one marked 5 V and 2 A supplies 5 × 2 = 10 W
- An electric car — a 60 kWh battery charging at 7.2 kW takes about 60 ÷ 7.2 ≈ 8 hours to fill from nearly empty
- 1. The words, first
- 2. The chain, in one line
- 3. Link 1: Ohm's law
- 4. Link 2: power
- 5. Link 3: energy is power × time
- 6. Link 4: from kilowatt-hours to dollars
- 7. Efficiency: paying for what you did not get
The words, first
The idea: Power is a rate and energy is an amount. Every unit trap in this toolkit comes from mixing those two up.
| Word | What it means |
|---|---|
| Current, voltage, resistance | Current (I) is how fast charge flows, in amperes (A). Voltage (V) is the push, in volts. Resistance (R) opposes the current, in ohms (Ω). |
| Power (P) | How fast energy is used, in watts (W). 1 W is 1 joule every second; 1 kilowatt (kW) is 1000 W. |
| Energy (E) | How much is used, in joules (J) or kilowatt-hours (kWh). |
| Kilowatt-hour | The energy used by 1 kW for 1 hour: 1000 W × 3600 s = 3 600 000 J. It is what the meter outside your house counts. |
| Circuit breaker | A switch that cuts off a circuit automatically when the current gets too high. |
| Standby power | Power a device draws while switched off but still plugged in. |
| Efficiency | Useful energy out ÷ total energy in × 100%. |
The chain, in one line
The idea: Each formula feeds the next. Knowing which link you are on tells you which formula comes next.
V = I × R → P = I × V → E = P × t → cost = E (kWh) × price
Questions start at different links. A label that gives watts starts you at P. A circuit diagram with a resistance starts you at R. A bill starts you at the end and asks you to work backwards. The formulas are not the hard part; knowing which link you are on, and keeping the units straight as you move between links, is.
Link 1: Ohm's law
The idea: The voltage and the resistance decide how much current flows.
Worked. A kettle's heating element has a resistance of 9.6 Ω and is plugged into 120 V. I = V ÷ R = 120 ÷ 9.6 = 12.5 A.
Lower resistance, more current, at the same voltage. That surprises people: the element that makes the most heat has a low resistance, because it lets a large current through, and the next link shows power grows with current.
Why a short circuit is dangerous. When a bare wire touches another, the path has almost no resistance, so I = V ÷ R becomes enormous. The wire heats fast, and the breaker is there to cut the circuit before it starts a fire.
Link 2: power
The idea: Power is how fast the energy is used: current times voltage.
Worked. The same kettle: P = I × V = 12.5 × 120 = 1500 W, or 1.5 kW — the number printed on its base.
Worked: the tripped breaker. A kitchen circuit has a 15 A breaker. The kettle draws 12.5 A. A 900 W toaster draws I = P ÷ V = 900 ÷ 120 = 7.5 A. Together: 12.5 + 7.5 = 20 A, more than 15 A, so the breaker trips. Everything plugged into a house is in parallel, so the currents add — just as the branch currents did in a Unit D circuit.
Link 3: energy is power × time
The idea: The bill charges for energy, not power. A small device left on for a long time can use more than a big one used briefly.
E (kWh) = P (kW) × t (h) · E (J) = P (W) × t (s) · 1 kWh = 3 600 000 J
Worked. The 1.5 kW kettle runs for 3 minutes, which is 3 ÷ 60 = 0.05 h: E = 1.5 × 0.05 = 0.075 kWh. In joules, 1500 W × 180 s = 270 000 J, and 270 000 ÷ 3 600 000 = 0.075 — the same answer.
Worked: small but always on. A 10 W LED bulb on 5 hours a day for a year uses 0.010 kW × 5 h × 365 = 18.25 kWh. A device drawing 5 W on standby, all day every day, uses 0.005 × 24 × 365 = 43.8 kWh — more than twice the bulb, while doing nothing useful.
The two unit traps. Watts × hours gives watt-hours, so divide by 1000 for kilowatt-hours. And minutes are not hours: 3 minutes is 0.05 h, not 0.3 h.
Link 4: from kilowatt-hours to dollars
The idea: Multiply the energy by the price. The price changes; the method does not.
cost = energy (kWh) × price per kWh
Prices differ between plans and change over time, so a question will give you one. These examples use 15¢ per kWh.
Worked: two bulbs. A 60 W incandescent bulb and a 10 W LED giving about the same light, each on 4 hours a day for a year:
| Bulb | Energy in a year | Cost |
|---|---|---|
| 60 W incandescent | 0.060 × 4 × 365 = 87.6 kWh | $13.14 |
| 10 W LED | 0.010 × 4 × 365 = 14.6 kWh | $2.19 |
The LED saves $13.14 − $2.19 = $10.95 a year, for one bulb.
Worked: the whole chain from a label. A space heater's label reads 120 V, 12.5 A, and it runs 6 hours a day for 30 days.
- Power: P = 12.5 × 120 = 1500 W = 1.5 kW.
- Energy: E = 1.5 kW × 6 h × 30 = 270 kWh.
- Cost: 270 × $0.15 = $40.50 for the month.
A real bill is more than this. In Alberta, charges for delivering the power and other fees are added to the energy charge, and together they can be as large as the energy charge itself. The chain gives you the part you control by switching things off.
Efficiency: paying for what you did not get
The idea: Every joule you pay for gets used, but not every joule does the job you wanted. Efficiency is the share that did.
efficiency = useful energy out ÷ total energy in × 100%
Worked. The kettle uses 270 000 J in its 3 minutes, and the water gains 229 500 J. Efficiency = 229 500 ÷ 270 000 × 100% = 85%. The other 40 500 J warmed the kettle and the kitchen air.
Why LED bulbs cost less to run. A 60 W incandescent bulb is only about 5% efficient at making light: each second, 3 J of light and 57 J of heat. The LED makes about the same light from roughly 10 W, which is where the $10.95 comes from.
When heat is the point. For a space heater, thermal energy is the useful output, so it is close to 100% efficient. Efficiency always depends on what you wanted the device to do.
Nothing goes over 100%. An answer above 100% means the fraction is upside down. The input is always the bigger number.