Energy and Kilowatt-Hours

In one sentence: Energy is power multiplied by time, so a heater that uses 2 kilowatts for 3 hours uses 6 kilowatt-hours of energy.

$E = P \, t$

Read it out loud: “E equals P times t. Energy equals power times time.”

What each symbol means

Symbol	Meaning
$E$	Energy, the total amount of electricity used or delivered. Measured in joules (J), or in watt-hours (Wh) and kilowatt-hours (kWh) for electricity bills.
$P$	Power, how fast energy is used, measured in watts (W).
$t$	Time, how long the device runs, measured in seconds (s) or hours (h), as long as the units match the power.

What the formula is doing

Energy is the total amount of work electricity can do or the total heat it can make. The SI unit of energy is the joule (J). One joule is the energy used by a 1 W device running for one second.

Power $P$ is how fast energy is used, in watts (W). Time $t$ is how long the device runs. Multiply power by time and you get the total energy used: $E = P\,t$ . If the time doubles at the same power, the energy doubles too.

A watt-hour (Wh) is energy measured in a bigger unit: one watt running for one hour. One hour is 3600 seconds, so 1 Wh = 3600 J. A kilowatt-hour (kWh) is 1000 Wh. Power companies bill in kilowatt-hours, because a joule is far too small a unit to count on a bill: 1 kWh is 3.6 million joules, or 3.6 MJ.

Watts and kilowatt-hours answer different questions. Watts describe a rate, like speed. Kilowatt-hours describe a total, like the distance traveled. A 2000 W heater uses 2 kWh in one hour and 6 kWh in three hours. Heaters run on mains voltage, so do not open them, and keep them away from water.

An everyday picture

Think of a tap filling a bucket. The flow rate, in liters per minute, is like power, because it says how fast water arrives. The amount of water in the bucket is like energy. Run the tap twice as long and the bucket holds twice as much. A kilowatt-hour is a standard-size bucket of electricity, and the power company counts how many of those buckets you used.

Worked example: A heater, an LED bulb and what they cost

A 2000 W heater runs for 3 hours on a mains supply. A 10 W LED bulb runs for 10 hours. For the cost, use an example price of 0.15 dollars per kWh. This price is only for the arithmetic: it is not a real tariff, and real prices vary by country and provider. Heaters run on mains voltage, so do not open one to check it.

Heater energy: $E = P\,t = 2000\ \text{W} \times 3\ \text{h} = 6000\ \text{Wh} = 6\ \text{kWh}$ .
Heater cost at the example price: $6\ \text{kWh} \times 0.15\ \frac{\text{dollars}}{\text{kWh}} = 0.90\ \text{dollars}$ .
LED energy: $E = 10\ \text{W} \times 10\ \text{h} = 100\ \text{Wh} = 0.1\ \text{kWh}$ .
Check the unit: $1\ \text{kWh} = 1000\ \text{W} \times 3600\ \text{s} = 3\,600\,000\ \text{J} = 3.6\ \text{MJ}$ .

Answer: The 2000 W heater uses 6 kWh and costs 0.90 dollars at the example price. The 10 W LED bulb uses only 0.1 kWh, even though it ran for longer. Power sets how fast energy is used, and time sets how much is used in total.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Confusing watts with watt-hours. A watt is a rate, while a watt-hour is an amount. A 2000 W heater uses 2 kWh in one hour, so the rate and the total are different numbers.
Forgetting to convert minutes to hours. Thirty minutes is 0.5 h, so a 2000 W heater for 30 minutes uses 1 kWh.
Multiplying the price by the power instead of the energy. Bills charge per kWh, so multiply the kWh used by the price per kWh.
Treating the example price as a real tariff. Electricity prices vary by country, provider and time of day, so check your own bill.

Where you will meet it

Electricity bills, which charge for the kilowatt-hours used in each billing period.
Appliance energy labels, where a fridge or washing machine is compared by its energy use over a year, in kilowatt-hours.
Battery capacity, where home and car batteries are rated in kilowatt-hours.
Physics and engineering, where energy is counted in joules and 1 kWh = 3.6 MJ links the two units.

Questions people ask

What is the difference between a watt and a watt-hour?

A watt (W) is a rate: how fast energy is used. A watt-hour (Wh) is an amount: 1 W running for 1 hour. A 60 W bulb uses 60 Wh in one hour.

Why do bills use kilowatt-hours and not joules?

A joule is a tiny amount. One kilowatt-hour is 3.6 million joules, so kilowatt-hours give manageable numbers, such as 6 kWh for a heater running 3 hours.

How do I work out what an appliance costs to run?

Multiply the power in kilowatts by the hours used to get kWh. Then multiply the kWh by your price per kWh, and check your bill for the real price.

Is energy the same as power?

No. Power is a rate and energy is a total. Two devices can have the same power, but the one that runs longer uses more energy.

Electric Power: Electric power tells you how fast a device uses energy, and you get it by multiplying the volts by the amps to get watts.
Ohm's Law: Ohm’s law says the push on a wire, the voltage, equals the flow through it, the current, times how hard that flow is to get through, the resistance.
Planck-Einstein Relation (E = hf): E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.