The kilowatt hour and paying for electricity

In a nutshell

Power is defined as the rate of energy transfer. The kilowatt-hour is a unit used for electrical bills and is defined as the energy transferred by a device with one kilowatt for one hour.

Equations

Variable definitions

Power

Power is defined as the rate of energy transfer or how much energy has been transferred per second. It is symbolised by the letter $$P$$ and is measured in watts $$W$$. To calculate power you can use the general equation:

$$P=\dfrac{W}{t}$$​​

In an electrical circuit however, the electrical power can be calculated with this equation instead:

​$$P=IV$$​​

Current and potential difference are also related through the equation:

​$$V=IR$$​​

This means that you can substitute them into the equation for power and have it in terms of resistance. By doing so you will end up with the following equations:

​$$\begin{aligned}P&=I^2 R \\P&=\dfrac{V^2}{R}\end{aligned}$$​​

Example

A component in a circuit with a resistance of $$5\ \Omega$$ has a current of $$3\ A$$ flowing through it, what is the power of the component?

Firstly, write down all the known values:

$$\begin{aligned}R&=5\ \Omega \\I&=3\ A\end{aligned}$$​​

Next, write down the equation needed. Since you only have resistance and current the equation you need to use is:

​$$P=I^2R$$​​

Now substitute all the values and calculate the power of the component:

$$P=3^2\times5=45\ W$$​​

The component has a power of $$\underline{45\ W}$$.

Energy transferred

By rearranging the general equation for power one finds that the energy transferred is:

​$$W=Pt$$​​

You can now substitute the equation for power into this equation to find the energy transferred in terms of current and voltage:

​$$W=IVt$$​​

The kilowatt-hour

Electricity companies bill you based on the energy that has been transferred from the national grid to your home which is recorded on an electricity meter. Since recording the energy in joules would give very high numbers, companies record the energy transferred in kilowatt-hours.

The kilowatt-hour ($$kW\ h$$​) is a unit defined as the energy transferred by a device with $$1\ kW$$ for $$1$$ hour. $$1\ kW\ h$$ corresponds to $$3.6\ MJ$$. Electricity companies charge an amount of money for each kilowatt-hour, e.g. $$10\ p$$ per $$kW\ h$$. To find how much an amount of energy costs you can use the following equation:

​$$cost=units\ used \times price\ per\ unit$$​​

Example

A company charges $$10\ p$$ per $$kW\ h$$. A $$1500\ W$$ washing machine was used for $$45$$​ min, how much did it cost to run the washing machine.

Firstly write down all the known values:

$$price\ per\ unit= 10p /kW\ h \newline P= 1500\ W \newline t=45\ min$$​​

Since you are using kilowatt-hours you have to convert the power and time into $$kW$$​ and $$h$$:

$$P=1500\ W =1.5\ kW \newline t=45\ min= 0.75\ h$$​​

Now write down the equation for energy transferred:

​

​$$W=Pt$$​​

Substitute the numbers and calculate the energy transferred in kilowatt-hours:

$$W=1.5\times0.75=1.125\ kW\ h$$​​

Next you need to multiply the energy transferred by the cost per kilowatt-hour to find how much it cost to run the washing machine:

$$1.125\times10=11.25\ p$$​​​

Running the washing machine cost $$\underline{11.25\ p}$$.