Electric power

What do the numbers on a bulb mean?

Look at an electric bulb. It is marked something like 220 V, 100 W. The first number is the potential difference at which it is made to work. The second is its power. What does this mean?

Power

The power of an appliance is the rate at which it uses electrical energy:

The unit of power is the watt (W): $1\ \text{W} = 1\ \text{J/s}$. A bulb of $100\ \text{W}$ changes $100\ \text{J}$ of electrical energy into light and heat every second. Larger units are the kilowatt: $1\ \text{kW} = 1000\ \text{W}$.

Power from the voltage and the current

The energy used in a time $t$ is $W = VIt$. So

With Ohm's law, $V = IR$ and $I = \dfrac{V}{R}$, we get two more forms:

When the current and the resistance are known, use $P = I^2R$. When the voltage and the resistance are known, use $P = \dfrac{V^2}{R}$.

A worked example

A bulb is marked $220\ \text{V}, 100\ \text{W}$. Find the current that it takes, and the resistance of its filament.

Step 1. We want $I$, so from $P = VI$, $I = \dfrac{P}{V} = \dfrac{100\ \text{W}}{220\ \text{V}} = 0.45\ \text{A}$.

Step 2. We want $R$, so from $P = \dfrac{V^2}{R}$, $R = \dfrac{V^2}{P} = \dfrac{(220\ \text{V})^2}{100\ \text{W}} = 484\ \Omega$.

Check. $V = IR = 0.4545\ \text{A} \times 484\ \Omega = 220\ \text{V}$. Good.

What if the voltage is not the rated voltage?

The rating tells us the power only at the rated voltage.

You should find that on $110\ \text{V}$ every bulb gives a quarter of its marked power, and on a little more than $220\ \text{V}$ it gives more, and wears out sooner. On a lower voltage, the bulb takes a smaller current and makes less power. On half the voltage, the same filament makes only a quarter of the power ($P \propto V^2$). If the supply is $110\ \text{V}$, the $100\ \text{W}$ bulb of the example makes $\dfrac{(110\ \text{V})^2}{484\ \Omega} = 25\ \text{W}$.

Power in the whole house

All the appliances of a house are in parallel, and they all work on the same voltage. The total current is the sum of the currents that they take. A house with a fuse of $10\ \text{A}$ on a $220\ \text{V}$ supply can take at most $220\ \text{V} \times 10\ \text{A} = 2200\ \text{W}$ at one time. If more appliances are switched on, the fuse melts.

Practise with new numbers

Try it in Python

The program finds the current and the resistance of six appliances that work on $220\ \text{V}$.

Remember these

• Power is the rate of use of energy: $P = \dfrac{W}{t}$. The unit is the watt ($1\ \text{W} = 1\ \text{J/s}$).
• $P = VI = I^2R = \dfrac{V^2}{R}$.
• The rating of an appliance is true at its rated voltage only.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. An appliance takes a current of $5\ \text{A}$ from a $220\ \text{V}$ supply. What is its power?

2. What does $60\ \text{W}$ on a bulb mean?

3. Which has the larger resistance, a $40\ \text{W}$ bulb or a $100\ \text{W}$ bulb, both for $220\ \text{V}$?

4. A heater marked $220\ \text{V}$, $1000\ \text{W}$ runs on $200\ \text{V}$. What power does it give?