The heating effect of electric current

A warm wire

Touch a charger after it has been working for some hours. It is warm. Touch the element of a heater. It is hot. When a current flows through a wire, the wire gets warm. This is the heating effect of electric current.

Where does the heat come from?

The free electrons in a wire move under the push of the potential difference. On the way, they hit the atoms of the wire, and they give them some of their energy. The atoms vibrate more strongly, and the wire becomes hotter. A resistance is what makes this happen. A battery keeps giving energy to the charge, and the resistor changes it all into heat.

How much heat?

The work done to move a charge $Q$ through a potential difference $V$ is $W = VQ$. With $Q = It$, $W = VIt$. Ohm's law gives $V = IR$, so the heat produced in a time $t$ is

This is Joule's law of heating: the heat produced in a resistor is

• proportional to the square of the current ($I^2$),
• proportional to the resistance ($R$),
• proportional to the time ($t$) for which the current flows.

The unit of $H$ is the joule. With Ohm's law, we can also write $H = VIt$ and $H = \dfrac{V^2}{R}\,t$. Use the form that matches the quantities that you know.

If the current is doubled, the heat becomes four times as large, because of the square.

You should find that the hollow point, at twice the current, always sits four times as high as yours.

A worked example

A heater coil has a resistance of $50\ \Omega$ and carries a current of $2\ \text{A}$. How much heat does it produce in $10\ \text{s}$?

We use $H = I^2Rt = (2\ \text{A})^2 \times 50\ \Omega \times 10\ \text{s} = 2000\ \text{J}$.

Uses of the heating effect

• Heaters, toasters, irons and kettles. The element is a coil of an alloy such as nichrome. It has a high resistivity, so it makes a lot of heat, and it does not oxidise easily when it is red hot.
• The filament bulb. The filament is a thin coil of tungsten. It becomes white hot and gives light, but it does not melt, because tungsten has a very high melting point (about $3400\ ^\circ\text{C}$). The bulb is filled with an unreactive gas such as nitrogen or argon, so that the filament lasts longer. Most of the energy of a filament bulb becomes heat, and only a small part becomes light.
• The fuse. A fuse is a short piece of wire with a low melting point, in series in the circuit.

When the current is too large (for example, in a short circuit, or when too many appliances are on), the fuse wire becomes hot, melts and breaks the circuit. This protects the appliances and the wires of the house from fire. The fuse is placed in the live wire.

Why does the bulb glow and the wire does not?

The bulb and the wires that bring the current carry the same current. The heat in each is $H = I^2Rt$, so it depends on the resistance. The filament has a very large resistance, so it gets very hot. The copper wires have a very small resistance, so they stay cool.

Practise with new numbers

Try it in Python

The program finds the heat made in $10\ \text{s}$ in a $5\ \Omega$ resistor for four currents.

Remember these

• The heat produced is $H = I^2Rt = VIt = \dfrac{V^2}{R}t$.
• It is proportional to the square of the current. A double current gives four times the heat.
• Uses: heaters (nichrome), bulbs (tungsten), fuses (a wire that melts easily).

Check yourself

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

1. A $4\ \Omega$ resistor carries $3\ \text{A}$ for $6\ \text{s}$. How much heat is produced?

2. What happens to the heat if the current is made three times as large, with the same resistance and time?

3. Why is the fuse put in the live wire, and not in the neutral wire?

4. Why does the cable of a heater stay cool while its element glows?