Ohm's law

How does the current change with the push?

Take a wire and a variable source. Join an ammeter in series and a voltmeter in parallel with the wire. Raise the voltage, and note the current each time.

The result is a pattern that the scientist Georg Ohm found in 1827. When the potential difference across the wire goes up, the current goes up in the same proportion.

Ohm's law

At a constant temperature, the current through a conductor is proportional to the potential difference across its ends.

In symbols, $V \propto I$. So the ratio $\dfrac{V}{I}$ is a constant for the conductor. We call this constant its resistance, $R$:

Use $V = IR$ as it is when you want $V$. When the unknown is the current, we write $I = \dfrac{V}{R}$, and when it is the resistance, $R = \dfrac{V}{I}$.

Resistance

Resistance is the property of a conductor to oppose the flow of charge through it. A conductor of a larger resistance lets less current flow for the same potential difference. The unit of resistance is the ohm, written $\Omega$ (the Greek letter omega):

A conductor has a resistance of one ohm if a potential difference of $1\ \text{V}$ across its ends makes a current of $1\ \text{A}$ flow through it.

The V–I graph

If we draw the potential difference $V$ against the current $I$ for a conductor that obeys Ohm's law, the graph is a straight line through the origin. The slope of the line is $\dfrac{V}{I} = R$, the resistance.

Some devices do not obey Ohm's law, and their graph is not a straight line. The filament of a bulb is one of these. It becomes very hot, and its resistance rises as it gets hotter.

You should find that the resistor's points always lie on one straight line, while the bulb's curve bends upward: the hotter it gets, the more it resists.

A variable resistor

A variable resistor (a rheostat) is a resistor whose resistance we can change. We use it to change the current in a circuit, with the same battery. When the resistance is made larger, the current becomes smaller, as Ohm's law says.

A worked example

A potential difference of $11\ \text{V}$ is put across a resistor, and a current of $2\ \text{A}$ flows. What is its resistance?

We want $R$, so $R = \dfrac{V}{I} = \dfrac{11\ \text{V}}{2\ \text{A}} = 5.5\ \Omega$.

And what current will flow if the potential difference is made $5.5\ \text{V}$? The resistance does not change, so $I = \dfrac{V}{R} = \dfrac{5.5\ \text{V}}{5.5\ \Omega} = 1\ \text{A}$. Half the push gives half the current.

Practise with new numbers

Try it in Python

The program finds the resistance from four pairs of readings, and it checks if the resistance stays the same.

Remember these

• Ohm's law: at a constant temperature, $V \propto I$. So $V = IR$.
• Resistance $R = \dfrac{V}{I}$. The unit is the ohm ($1\ \Omega = 1\ \text{V/A}$).
• The V–I graph of a conductor that obeys Ohm's law is a straight line through the origin, with the slope $R$.

Check yourself

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

1. A current of $0.5\ \text{A}$ flows through a $12\ \Omega$ resistor. What is the potential difference across it?

2. What current flows through a $6\ \Omega$ resistor at $3.6\ \text{V}$, and at $7.2\ \text{V}$?

3. How do you find the resistance from a V-I graph?

4. A rheostat in a circuit is turned up to a larger resistance. The battery is the same. What happens to the current?