Ohm’s law

What is it about?

If a voltage $U$ is applied to a component, an electric current $I$ flows. This is how you measure:

• The ammeter is connected in series with the component – the whole current has to pass through it.
• The voltmeter is connected in parallel with the component – it measures the voltage between its two terminals.

The electrical resistance tells you how strongly a component opposes the current:

Ohm’s law: For metal wires at a constant temperature, the current is proportional to the voltage, $I \propto U$. The resistance $R$ is then constant and $U = R \cdot I$. The characteristic in an $I$–$U$ graph is a straight line through the origin – the larger $R$, the flatter the line. In a $U$–$I$ graph the slope of the line equals $R$.

Light bulb: The tungsten filament gets hotter than $2000\,°\text{C}$. As its temperature rises, so does its resistance – from about $3\,\Omega$ (cold) to $48\,\Omega$ at $12\,\text{V}$. Its characteristic is therefore curved; Ohm’s law does not apply to the bulb.

Resistance of a wire: It increases with the length $l$ and decreases with the cross-sectional area $A$:

The resistivity $\rho$ depends on the material. The resistance of constantan hardly changes when it warms up.

Try it

1. Turn the knob of the power supply and watch the pointers. Record data points at different voltages – or start the automatic series.
2. Measure different resistors one after the other and compare the lines in the graph. 47 Ω · 150 Ω
3. Record the characteristic of the light bulb and watch the filament. Light bulb
4. Switch to the U–I graph: now the slope of the line is the resistance. U–I graph
5. Investigate the constantan wire: double the length, then the cross-section. 1 m · 2 m · double cross-section

Tasks

Determine resistor B from a series of measurements. Draw the best-fit line yourself.

Task 2: Predict the current

A voltage of $6\,\text{V}$ is applied to a resistor of $100\,\Omega$. How large is the current?

Task 3: Set the voltage

Which voltage do you need so that exactly $0.2\,\text{A}$ flows through a resistor of $47\,\Omega$?

Calculate the resistance of the bulb at $2\,\text{V}$ and at $12\,\text{V}$. Why are the values different?

Task 5: Longer and thicker wire

A constantan wire with $l = 1\,\text{m}$ and $A = 0.1\,\text{mm}^2$ has $4.9\,\Omega$. What is the resistance of a wire that is $2\,\text{m}$ long with a cross-section of $0.2\,\text{mm}^2$?

For the wire ($1\,\text{m}$, $0.1\,\text{mm}^2$) the power supply is set to $12\,\text{V}$, but the voltmeter only shows about $9.8\,\text{V}$. Explain.

Ideas for the classroom

• Use: resistance, Ohm’s law and characteristics (grades 8/9), the wire resistance $R = \rho \cdot \frac{l}{A}$ as an extension. Data are produced by reading the meters, like in the lab; with fit=0 students draw the best-fit line themselves. Since textbooks use both conventions, you can switch between the $I$–$U$ characteristic and the $U$–$I$ graph. Several series stay in the graph for comparison.
• Meters: The analogue meters choose their range automatically (it is shown below the scale; small currents in mA). With real meters the range is set by hand – starting with the largest range.
• Model: power supply $0$ to $12\,\text{V}$ with a current limit of $2\,\text{A}$; ideal meters. Bulb $12\,\text{V}\,/\,0.25\,\text{A}$: balance between the electrical power supplied and the losses by radiation and heat conduction, resistance of tungsten $R \propto T^{1.2}$. Heating of the constantan wire is neglected. The filament temperature is a model value.
• Common stumbling blocks:
  • “$R = U / I$ is Ohm’s law.” – This is the definition of resistance and holds for every component. Ohm’s law says that $R$ is constant ($I \propto U$).
  • “A large resistance lets a lot of current through.” – It is the other way round.
  • “The ammeter can also be connected in parallel.” – In parallel it causes almost a short circuit.