State and apply Ohm's law, define resistance from it, and recognise ohmic and non-ohmic components from their V-I graphs

What this dot point is asking

SEAB wants you to state Ohm's law, use it to calculate any one of voltage, current or resistance from the other two, define resistance as the ratio of voltage to current, and read a voltage-current ($V$-$I$) graph to tell an ohmic component from a non-ohmic one. The central insight is that for a metal conductor at constant temperature the current is directly proportional to the voltage, so its $V$-$I$ graph is a straight line through the origin.

The answer

Ohm's law

Ohm's law states that, for a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across it. Written as an equation:

where $V$ is the voltage in volts, $I$ the current in amperes and $R$ the resistance in ohms. The same equation rearranges to find any quantity:

Resistance defined from the law

Resistance is defined as the ratio of the potential difference across a component to the current through it, $R = V/I$. For an ohmic component this ratio is constant, so doubling the voltage doubles the current and the resistance stays the same. The unit is the ohm ($\Omega$): one ohm is one volt per ampere.

Ohmic components

An ohmic component obeys Ohm's law: its $V$-$I$ graph is a straight line through the origin, and its gradient ($\Delta I / \Delta V$) is constant. A fixed resistor or a length of metal wire at constant temperature behaves this way. The constant resistance is the reason fixed resistors are so useful for setting currents and voltages.

Non-ohmic components

A non-ohmic component does not give a straight-line $V$-$I$ graph. Two common examples in the syllabus:

• A filament lamp gives a curve that bends towards the voltage axis. As more current flows, the filament heats up, its resistance rises, and so the current increases less for each extra volt.
• A diode conducts almost no current in reverse and only conducts in the forward direction once the voltage exceeds about $0.7\ \text{V}$, giving a sharply rising curve in one direction only.

Reading the graph

On a $V$-$I$ graph with voltage on the horizontal axis, the resistance at any point is voltage divided by current, which is $1$ divided by the gradient. A steep line means a small resistance; a shallow line means a large resistance. A straight line through the origin is the signature of an ohmic component.

Examples in context

Example 1. Setting an LED current. To run an LED safely you choose a series resistor using Ohm's law. If the supply is $5\ \text{V}$, the LED drops about $2\ \text{V}$, and you want $10\ \text{mA}$, the resistor must drop $3\ \text{V}$ at $10\ \text{mA}$, so $R = 3 / 0.010 = 300\ \Omega$. This single application of $V = IR$ appears in almost every practical circuit.

Example 2. A heating element. The element of a hair dryer is a metal coil designed to have a fixed resistance. Connected to the mains voltage, $I = V/R$ fixes the current and therefore the heat produced. Because the designer wants a predictable current, an ohmic material is chosen so the resistance does not wander as conditions change.

Try this

• Cue. A $1.5\ \text{k}\Omega$ resistor carries $4.0\ \text{mA}$. Find the voltage across it. Convert units and use $V = IR = 0.0040 \times 1500 = 6.0\ \text{V}$.

• Cue. State the conditions under which Ohm's law applies. It applies to a metallic conductor kept at constant temperature, for which current is directly proportional to voltage.

• Cue. Sketch the $V$-$I$ graph for an ohmic resistor and for a filament lamp on the same axes. The resistor is a straight line through the origin; the lamp is a curve through the origin that bends towards the voltage axis as it heats.