Resistance

Push a crowd of people down a wide, empty corridor and they stream through easily. Send the same crowd through a narrow, cluttered doorway and they bunch up, slow down, and only a trickle gets through at a time. The doorway resists the flow.

Electricity is the same. When current — the flow of charge — travels round a circuit, every wire and component pushes back against it a little. That opposition is called resistance. A thin filament, a heating element or a fixed resistor pushes back a lot; a fat copper connecting wire barely pushes back at all.

Resistance decides how much current a potential difference can drive. For the same pd, a large resistance lets only a small current through, while a small resistance lets a large current pour through. We measure resistance in ohms, written with the Greek capital omega, \Omega.

Ohm's law: linking pd, current and resistance

The three quantities are tied together by one short, famous equation. The potential difference V across a component (in volts), the current I through it (in amps) and its resistance R (in ohms) obey:

V = I\,R

For a resistor kept at constant temperature, the current through it is proportional to the potential difference across it. In symbols, with volts, amps and ohms:

V = I\,R — the pd equals current times resistance.
R = \dfrac{V}{I} — rearranged to find a resistance.
I = \dfrac{V}{R} — rearranged to find a current.

All three are the same fact written three ways — cover the quantity you want in the triangle \dfrac{V}{I\ R} and the other two tell you how to get it. Read the last form out loud: current equals pd divided by resistance. Hold the pd steady and make R bigger, and I must get smaller — more resistance, less current. That is resistance doing its job.

Play with the circuit

Here is a simple circuit: a cell provides the potential difference V, a resistor provides the resistance R, an ammeter (marked A) reads the current and a voltmeter (marked V) reads the pd across the resistor. Drag the two sliders and watch the current I = V / R respond — turn the resistance up and the current bar shrinks even though the pd hasn't changed.

Worked examples

Every one of these is just V = I\,R, rearranged to leave the unknown on its own.

Example 1 — find the resistance. A pd of 12\ \text{V} drives a current of 3\ \text{A} through a resistor. What is its resistance?

R = \frac{V}{I} = \frac{12\ \text{V}}{3\ \text{A}} = 4\ \Omega.

Example 2 — find the current. A 6\ \text{V} battery is connected across a 150\ \Omega resistor. What current flows?

I = \frac{V}{R} = \frac{6\ \text{V}}{150\ \Omega} = 0.04\ \text{A} = 40\ \text{mA}.

Example 3 — find the pd. A current of 0.5\ \text{A} flows through a 20\ \Omega resistor. What pd is across it?

V = I\,R = 0.5\ \text{A} \times 20\ \Omega = 10\ \text{V}.

Notice the pattern: write the equation, put in what you know with units, and the answer's unit falls out automatically (volts, amps or ohms). Always start from V = I\,R and rearrange — never guess which number goes on top.

Ohmic conductors: a straight line through the origin

A resistor (or a metal wire) kept at a constant temperature has a fixed resistance. Double the pd and the current doubles; treble the pd and the current trebles. Current is proportional to potential difference — a component that behaves this way is called an ohmic conductor.

Plot the current I against the potential difference V and you get a straight line through the origin. Because I = V/R, the line's gradient is 1/R: a steeper line means a smaller resistance (current climbs quickly), and a shallow line means a large resistance. Drag the resistance slider to tilt the line, and the pd slider to read a current straight off it.

Three mistakes trip up almost everyone here:

Ohm's law only holds at constant temperature. Push a big current through a filament lamp and the wire gets hot, which raises its resistance, so the current no longer keeps up with the pd. Its I–V graph bends over into an S-shape — a filament lamp is non-ohmic. The nice straight line is only for a resistor held at a steady temperature.
Resistance is R = V/I, not I/V. Volts on top, amps underneath. Flip them and your answer will be upside-down (and its units will be 1/\Omega, a clue you've gone wrong).
Resistance is not the same as current. They pull opposite ways: for a fixed pd, more resistance means less current. Don't say "the resistance is high so the current is high" — it's exactly backwards.

What makes resistance bigger?

Resistance comes from charge carriers bumping into the vibrating atoms of the material as they drift along. Anything that means more collisions means more resistance. Three things matter most:

Length. A longer wire has more atoms to squeeze past, so a longer wire has a higher resistance — double the length, double the resistance.
Thickness. A thinner wire is a narrower doorway, so it has a higher resistance; a fat wire lets charge flow more freely and has a lower resistance.
Temperature. A hotter wire has atoms jiggling harder and getting in the way more, so for most metals resistance rises as the wire heats up. (This is exactly why the filament lamp above is non-ohmic.)

This is why the heating element in a toaster or kettle is a long, thin coil of special high-resistance wire: a big resistance turns lots of electrical energy into heat. The thick copper wires that carry the current to it are chosen to have almost no resistance, so they stay cool.

Measuring a resistance

You can find any resistance experimentally with two meters, then use R = V/I:

Connect an ammeter in series with the component — right in the loop — so the same current that flows through the component also flows through the ammeter and is measured. (A good ammeter has almost zero resistance so it doesn't change that current.)
Connect a voltmeter in parallel, straight across the component, so it reads the potential difference between the component's two ends. (A good voltmeter has a huge resistance so it siphons off almost no current.)

Read the pd V off the voltmeter and the current I off the ammeter — exactly the two meters in the circuit above — and divide: R = V/I. Take several pairs of readings at different pds and plot I against V; for an ohmic resistor the gradient 1/R gives you the resistance more accurately than any single pair.

Astonishingly, yes. Cool certain materials below a critical temperature — often colder than -260\,^{\circ}\text{C} — and their resistance doesn't just fall, it drops to exactly zero. They become superconductors. A current started in a superconducting ring will circle round and round for years without any battery, because nothing opposes it.

With no resistance there's no heat wasted, so superconductors carry enormous currents and make the ferociously strong magnets inside MRI scanners and particle accelerators — and can even make a magnet hover in mid-air. The catch, for now, is keeping them cold enough; finding a material that superconducts at room temperature is one of the great prizes of physics.