Potential Difference

Charge on its own does nothing useful. To light a bulb, spin a motor or warm a heater you have to make that charge carry energy — pick it up at the battery, hand it energy, then let it dump that energy into the component as it passes through. The question every circuit really asks is: how much energy does each scrap of charge carry?

That is potential difference — usually just called voltage. It is the energy transferred for every coulomb of charge that moves between two points. A big voltage means each coulomb is loaded up with lots of energy; a small voltage means each coulomb carries only a little. Think of it as the push per passenger: not how many charges flow, but how hard each one is being shoved and how much energy it hauls along.

A cell or battery is a little energy pump: it provides a potential difference, loading up every coulomb that leaves it. A bulb is the opposite — there is a potential difference across it because charge gives up its energy there, and the bulb glows.

Volts = joules per coulomb

Because potential difference is energy shared out per unit of charge, the definition is a simple fraction: take the energy transferred and divide by the charge that carried it.

V = \dfrac{E}{Q}

Here V is the potential difference in volts (\text{V}), E is the energy transferred in joules (\text{J}), and Q is the charge in coulombs (\text{C}). Read the units straight off the formula and you get the real meaning of a volt:

1\ \text{V} = 1\ \dfrac{\text{J}}{\text{C}} = \text{one joule of energy for every coulomb of charge.}

The potential difference between two points is the energy transferred per unit charge that moves between them: V = \dfrac{E}{Q}.
It is measured in volts, where 1\ \text{V} = 1\ \text{J/C}.
Rearranged, the same relationship gives the energy transferred, E = QV, and the charge moved, Q = \dfrac{E}{V}.

Worked example 1 — finding the voltage. A lamp transfers 36\ \text{J} of energy as a charge of 4\ \text{C} passes through it. What is the potential difference across the lamp?

V = \dfrac{E}{Q} = \dfrac{36\ \text{J}}{4\ \text{C}} = 9\ \text{V}.

Each coulomb handed 9\ \text{J} to the lamp, so the pd across it is 9\ \text{V}.

Working the formula three ways

The very same equation answers three different questions — cover up the quantity you want and the triangle of E, Q and V tells you what to do.

Worked example 2 — finding the energy. A motor has a potential difference of 12\ \text{V} across it. How much energy does it transfer when a charge of 5\ \text{C} flows through?

E = QV = 5\ \text{C} \times 12\ \text{V} = 60\ \text{J}.

Every coulomb carries 12\ \text{J}, and five coulombs pass, so 60\ \text{J} is delivered to the motor.

Worked example 3 — finding the charge. A heater transfers 2400\ \text{J} of energy while connected across a 240\ \text{V} supply. How much charge flowed through it?

Q = \dfrac{E}{V} = \dfrac{2400\ \text{J}}{240\ \text{V}} = 10\ \text{C}.

With 240\ \text{J} handed over by each coulomb, delivering 2400\ \text{J} needs 10 coulombs of charge.

See the energy handed over

Below is a single loop: a cell on the left pumps charge round to a lamp on the right, and a voltmeter sits in parallel across the lamp to read the pd. Turn up the cell voltage and watch two things grow together — the energy each coulomb collects, and the voltmeter's reading. Then set how much charge Q flows and read off the total energy E = QV delivered to the lamp.

Notice the voltmeter always reads the same number as the cell's push here — every joule a coulomb picks up at the cell it gives straight back to the lamp. The charge Q does not change the voltage; it changes the total energy, because more coulombs each carrying the same energy add up to more joules.

Measuring pd: the voltmeter goes in parallel

Potential difference is a difference between two points — the energy a coulomb has on one side of a component compared with the other. So to measure it you must touch both sides of the component at once. That is why a voltmeter is connected in parallel, straddling the component like a bridge, never broken into the main loop.

This is the opposite of an ammeter, which measures the current through a component and so must sit in series, in the line of flow. A neat way to remember it: current flows through, voltage sits across.

Voltage is across, current is through. Current is the flow of charge through a component, measured in series; potential difference is measured across it, in parallel. They are different quantities with different meters — never mix them up.
Voltage is not "used up" like fuel. A coulomb does not slowly leak voltage as it travels. It carries its energy along an (ideal) wire unchanged, then hands the lot over in one go at the component that has a pd across it. The pd across a plain connecting wire is essentially zero, because no energy is given up there.
A voltmeter reads a difference, not a "level". There is no such thing as the voltage at a single wire on its own — pd always compares two points. Break a voltmeter into the main loop (in series) and it reads almost nothing, because hardly any charge can get through it.

The uphill-water picture

A circuit is a lot like water flowing round a loop of pipes, and the picture sorts out the two ideas that beginners tangle up — current and voltage.

The cell is a pump that lifts water up to a height. The higher it lifts each litre, the more energy that litre carries — that height is the potential difference.
The current is how much water rushes past each second — a completely separate thing from how high it was lifted.
The lamp is a waterwheel: the water drops back down through it, giving up its energy to make it spin. The drop across the wheel is the pd across the lamp.

You can have a tiny trickle lifted very high (high voltage, low current) or a great flood lifted only a little (low voltage, high current). Voltage is the push per litre; current is the litres per second. Keeping those apart is half of understanding electricity.

The unit honours Alessandro Volta, who in 1800 built the first true battery — the "voltaic pile", a stack of copper and zinc discs separated by brine-soaked cloth. For the first time scientists had a steady potential difference to experiment with, instead of the brief crackle of static. Volta's pile launched the whole science of current electricity, and two centuries later every battery you buy still has its voltage — its joules per coulomb — stamped on the side. A single AA cell reads 1.5\ \text{V}: each coulomb leaving it carries 1.5\ \text{J} of energy.