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# Potential difference

## Potential difference definition

The potential difference or voltage (p.d. or $V$) between two points in a circuit is equal to the energy that is lost in moving a charge of one coulomb between two points in a circuit.

This energy loss is due to the electrical resistance of wires and circuit components (like resistors).

If there are no circuit components then all of the energy loss happens inside the wires. This causes the wires to become very hot!

The potential difference is measured in volts $(\text{V})$ or joules per coulomb $(\text{J }/ \text{ C})$.

A digital multimeter allows you to measure the current or potential difference across a particular electrical component.

## Potential difference formulae

The potential difference (or voltage) $V$ can be expressed in terms of energy $W$ (sometimes $E$) and charge $Q$ using:$$V=\frac{W}{Q}$$ Recall that voltage is the work done per unit charge.

The symbol $W$ is used to refer to energy in formulae in the study of electricity and magnetism to avoid confusion because $E$ is also used to symbolise the e.m.f.

Voltage can also be expressed in terms of power $P$ and current $I$ by dividing the numerator and denominator of the formula above by time $t$: \begin{align*} V&=\frac{W/t}{Q/t}\\ &=\frac{P}{I} \end{align*}The potential difference can therefore also be defined as the power loss per unit current, or the rate of energy loss per unit current. All of these definitions are equivalent.

## Potential difference and electromotive force

Potential difference (p.d.) and electromotive force (e.m.f.) are closely related. Both are measured in volts.

The e.m.f. describes the energy supplied to a coulomb of charge in order to move it around the circuit.

The p.d. describes the energy lost between two points in a circuit (such as between the two ends of a resistor) by one coulomb of charge as it moves around the circuit.

Conservation of energy implies that: $$\Tred{\text{total e.m.f. }} = \Tblue{\text{total p.d.}}$$

If this were not the case, the charges in the circuit would gain or lose energy and the current would either increase or decrease.

The potential difference across the light bulb is equal to the e.m.f. supplied by the cell.