Electrical resistance

Electrical resistance describes how much a component opposes the flow of electric current. It relates the potential difference across the component to the current through it. For the same potential difference, a component with greater resistance carries a smaller current.

Resistance is calculated by dividing potential difference by current:

Here, $$R$$ is resistance in ohms ($$\Omega$$), $$V$$ is potential difference in volts ($$\mathrm{V}$$), and $$I$$ is current in amperes ($$\mathrm{A}$$). A resistance of $$1\ \Omega$$ means that a potential difference of $$1\ \mathrm{V}$$ gives a current of $$1\ \mathrm{A}$$.

For example, if the potential difference across a component is $$12\ \mathrm{V}$$ and the current through it is $$3\ \mathrm{A}$$, its resistance is:

Calculating potential difference or current

The resistance equation can be rearranged to find either of the other quantities:

Use $$V = IR$$ when current and resistance are known, or $$I = V/R$$ when potential difference and resistance are known.

The calculated resistance applies at the conditions under which the potential difference and current were measured. A component’s resistance can change when its conditions change, so a value calculated from one set of measurements does not necessarily apply under different conditions.

Measuring resistance with a voltmeter and ammeter

The resistance of a component can be found by measuring the current through it and the potential difference across it. The resistance is the potential difference divided by the current.

Connect the component and an ammeter in series with a power supply, so that the current through the component also passes through the ammeter. Connect a voltmeter in parallel across the component, with one connection on each side of it.

With the power supply connected, record the current and the potential difference. The two readings must refer to the same operating conditions: do not change the circuit settings between taking them.

Here, $$R$$ is resistance in ohms ($$\Omega$$), $$V$$ is potential difference in volts ($$\mathrm{V}$$), and $$I$$ is current in amperes ($$\mathrm{A}$$). Use the readings in volts and amperes to obtain resistance in ohms.

For example, if the potential difference is $$6,\mathrm{V}$$ and the current is $$0.3,\mathrm{A}$$ under the same conditions, the resistance is $$6 \div 0.3 = 20,\Omega$$.

Effect of wire dimensions on resistance

A wire’s resistance describes how strongly it opposes the flow of electric current. For metallic wires made of the same material and at the same temperature, resistance depends on both length and cross-sectional area.

Wire length and resistance

A longer wire has greater resistance than a shorter wire of the same cross-sectional area. Charge carriers travelling through the longer wire encounter more obstacles along their path, so the wire offers greater opposition to current.

Cross-sectional area and resistance

The cross-sectional area is the area of the wire’s end when it is cut straight across. A thicker wire has a larger cross-sectional area.

A wire with a larger cross-sectional area has lower resistance than a narrower wire of the same length. The wider cross-section provides more space for charge carriers to flow through the wire at the same time.

Resistance scaling with wire dimensions

A longer wire offers more resistance to current than a shorter wire of the same cross-sectional area: charges must travel through more material. A wire with a larger cross-sectional area offers less resistance when its length is unchanged: there is more space across the wire for charge to flow.

For metallic wires made of the same material and kept at the same temperature, resistance depends on length and cross-sectional area as follows:

Here, $$R$$ is resistance, $$L$$ is length and $$A$$ is cross-sectional area—the area of the wire’s cut end.

• Changing length: If the area stays the same, multiplying the length by a factor multiplies the resistance by the same factor. For example, doubling the length doubles the resistance.
• Changing area: If the length stays the same, multiplying the area by a factor divides the resistance by that factor. For example, doubling the area halves the resistance.

When both wire dimensions change

Combine the two effects: multiply the original resistance by the length factor, then divide by the area factor. For an original wire labelled 1 and a changed wire labelled 2:

For example, if the length is tripled and the cross-sectional area is doubled, the new resistance is $$3 \div 2 = 1.5$$ times the original resistance. The increase in length raises the resistance, while the increase in area partly offsets it.

Current–voltage characteristic of a constant resistor

For a resistor whose resistance stays constant, the current through it is directly proportional to the potential difference across it. If the potential difference doubles, the current doubles; if it is halved, the current is halved.

The resistance is the ratio of potential difference to current:

Here, $$R$$ is resistance, $$V$$ is potential difference and $$I$$ is current. The ratio $$V/I$$ remains constant for non-zero current.

The current–voltage graph

With current on the vertical axis and potential difference on the horizontal axis, the characteristic is a straight line through the origin. The origin shows that when the potential difference is zero, the current is also zero.

Reversing the potential difference reverses the current. The straight line therefore continues through the origin into the opposite direction, where both potential difference and current have negative values.

Current–voltage characteristic of a filament lamp

A filament lamp has a curved current–voltage characteristic: as the potential difference across the lamp increases, the current increases, but not in direct proportion.

At a low potential difference, the filament is relatively cool and has a low resistance. As the potential difference and current increase, the filament heats up. The particles in the filament vibrate more strongly, making it harder for electrons to move through it. The filament’s resistance increases.

With greater resistance, each further increase in potential difference produces a smaller increase in current. On a graph with potential difference on the horizontal axis and current on the vertical axis, the curve therefore becomes less steep as the potential difference increases.

If the polarity is reversed, the current flows in the opposite direction. Increasing the magnitude of the reversed potential difference again heats the filament and increases its resistance. The characteristic curves similarly on the opposite side of the graph.

Current–voltage characteristic of a diode

A diode conducts current mainly in one direction. Its current–voltage characteristic shows how the current through it changes as the potential difference across it changes.

When the potential difference is applied in the forward direction, very little current flows at first. As the potential difference reaches a threshold, the current begins to increase rapidly. Above this threshold, a small increase in forward voltage produces a much larger increase in current.

When the potential difference is reversed, the diode is in the reverse direction. Very little current flows, even as the reverse voltage increases.

On a current–voltage graph, the forward part therefore stays close to zero current before rising steeply. The reverse part stays close to zero current. The characteristic is non-linear: current does not increase in direct proportion to voltage. It also has markedly different shapes for the two polarities.