B.5 Current and circuits

Every electrical device, from a phone to a train, works by moving charge through a closed path called a circuit. A source such as a cell gives energy to the charges, and the components they pass through transfer that energy to light, motion or internal energy. The physics of circuits explains why a phone battery runs down, why a kettle element gets hot while the cable feeding it stays cool, and how a sensor can switch on a street lamp at dusk.

A small set of quantities describes every circuit: current, potential difference, emf, resistance and power. Each has a precise definition, and the rules for combining components follow from two conservation laws: charge is conserved at every junction, and energy is conserved round every loop.

Electric current and charge carriers

Charge is carried through a material by charge carriers: charged particles that are free to move. In a metal the carriers are free electrons, roughly one or two per atom, that are not bound to any particular atom. In a salt solution (an electrolyte) the carriers are positive and negative ions, and in the glowing gas of a fluorescent lamp they are positive ions and free electrons. When a potential difference is applied, the carriers drift through the material and charge flows.

The coulomb is the unit of charge. One coulomb is the charge that passes a point when a current of 1 A flows for 1 s, so $\Delta q = I\Delta t$. The charge of one electron is $-e$, where $e = 1.60\times10^{-19}\ \text{C}$ is the elementary charge, so a current of 1 A is a flow of about $6.25\times10^{18}$ electrons every second.

A direct current (dc) flows in one direction only, as it does in a circuit powered by a cell or a battery. A current is measured with an ammeter placed in the circuit so that the current passes through it.

Conventional current and electron flow

Electric circuits were studied long before the electron was discovered, and current was defined as the flow of positive charge from the positive terminal of a source, round the circuit, to the negative terminal. This direction is called conventional current, and every circuit rule and diagram uses it. In a metal the carriers are negative electrons, so they actually move the opposite way, from the negative terminal towards the positive terminal. A flow of negative charge one way has exactly the same electrical effect as a flow of positive charge the other way, so the convention never gives a wrong answer.

Potential difference and emf

As charge carriers move through a component such as a lamp, they transfer energy to it. The energy transferred per unit charge is the potential difference (pd) across the component. Because the definition is "per unit charge", the pd does not depend on how much charge flows, only on the energy each coulomb delivers.

A pd of 6 V across a lamp means that each coulomb passing through the lamp transfers 6 J of energy to it. Potential difference is measured with a voltmeter connected across the component, with one lead on each side of it.

The same definition gives a unit of energy suited to single particles. The electronvolt (eV) is the energy transferred when one electron moves through a pd of 1 V: $1\ \text{eV} = 1.60\times10^{-19}\ \text{J}$. An electron moving through a pd of 9.0 V gains 9.0 eV, or $1.44\times10^{-18}\ \text{J}$.

Electromotive force

A source such as a cell does the opposite of a lamp: it gives energy to the charges that pass through it, by converting chemical or light energy into electrical energy. The energy given per unit charge is the emf of the source. Despite its name ("electromotive force"), emf is not a force; it is measured in volts like a pd.

Emf and pd both measure energy per unit charge, but in opposite directions of transfer. Emf describes energy going into the circuit at a source; pd describes energy leaving the circuit at a component. In a complete circuit, energy conservation means that the total energy given to each coulomb by the sources equals the total energy it transfers in the components.

Sources of emf: chemical cells and solar cells

A chemical cell converts chemical energy into electrical energy. Reactions at its two electrodes separate charge, leaving one terminal positive and the other negative. A primary cell, such as an alkaline AA cell (emf about 1.5 V), is used once and thrown away when its reactants are used up. A secondary cell, such as a lithium-ion cell (about 3.7 V) or a lead–acid cell (about 2 V), is rechargeable: driving a current backwards through it reverses the reactions and stores energy again. A battery is two or more cells connected in series; a 12 V car battery is six lead–acid cells.

A solar cell (photovoltaic cell) converts the energy of light directly into electrical energy. Absorbed photons free electrons in a layer of semiconductor, usually silicon, and the structure of the cell drives them round the external circuit. One silicon cell gives an emf of only about 0.6 V, so a panel connects dozens of cells in series. Commercial silicon panels convert roughly a fifth to a quarter of the incident light energy into electrical energy. Even the best laboratory silicon cell, made by LONGi and certified by the ISFH institute in Germany in 2026, reaches only 28.3 %.

The amount of charge a cell can deliver is often quoted as its capacity in milliampere-hours (mA h). Smartphone batteries typically have capacities of a few thousand mA h: a 3000 mA h battery can supply 3000 mA for one hour, or a smaller current for longer.

Circuit diagrams

A circuit diagram shows how components are connected, using standard symbols and straight wires drawn as horizontal and vertical lines. It does not show the physical layout or the length of the wires. Connecting wires are treated as having zero resistance, so there is no pd along a wire, and a dot marks a junction where wires join.

An ammeter is connected in series, so that the current to be measured passes through it. A voltmeter is connected in parallel across the component, so that it compares the energy per unit charge on the two sides of it. The simple circuit with a cell, a switch, a lamp and two meters shows both arrangements, together with the direction of conventional current.

Conductors, insulators and semiconductors

Whether a material conducts depends on how many charge carriers it has and how freely they can move, a property called their mobility. A conductor contains a large number of mobile charge carriers. In metals such as copper, the outer electrons of each atom are not tied to that atom but move freely through the whole lattice of positive ions, so even a small pd produces a large current.

In an insulator, such as glass, rubber or most plastics, almost every electron is tightly bound to its atom or shared in a bond. There are almost no mobile charge carriers, so the current is negligible even at a large pd. Insulators are used to cover wires and to make plugs and switch casings safe to touch.

A semiconductor, such as silicon, lies in between. At low temperature it has very few mobile carriers, but supplying energy, by heating it or shining light on it, frees more of them. Its resistance therefore falls as its temperature or the light intensity rises, the opposite of a metal. Thermistors, LDRs, LEDs and solar cells are all made of semiconductors.

Resistance and its origin

For a given pd, some components let much more current through than others. The ratio of the pd to the current is the resistance of the component.

A resistor of 1 kΩ with 5.0 V across it carries $I = V/R = 5.0/1000 = 5.0\times10^{-3}\ \text{A}$, or 5.0 mA. The definition applies at each instant: if the resistance changes with the current, $R$ is still $V/I$ at that particular pd and current.

Why metals have resistance

In a metal the free electrons move rapidly in random directions even with no current. When a pd is applied, the electric field accelerates every free electron towards the positive end. After a very short time the electron collides with one of the positive ions of the lattice, transfers energy to it, and changes direction. The field then accelerates it again. The result is a slow net drift of electrons along the wire, superimposed on their random motion.

These collisions are the origin of resistance: they oppose the drift of the carriers, so a larger pd is needed to drive a given current. They also explain the heating effect of a current. Each collision transfers kinetic energy from an electron to the lattice ions, which vibrate with a larger amplitude. The internal energy of the metal increases and its temperature rises. This is how a heating element, a toaster and the filament of a lamp work.

The effect of temperature

When a metal gets hotter, its ions vibrate with a larger amplitude, so the electrons collide with them more often. The drift is opposed more strongly, and the resistance of a metal increases with temperature. In a semiconductor, a temperature rise also frees many more charge carriers, and this effect is much larger than the increase in collisions, so the resistance of a semiconductor falls as it gets hotter.

Resistivity

Two wires made of the same metal can have very different resistances. A longer wire has more lattice ions in the path of each electron, so its resistance is larger: resistance is proportional to length. A thicker wire provides more parallel paths for the electrons, like more lanes on a road, so its resistance is smaller: resistance is inversely proportional to the cross-sectional area. The remaining factor depends only on the material and its temperature, and is called its resistivity.

Resistivity is a property of the material, while resistance is a property of a particular object. A copper wire and a copper bar have the same resistivity but very different resistances. The values of resistivity cover an enormous range, more than 25 orders of magnitude from silver to fused quartz, which is why materials divide so sharply into conductors and insulators.

Values at 20 °C. Data: HyperPhysics resistivity table, Georgia State University (values from Giancoli, Physics, 4th edition, 1995, and the CRC Handbook).

Comparing wires by proportion

Many problems compare wires of the same material. Since $R = \rho L/A$ and $A = \pi d^2/4$, the resistance is proportional to $L/d^2$. Doubling the length doubles the resistance; doubling the diameter makes the area four times larger and the resistance four times smaller.

Measuring resistivity

In a school laboratory, resistivity is found by measuring the resistance of different lengths of one wire, using an ammeter and a voltmeter, and the diameter with a micrometer at several points along the wire. A graph of $R$ against $L$ is a straight line through the origin with gradient $\rho/A$, so $\rho = \text{gradient} \times A$. Using a small current, and switching off between readings, keeps the wire at a constant temperature so that its resistivity does not change during the experiment.

Ohm’s law and I–V characteristics

In 1827 Georg Ohm reported that, for metal wires, the current is proportional to the pd. This pattern is now known as Ohm’s law. Like many laws in physics it describes how some materials behave without explaining why, and it holds only under stated conditions.

A component that obeys Ohm’s law is an ohmic conductor: its resistance is the same at every pd. A metal at constant temperature is ohmic. A component whose resistance changes with the current or pd is a non-ohmic conductor. The equation $R = V/I$ defines resistance for every component, ohmic or not; Ohm’s law is the extra statement that $R$ is constant.

Measuring an I–V characteristic

The way the current through a component varies with the pd across it is its I–V characteristic. To measure it, the component is connected in a circuit in which the pd can be varied smoothly from zero, with an ammeter in series and a voltmeter across the component. A potentiometer used as a potential divider gives the full range from 0 V upwards.

For each setting, the current and pd are recorded. The connections to the component are then reversed to obtain readings with negative pd and current, so that the characteristic is plotted in both directions.

Ohmic resistor and filament lamp

For an ohmic resistor the characteristic is a straight line through the origin, and $V/I$ is the same at every point. For a filament lamp the line curves towards the $V$ axis. As the current increases, more energy is transferred to the tungsten filament, its temperature rises to about 2500–3000 K and its ions vibrate more, so the electrons collide more often. Each extra volt then produces a smaller increase in current. The hot resistance of a lamp filament is commonly more than ten times its cold resistance. The curve is symmetric about the origin because the filament heats up equally whichever way the current flows.

Light-emitting diode

A light-emitting diode (LED) is a semiconductor device that conducts in one direction only and emits light when it does. In the reverse direction its resistance is extremely large and the current is almost zero. In the forward direction almost no current flows until the pd reaches a threshold, about 1.8 V for a red LED and higher for blue; above this the current rises steeply.

The resistance of an LED is therefore very large below the threshold and small above it. Because the current rises so steeply, an LED is always used with a resistor in series to limit the current; connected straight across a battery it would carry a large current and be damaged.

Electrical power and energy

The power of a component is the rate at which it transfers energy. In a time $\Delta t$, a charge $\Delta q = I\Delta t$ passes through a component with pd $V$ across it, and the energy transferred is $W = qV = VI\Delta t$. Dividing by the time gives the power:

The second and third forms come from substituting $V = IR$ and $I = V/R$. All three are always equivalent; the choice depends on which quantities are known. For a resistor carrying a known current, $I^2R$ is quickest; for a component on a fixed supply pd, $V^2/R$ is quickest. Power is measured in watts, $1\ \text{W} = 1\ \text{J s}^{-1}$, and the energy transferred in time $t$ is $E = Pt$.

Heating elements

A heating element is a resistor designed to transfer as much energy as possible to its surroundings. It is usually a coil of nichrome wire, which has a high resistivity and forms a thin protective oxide layer, so it does not burn away when red hot. The cable that feeds a heater carries the same current, but its copper wires have a very small resistance, so by $P = I^2R$ the power transferred in the cable is tiny compared with that in the element.

Brightness of lamps

The brightness of a filament lamp depends on its power: the larger the power, the hotter the filament and the brighter the lamp. Comparing brightness in a circuit therefore means comparing $IV$ for each lamp, not just the current or the pd alone.

The kilowatt-hour

Electricity suppliers charge for energy in kilowatt-hours (kW h). One kilowatt-hour is the energy transferred by a 1 kW device in one hour: $1\ \text{kW h} = 1000\ \text{W} \times 3600\ \text{s} = 3.6\times10^{6}\ \text{J}$. The cost of running a device is its energy in kW h multiplied by the price per kW h.

Series and parallel circuits

Components can be connected in two basic ways. In series, they are joined end to end so that there is only one path for the current. In parallel, they are connected across the same two points so that the current divides between several branches. The rules for each follow from conservation of charge and conservation of energy.

Components in series

With only one path, every charge carrier that passes through one component must pass through the next, and charge cannot pile up anywhere in the circuit. The current is therefore the same at every point in a series circuit. Each coulomb gives up some of its energy in each component, and the energy given up in total equals the energy it gained in the source, so the pds add up to the total pd.

Dividing the pd equation by the common current, $V/I = V_1/I + V_2/I + \dots$, gives the total resistance of resistors in series:

In the series circuit with a 12 V battery, the total resistance is $2.0 + 4.0 + 6.0 = 12\ \Omega$, so $I = 12/12 = 1.0\ \text{A}$. Each pd is then $V = IR$: 2.0 V, 4.0 V and 6.0 V. The pd is shared in proportion to the resistances, and the shares add up to 12 V. Adding a resistor in series always increases the total resistance and reduces the current.

Components in parallel

Each branch of a parallel circuit is connected between the same two points, so each coulomb passing through any branch transfers the same energy: the pd is the same across every branch. At a junction, the charge flowing in each second must equal the charge flowing out, so the branch currents add up to the total current.

Dividing the current equation by the common pd, $I/V = I_1/V + I_2/V + \dots$, gives the total resistance of resistors in parallel:

For the 3.0 Ω and 6.0 Ω resistors in parallel, $1/R_\text{p} = 1/3.0 + 1/6.0 = 0.50\ \Omega^{-1}$, so $R_\text{p} = 2.0\ \Omega$ and the battery supplies $6.0/2.0 = 3.0\ \text{A}$. Adding a branch in parallel gives the current another path, so the total resistance always falls. The total resistance of a parallel combination is always less than the smallest resistance in it.

Household lighting circuits are wired in parallel so that each lamp has the full supply pd and can be switched on and off independently.

Series–parallel networks

A network that mixes series and parallel connections is solved in stages. First replace each parallel group by its equivalent resistance, then add the series resistances to find the total. The total current follows from $I = V/R$, and the current and pd for each component are found by working back through the network.

Emf and internal resistance

A real cell is not a perfect source. The charges passing through it must move through its own materials, which have resistance. This internal resistance $r$ means that some of the energy given to each coulomb is transferred inside the cell, which warms it, before the charge reaches the external circuit. A real cell is modelled as an ideal emf $\varepsilon$ in series with a resistor $r$.

With an external resistance $R$, the total resistance of the circuit is $R + r$, so the current is $I = \varepsilon/(R + r)$, or

Expanding gives $\varepsilon = IR + Ir$. The term $IR$ is the pd across the external circuit, which is the pd between the terminals of the cell, the terminal potential difference $V$. The term $Ir$ is the pd across the internal resistance, sometimes called the "lost volts": energy per coulomb transferred inside the cell. So

The terminal pd equals the emf only when no current flows, which is why a high-resistance voltmeter connected alone across a cell reads very nearly its emf. The larger the current drawn, the more the terminal pd falls below the emf.

Short circuits and maximum current

If the terminals are joined by a wire of negligible resistance, a short circuit, then $R = 0$ and the current takes its largest possible value, $I_\text{max} = \varepsilon/r$. The terminal pd is zero and all the energy is transferred inside the cell, which can become dangerously hot. For the 9.0 V battery with $r = 1.5\ \Omega$, $I_\text{max} = 6.0\ \text{A}$.

Internal resistance matters most where large currents are needed. A car starter motor draws a current of the order of 100 A, so a car battery must have an internal resistance of the order of hundredths of an ohm; otherwise most of its emf would be lost inside it. A small alkaline cell, with an internal resistance of a fraction of an ohm, could never supply such a current.

Finding emf and internal resistance from a graph

To measure $\varepsilon$ and $r$, a cell is connected to a variable resistor and an ammeter, with a voltmeter across the cell's terminals. The resistance is changed to give a range of currents, and the terminal pd is recorded for each.

Rearranging $V = \varepsilon - Ir$ as $V = (-r)I + \varepsilon$ shows that a graph of $V$ against $I$ is a straight line, of the form $y = mx + c$. The intercept on the $V$ axis is the emf, and the gradient is $-r$. Extending the line to the $I$ axis gives the short-circuit current $\varepsilon/r$.

Variable resistors and potential dividers

Many circuits need a resistance that changes, either by hand or in response to the surroundings. Three components do this: the thermistor, whose resistance changes with temperature; the light-dependent resistor (LDR), whose resistance changes with light intensity; and the potentiometer, a resistor with a sliding contact. Each is most useful in a potential divider.

The potential divider

A potential divider is two resistors in series across a supply, with the output taken across one of them. The current is the same in both, $I = V_\text{in}/(R_1 + R_2)$, so the pd across $R_2$ is $IR_2$:

The input pd is shared in the ratio of the resistances: $V_1 : V_2 = R_1 : R_2$. Only the ratio matters, so 1 kΩ and 2 kΩ divide a pd in the same way as 1 MΩ and 2 MΩ. Making $R_2$ larger, or $R_1$ smaller, increases the output.

Thermistors and light-dependent resistors

A thermistor is made of a semiconductor. The common type used in temperature sensors has a resistance that falls as its temperature rises, because heating frees more charge carriers. The fall is steep and non-linear, often by a factor of about two for every 15–20 °C.

An LDR is also a semiconductor device. Light falling on it frees charge carriers, so its resistance falls as the light intensity rises: a typical LDR has a resistance of the order of 1 MΩ in darkness and a few hundred ohms in bright light.

Placed in a potential divider with a fixed resistor, either sensor turns a change in temperature or light into a change in output pd, which can switch on a lamp, a heater or an alarm. Where the sensor sits decides whether the output rises or falls.

• Light sensor (a). The LDR is the lower resistor. As the light dims, its resistance rises, its share of the total resistance grows and $V_\text{out}$ across it rises. This can switch on a street lamp at dusk.

• Temperature sensor (b). The thermistor is the upper resistor and the output is across the fixed resistor. As the temperature rises, the thermistor's resistance falls, the fixed resistor takes a larger share of the pd and $V_\text{out}$ rises. This can trigger a fire alarm or switch on a cooling fan.

Potentiometers and variable resistors

A potentiometer is a resistor with a sliding contact. With the supply across its whole length and the output taken between the slider and one end, it is a potential divider whose ratio is set by the position of the slider. It can give any output from zero to the full supply pd, which is how volume controls and dimmer controls work, and why it is used to measure I–V characteristics.

The same component with only one end and the slider connected acts as a variable resistor in series with the load. This can reduce the current but never to zero: the smallest pd across the load is set by the largest resistance of the variable resistor.

Ideal and non-ideal meters

A meter should measure a circuit without changing it. An ideal ammeter has zero resistance, so placing it in series adds no resistance and leaves the current unchanged. An ideal voltmeter has infinite resistance, so connecting it in parallel draws no current from the circuit and leaves the pd unchanged. Problems assume ideal meters unless they say otherwise.

A real meter has a fixed resistance of its own and is treated as a resistor connected where the meter is. A real ammeter has a small resistance, which slightly reduces the current; with an ammeter of 0.50 Ω in series with a 2.0 Ω resistor on an ideal 1.5 V cell, the current is $1.5/2.5 = 0.60$ A instead of 0.75 A. A real voltmeter has a large but finite resistance, and its effect matters most when it is connected across a large resistance.

Applying the ideas in exam questions

Paper 1A: comparing two wires

Paper 1A: internal resistance

Paper 1A: a series–parallel network

Paper 1B: emf and internal resistance of a cell

A group of students connects a cell to a variable resistor and an ammeter, with a voltmeter across the cell. They record the terminal pd for five currents.

Paper 2: a thermistor fire alarm

Paper 2: a kettle heating element

Self-check

• To do:

  I can compare chemical cells and solar cells as sources of emf, with advantages and disadvantages of each.

• To do:

  I can draw and recognise the circuit symbols, and place ammeters in series and voltmeters in parallel.

• To do:

  I can use I = Δq/Δt to find charge, current or time, and count electrons using e.

• To do:

  I can define potential difference and emf as energy per unit charge and explain the difference between them.

• To do:

  I can use the electronvolt as a unit of energy.

• To do:

  I can explain conductors, insulators and semiconductors in terms of the number and mobility of charge carriers.

• To do:

  I can explain resistance and the heating effect of a current using collisions between electrons and lattice ions.

• To do:

  I can use R = V/I and ρ = RA/L, including comparing wires of different length and diameter.

• To do:

  I can state Ohm’s law and sketch and explain the I–V characteristics of an ohmic resistor, a filament lamp and an LED.

• To do:

  I can calculate power with P = IV = I²R = V²/R and energy in joules and kilowatt-hours.

• To do:

  I can apply the series and parallel rules and find the current and pd for every resistor in a series–parallel network.

• To do:

  I can use ε = I(R + r), explain terminal pd and short-circuit current, and find ε and r from a V–I graph.

• To do:

  I can calculate the output of a potential divider and explain how LDR and thermistor sensor circuits respond.

• To do:

  I can explain the difference between a potentiometer and a variable resistor in series.

• To do:

  I can calculate the effect of a non-ideal ammeter or voltmeter on a circuit.

The answer is C.

And 20 more in the data booklet.

Total resistance and current. $R = 6.0 + 2.0 = 8.0\ \Omega$, so $I = 12/8.0 = 1.5\ \text{A}$. All of this current passes through $R_1$.

Share the pd. $V_1 = 1.5 \times 6.0 = 9.0\ \text{V}$, which leaves $12 - 9.0 = 3.0\ \text{V}$ across the parallel pair.

Branch currents. $I_2 = 3.0/6.0 = 0.50\ \text{A}$ and $I_3 = 3.0/3.0 = 1.0\ \text{A}$. As a check, $0.50 + 1.0 = 1.5\ \text{A}$, the current through $R_1$.

One student suggests that the efficiency of the motor increases as the load increases. Discuss whether the data support this suggestion.

[2]2 marks