The Straight Line Behind Every Circuit

Electricity has become such an ordinary, controllable part of daily life, in homes, schools and hospitals alike, that it is easy to stop asking what is actually happening inside a wire when a bulb lights up. In a metallic wire, a flow of electrons is what carries charge from one place to another, and whenever charge flows through a conductor, that flow is called an electric current. But electrons were not even known to exist when electricity was first studied seriously, so a historical convention was fixed early on and has stuck ever since: electric current is treated as the flow of positive charge, and its direction is taken as opposite to the actual direction electrons move in. Precisely, if a net charge Q flows across a cross-section of a conductor in time t, the current I through that cross-section is defined as the rate of flow of charge, I = Q/t. The SI unit of current, the ampere (A), is named after the French scientist Andre-Marie Ampere, and one ampere is exactly one coulomb of charge flowing per second. Current is measured using an ammeter, which must always be connected in series, directly in the path of the current it is measuring, so that every bit of charge passing through the circuit is forced to pass through the ammeter too.

Charges do not spontaneously flow through a copper wire any more than water flows through a perfectly horizontal, level pipe. Water only moves through a pipe when there is a pressure difference between its two ends, say by connecting one end to a tank held higher than the other. Charges in a wire behave the same way, driven not by gravity, but by a difference of electric pressure between two points, called the potential difference. This difference is set up by a battery: the chemical reaction inside a cell generates a potential difference across its two terminals, even before any current is drawn at all, and it is exactly this potential difference that sets the charges moving once a circuit is closed. Precisely, the potential difference V between two points is defined as the work done, W, to move one unit of charge, Q, between them: V = W/Q. The SI unit of potential difference is the volt (V), named after Alessandro Volta, and one volt is the potential difference between two points when one joule of work moves one coulomb of charge between them. Potential difference is measured using a voltmeter, which, unlike an ammeter, is always connected in parallel, directly across the two points whose difference is being measured, rather than inserted into the current's path.

Set up a circuit with a nichrome wire, an ammeter, a voltmeter, and a variable number of identical cells, and for each number of cells, note down the current I flowing and the potential difference V across the wire. Calculate V divided by I for every single reading, and something clean and consistent emerges: the ratio V/I comes out approximately the same value every single time, whether one cell is used or four. Plotted on a graph, V against I forms a straight line passing through the origin. This relationship was established precisely in 1827 by the German physicist Georg Simon Ohm: the potential difference V across a given metallic conductor is directly proportional to the current I flowing through it, provided its temperature stays constant, a relationship now called Ohm's law. Since V/I is constant for a given conductor, that constant is named the conductor's resistance, R, giving the working equation V = IR, or equivalently R = V/I. The SI unit of resistance is the ohm (Ω), and a conductor has a resistance of one ohm if a potential difference of one volt across it drives a current of exactly one ampere through it. Rearranging the same equation gives I = V/R, which shows directly that current is inversely proportional to resistance: double a conductor's resistance, and the current through it, for the same potential difference, is cut exactly in half.

Replace the nichrome wire in a test circuit with one twice as long, keeping everything else the same, and the ammeter reading drops to exactly half its previous value. Swap in a thicker wire of the same length and material instead, and the current rises. Swap the material entirely, say to copper instead of nichrome, keeping length and thickness the same, and the current changes again, often dramatically. Applying Ohm's law to interpret each of these results shows that a conductor's resistance depends on exactly three things: it is directly proportional to length (l), inversely proportional to cross-sectional area (A), and depends on the specific material it is made from. Combined, this gives R = (rho x l)/A, where rho, the resistivity of the material, is a fixed property, measured in ohm-metre, that varies enormously between substances: metals and alloys typically sit between 10⁻⁸ and 10⁻⁶ ohm-metre, making them good conductors, while insulators like rubber and glass sit between 10¹² and 10¹⁷ ohm-metre. This single property quietly explains a lot of everyday engineering choices: tungsten, with a high melting point, is used almost exclusively for lamp filaments; copper and aluminium, both excellent conductors with comparatively low resistivity, are used for power transmission lines; and alloys such as nichrome, despite having noticeably higher resistivity than pure metals, are deliberately chosen for heating elements, since alloys also resist oxidising (burning) at the high temperatures a heating element must survive.

Connect three resistors end to end in a single loop with a battery, and however you move an ammeter, before the first resistor, between two of them, or after the last, the reading never changes. Resistors connected this way are said to be in series, and this one observation, the same current everywhere, is the defining feature of a series circuit. Measure the potential difference across each resistor separately, then compare it with the potential difference across the whole series combination, and a second relationship appears: the total potential difference equals the sum of the potential differences across each individual resistor, V = V1 + V2 + V3. Applying Ohm's law to the whole combination and to each resistor separately, then combining both relationships, gives the equivalent resistance of resistors in series: Rs = R1 + R2 + R3, always larger than any single resistor in the group. A series circuit has two real, practical downsides worth knowing. First, since every component shares exactly the same current, a bulb and a heater, which need very different currents to work correctly, cannot sensibly be wired in series together. Second, and more visibly, if any single component in a series loop fails, the entire loop breaks and every component in it stops working at once, exactly why an electrician troubleshooting a string of festival lights sometimes has to test every single bulb individually to find the one that has fused.

Wire the same three resistors differently this time, all connected directly between the same two points instead of end to end, and the behaviour flips almost entirely. Every resistor now experiences the exact same potential difference, since each one is connected across the very same two points, and this shared voltage, not a shared current, is what defines a parallel combination. Measure the current through each resistor separately using an ammeter, then compare the sum of those three currents with the total current drawn from the battery, and the relationship that appears is I = I1 + I2 + I3, the total current splits across the branches and then recombines. Applying Ohm's law to each branch and combining gives the equivalent resistance of a parallel combination: 1/Rp = 1/R1 + 1/R2 + 1/R3, and this equivalent resistance always comes out smaller than the smallest individual resistor in the group, since adding another parallel path can only make it easier, never harder, for current to get through overall. This is exactly why household wiring uses parallel circuits rather than series ones: every appliance gets the same full mains voltage regardless of what else is switched on, each appliance can be switched on or off independently without affecting the others, and appliances needing very different currents, a bulb and a refrigerator, for instance, can share the same supply lines without conflict.

A cell's chemical reaction generates a potential difference that costs the cell energy to maintain, and some of that energy the cell expends never ends up as useful work at all, it simply becomes heat within the circuit's own resistance, warming the components. In a purely resistive circuit, one made only of resistors connected to a battery, all of the source's energy eventually dissipates as heat, an outcome called the heating effect of electric current. Working through the energy accounting precisely: moving a charge Q through a potential difference V takes work VQ, so a source supplying current I for time t supplies energy VIt in total, and since power is energy per unit time, the power delivered is P = VI. Using Ohm's law to substitute V = IR gives the heat produced in time t as H = I²Rt, a relationship called Joule's law of heating: heat produced is directly proportional to the square of the current, directly proportional to resistance, and directly proportional to time. This is not merely a formula to memorise; it directly explains real, everyday design choices, such as why the cord supplying an electric heater stays cool while the heater's own coil glows: the cord is deliberately made of thick, low-resistance copper, while the heating coil is deliberately made of thin, high-resistance nichrome, so nearly all the I²R heating happens exactly where it is wanted, in the coil, and almost none of it happens in the cord carrying current to it.

The heating effect of electric current is far from purely a nuisance; entire categories of everyday devices are built specifically to make use of it. Electric irons, toasters, ovens, kettles and heaters all rely directly on this same effect, deliberately generating heat exactly where it is wanted. Even an ordinary electric light bulb is really a heating-effect device pushed to an extreme: its filament must get hot enough to glow, which is exactly why tungsten, with an unusually high melting point around 3380 degrees Celsius, is used almost exclusively for bulb filaments, and why the filament is thermally isolated and surrounded by unreactive gas, to survive at that temperature without burning away. But the very same effect, allowed to happen somewhere it should not, is also the exact hazard that a fuse exists to prevent. A fuse is simply a short length of wire, made from a metal or alloy with a carefully chosen, appropriately low melting point, placed deliberately in series with the device or circuit it protects. If a current larger than the fuse's rated value ever flows, I²R heating raises the fuse wire's temperature past its melting point, the wire melts, and the circuit breaks automatically, well before that same excess current could overheat the wiring itself or start a fire. Domestic fuses are rated in fixed standard values, such as 1 A, 2 A, 3 A, 5 A or 10 A, and are chosen specifically to match the maximum current the protected circuit or appliance is meant to safely carry.

The rate at which a source delivers, or a circuit consumes, electrical energy is called electric power, P, and using the relationships already established in this chapter, it can be written three equivalent ways: P = VI, or, substituting from Ohm's law, P = I²R = V²/R. The SI unit of power is the watt (W), the power used by a device drawing one ampere of current at a potential difference of one volt, but a single watt is far too small a unit to describe most real appliances conveniently, which is why the kilowatt (1000 W) is used in practice instead. Since energy is simply power multiplied by time, the practical, everyday unit of electrical energy is not the joule at all, but the kilowatt-hour (kWh), commonly just called a 'unit' on an electricity bill: exactly the energy consumed by a 1-kilowatt appliance running continuously for one hour, equal to 3.6 x 10⁶ joule. An electricity bill, in other words, is not billing you for electrons, since electrons themselves are never consumed or used up at all, only moved through the circuit and back; what an electricity provider genuinely charges for is the energy required to keep those electrons moving against resistance, hour after hour, appliance after appliance, tracked in exactly this kilowatt-hour unit.