Resistance and Ohm's law

A voltage pushes and a current flows, but how much current flows for a given push depends on what the charge is being pushed through. A thick copper bar barely resists the flow; a thin coil of nichrome wire resists it strongly and glows red as a result. The property that decides how much current a given voltage produces is resistance, and the relation between the two is so simple and so central that most of practical circuit work is applying it.

Ohm's law

For a large class of materials, the current through a component is directly proportional to the voltage across it. Double the voltage and the current doubles. The constant of proportionality is the resistance, and the relation is Ohm's law:

with $V$ the voltage across the component in volts, $I$ the current through it in amperes, and $R$ the resistance in ohms, symbol $\Omega$. One ohm is one volt per ampere: a resistance of one ohm passes one ampere when one volt is placed across it. Rearranged, the law says $I=V/R$, which reads more directly as cause and effect: a fixed voltage drives a current that shrinks as the resistance grows. A component that obeys this proportionality is called ohmic, and a plain resistor is the deliberate embodiment of one, a part made to have a known, steady resistance and nothing else.

Not everything is ohmic, and it is worth saying so early to keep the law in its place. A filament lamp grows more resistive as it heats, so its current rises less than proportionally with voltage. A diode passes current one way and blocks it the other, obeying no such proportionality at all. Ohm's law is a superb description of metals and resistors at steady temperature, not a law of nature that everything must follow, and the skill is knowing which components it describes.

Example. A 220 ohm resistor has 5 V across it. What current flows through it?

$I=V/R=5/220=0.0227$ A, or 22.7 mA. Currents in small electronics are usually milliamps, thousandths of an ampere, which is why the milliamp is the everyday unit on a bench.

Now you. What voltage appears across a 1000 ohm resistor carrying 12 mA?

What resistance is made of

Resistance is not magic; it is the charge colliding with the material as it drifts. The free electrons pushed by the field do not sail cleanly through a metal but bump repeatedly against the vibrating atoms of the lattice, losing energy to them at each collision, and that constant obstruction is resistance. The energy lost heats the material, which is why a resistor warms and a filament glows, and it is the mechanism behind the power dissipation of the next lesson. Because the atoms vibrate more when hot, a metal collides with its electrons more often when heated and so grows more resistive with temperature, which is the filament lamp's behaviour explained.

The resistance of a particular piece of wire follows a clean geometric rule. It rises with the length, because a longer path means more collisions, and falls with the cross-sectional area, because a fatter conductor offers more lanes for the charge to travel in parallel:

Here $L$ is the length, $A$ the cross-sectional area, and $\rho$ the resistivity, a property of the material itself that says how resistive a unit cube of it is. Copper has a very low resistivity, which is why wires are made of it; nichrome has a high one, which is why heating elements are made of it. The formula is why long thin wires drop noticeable voltage while short fat ones do not, the fact that sized the battery leads in the quadcopter's power system, and it is the bridge from a material's nature to a component's behaviour.

Example. A copper wire 10 m long has a resistance of 0.5 ohms. A second copper wire of the same thickness is 30 m long. What is its resistance?

Resistance is proportional to length at fixed thickness, so tripling the length triples the resistance: $0.5\times 3=1.5$ ohms. The material and thickness are unchanged, so only the length ratio matters.

Now you. A wire is replaced by one of the same material and length but twice the cross-sectional area. What happens to its resistance?

Conductance, the other way to say it

Sometimes it is more natural to speak of how easily current flows rather than how strongly it is opposed, and that is conductance, simply the reciprocal of resistance, $G=1/R$, measured in siemens. A high conductance is a low resistance. The idea earns its place when components are combined in parallel a couple of lessons on, where conductances add as neatly as resistances add in series, but it is worth meeting now as a reminder that resistance and conductance are two views of one fact: the relationship between the push and the flow. That relationship, $V=IR$, together with the two conservation laws still to come, is enough to solve any resistive circuit ever drawn, and the next lesson turns to what the resistance is doing with the energy it takes from the charge.