Resistivity and superconductivity

From the object to the material

Resistance is a property of one particular object. Make the wire longer and R rises in proportion, because the carriers undergo more collisions with the lattice on the way through; make it fatter and R falls in proportion, because a larger cross-sectional area provides more parallel conducting paths. Divide the geometry out, the same move that turned a spring constant into the Young modulus, and the material's own number is left behind.

Resistivity ρ belongs to the material alone, at a stated temperature, and its unit is the ohm metre, Ω m, not ohms per metre. The values span an astonishing range. Copper sits near 1.7 × 10⁻⁸ Ω m, good insulators reach 10¹⁶ Ω m, and those two lie some twenty-four orders of magnitude apart.

Metals and temperature

Warm a metal and its resistivity rises. The lattice ions vibrate with greater amplitude, the drifting carriers collide with them more often, and the same pd drives less current. Over everyday ranges the rise is roughly linear. It is the same mechanism that curved the filament lamp's characteristic.

Required practical 5 pins the number down. Measure the wire's diameter with a micrometer at several places and average, take V and I readings for a series of lengths, then plot R against L. The gradient is ρ/A, so ρ comes out as gradient × A. Nearly all the uncertainty arrives through the diameter, because A depends on d² and so doubles whatever percentage error d carries.

Thermistors

A thermistor is a semiconductor component whose resistance falls steeply as it warms; only this negative temperature coefficient (ntc) type is considered. Heating a semiconductor frees extra charge carriers, and the flood of new carriers outweighs the extra lattice collisions, so the net resistance drops.

That steep, reliable fall is what makes the component useful. A thermistor works as a resistance thermometer. Put one where the temperature matters, a car engine, an incubator, a smart thermostat, and read the temperature off the resistance, with the component's own resistance-temperature graph as the calibration.

Light-dependent resistors

Change the stimulus and the same argument runs a second time. A light dependent resistor (LDR) is a wafer of semiconductor, and in the dark almost every one of its electrons is locked into a bond. The number density of conduction electrons, the n in I = nAvq, is minute, so the wafer's resistance is enormous, a hundred kilohms and upwards.

Light undoes that, one electron at a time. A photon absorbed in the wafer can transfer its energy to a bound electron and free it, so the brighter the illumination the more conduction electrons there are to carry a current. A larger n means more current for the same pd, which is to say a lower resistance: the same LDR that reads 100 kΩ in darkness reads about 1 kΩ in daylight. Cover it again and the freed electrons settle back into bonds within milliseconds, and the resistance climbs.

Notice what never entered that account. The lattice did not soften and nothing was heated. In a metal the carriers are all free already, so warming can only shake the ions harder and scatter them more, and the resistance rises. In both semiconductor components it is the supply of carriers that changes instead: heat frees them in the thermistor, light frees them in the LDR, and either way n climbs while R falls.

Superconductivity

Cool certain materials below a critical temperature and their resistivity does something no ordinary conductor manages. It drops to exactly zero. Not small, not negligible, zero. A current started in a superconducting loop circulates without measurable loss for years, which is how MRI magnets run. The zero has conditions beyond temperature that A-level does not examine: push the magnetic field or the current density past the material's own critical values and the superconductivity is lost too.

The critical temperature depends on the material, from a few kelvin for simple metals to above 130 K for certain ceramic compounds. Two applications carry the marks. Superconducting coils make the very strong magnetic fields inside MRI scanners and maglev systems, since currents up to the material's limit can flow without heating anything. Superconducting cables would allow power transmission with no resistive loss, because I²R vanishes when R does.