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Resistivity and superconductivity

Resistance belongs to a particular wire; reshape it and R changes with the geometry. Divide the shape out and what remains is resistivity, the material's own property, which temperature can push around and, in a few remarkable materials, switch off entirely.

Year 12AQA 3.5.1.3

Builds on Current, charge and the direction problem and Stress, strain and the Young modulus.

IN THIS TOPIC

  • Use ρ = RA/L, with the unit Ω m, to move between a wire's resistance and its material's resistivity.
  • Describe qualitatively how temperature affects the resistance of metals and of ntc thermistors, and give thermistor applications.
  • Describe superconductivity below a critical temperature, and its applications.

WHAT YOU PROBABLY THINK

A material has a resistance.

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 run the gauntlet for longer; make it fatter and R falls in proportion, because more lanes carry the traffic. The same move that turned the spring constant into the Young modulus works here: divide the geometry out.

ρ = RALON YOUR DATA SHEET
Resistance depends on the wire's shape: longer means more, fatter means less; resistivity divides the shape outLAdouble L: double Rdouble A: half R
FIG. 1A wire's resistance scales with its shape: proportional to length, inversely proportional to cross-sectional area. Resistivity is what remains when both are divided away.

Resistivity ρ belongs to the material alone, 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 while good insulators reach 1016 Ω m, 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, and it is the same mechanism that curved the filament lamp's characteristic.

Thermistors

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

Resistance against temperature: a metal rises gently, an ntc thermistor falls steeplytemperatureRmetalntc thermistor
FIG. 2Resistance against temperature: the metal climbs gently, the ntc thermistor falls steeply as heat frees additional charge carriers.

That steep, reliable fall is the application: a thermistor is a resistance thermometer. Put it where the temperature matters, a car engine, an incubator, a smart thermostat, and read temperature off the resistance, using the component's resistance-temperature graph as the calibration.

Superconductivity

Cool certain materials below a critical temperature, and their resistivity does something no ordinary conductor ever does: it drops to exactly zero. Not small, not negligible, zero. A current started in a superconducting loop circulates without loss, indefinitely.

A superconductor's resistivity drops to exactly zero at its critical temperaturetemperatureρcritical temperaturezero, exactlyordinary metal
FIG. 3An ordinary metal's resistivity falls smoothly as it cools; a superconductor's drops discontinuously to zero at the critical temperature and stays there.

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 produce the very strong magnetic fields inside MRI scanners and maglev systems, because enormous currents flow without heating; and superconducting cables promise power transmission without resistive loss, since I2R vanishes when R does.

THE EXAM BIT

  • The unit of resistivity is the ohm metre, Ω m. Writing Ω m−1 is a definition error, not a slip, and it is tested directly.
  • Keep the nouns attached: R belongs to the wire, ρ to the material. “The resistivity of the wire is 5 Ω” loses marks twice over.
  • Doubling a wire's diameter quarters its resistance: A depends on d2. The d-versus-r and the square are the two traps in one line.
  • The thermistor mechanism is about carrier numbers: heating frees more charge carriers. “The ions vibrate less” is the metal story told backwards, and scores nothing.
  • Superconductivity: resistivity is zero at and below the critical temperature, which depends on the material. Say zero, not very small.

CHECK YOURSELF

A wire of length 2.5 m and diameter 0.40 mm has a resistance of 1.2 Ω. Find the resistivity of its material.

Show a hint

Area from the diameter first, in metres, then rearrange the resistivity equation.

Show the answer

Area: A = πd2/4 = π × (0.40 × 10−3)2 / 4 = 1.26 × 10−7 m2.

ρ = RAL = 1.2 × 1.26 × 10−7 / 2.5 = 6.0 × 10−8 Ω m, a value typical of a pure metal.

R belongs to the wire.

ρ belongs to the material.

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