Understand resistivity as a material property, explore its temperature dependence, and discover the extraordinary phenomenon of superconductivity.
AQA A-Level Physics · Unit 5: Electricity
🔗Resistivity equation Apply R = ρL/A and know the unit of resistivity (Ω m)
📊Comparing materials Use resistivity values to compare conductors, semiconductors, insulators
🌡️Temperature dependence Explain how temperature affects resistivity in metals and semiconductors
❄️Superconductivity Define the transition temperature and zero resistivity
🔬Measuring resistivity Describe the required practical: micrometer + metre rule method
📐Geometry effects Explain how R depends on length and cross-sectional area
Resistivity: A Material Property
The resistance of a conductor depends on three things: the material it is made of, its length, and its cross-sectional area. Experiments show:
R is proportional to length L (longer wire → more resistance)
R is inversely proportional to cross-sectional area A (thicker wire → less resistance)
These observations are combined in the resistivity equation:
R = ρL / A
Symbol
Quantity
Unit
R
Resistance
Ω
ρ (rho)
Resistivity
Ω m
L
Length of conductor
m
A
Cross-sectional area
m²
Resistivity ρ: A material constant that measures how strongly the material resists electric current, independent of its shape or size. Unit: Ω m (ohm metres).
Rearranging: ρ = RA/L. So the resistivity can be determined by measuring the resistance, length, and cross-sectional area of a sample of the material.
Typical values of resistivity:
Silver: 1.59 × 10⁻⁸ Ω m (best conductor)
Copper: 1.7 × 10⁻⁸ Ω m
Nichrome: ~1.1 × 10⁻⁶ Ω m (used in heating elements)
Silicon: ~1.0 × 10³ Ω m (semiconductor)
Glass: ~10¹² Ω m (insulator)
Temperature Dependence of Resistivity
Metals (conductors): As temperature increases, the positive ions in the metallic lattice vibrate with greater amplitude. These vibrations cause more frequent collisions between conduction electrons and the lattice, impeding electron flow. This increases resistivity (and therefore resistance).
For metals: resistivity increases approximately linearly with temperature above room temperature. This is why a filament lamp has a much higher resistance when hot than when cold.
Metals: as temperature increases → more lattice vibrations → more electron-lattice collisions → higher resistivity.
Semiconductors (intrinsic): As temperature increases, more electrons gain enough thermal energy to break free from their parent atoms and enter the conduction band. The number density n of free charge carriers increases significantly with temperature. This more than compensates for the increased scattering, so resistivity decreases with increasing temperature.
Semiconductors: as temperature increases → more free carriers liberated → lower resistivity. (Opposite to metals.)
This difference in behaviour is exploited in thermistors: NTC (negative temperature coefficient) thermistors are made from semiconductor materials and have resistance that decreases with temperature. They are used in temperature sensors and thermostats.
Superconductivity
Some materials exhibit a remarkable phenomenon: when cooled below a critical temperature called the transition temperature (T_c), their electrical resistance drops to exactly zero. This phenomenon is called superconductivity.
Superconductor: A material that conducts electricity with exactly zero resistance below its transition (critical) temperature T_c.
Properties and implications of superconductivity:
Resistivity = 0 exactly below T_c (not just very small)
Zero resistance means zero energy loss as heat (I²R = 0)
Persistent currents can flow indefinitely without a voltage source
Transition temperatures for most superconductors are very low (near absolute zero, e.g. mercury at 4.2 K, lead at 7.2 K)
"High-temperature" superconductors (ceramic materials) have T_c up to ~130 K — still far below room temperature
Applications of superconductors:
MRI (Magnetic Resonance Imaging) scanner magnets — huge superconducting coils generate strong magnetic fields without energy loss
Particle accelerators (e.g. Large Hadron Collider) — superconducting bending magnets
Potential: lossless power transmission cables
Room-temperature superconductivity remains one of the most sought-after goals in materials physics. Current superconductors require expensive liquid nitrogen (77 K) or liquid helium (4 K) cooling.
Geometry and Wire Design
The relationship R = ρL/A has practical consequences for wire design:
Effect of length: Doubling the length doubles the resistance (R ∝ L). Long cables in the home have greater resistance than short ones — this is why extension leads with very thin wire can overheat.
Effect of cross-sectional area: Doubling the diameter quadruples the area, which halves the resistance (R ∝ 1/A). Thicker wires have lower resistance — domestic ring mains use thicker cables than lighting circuits because they carry more current.
Combined effect: For a fixed volume V = AL of material, longer and thinner wires have much higher resistance than shorter and fatter ones.
From R = ρL/A and I = nAvq (drift velocity), we can show:
ρ = 1 / (nqμ)
where μ is the mobility of the charge carriers. This links the macroscopic property (resistivity) to microscopic quantities (carrier density and mobility).
Resistivity is a material property. Resistance is a property of a specific sample. R = ρL/A connects the two, depending on geometry.
A copper wire (ρ = 1.7 × 10⁻⁸ Ω m) has length 5.0 m and diameter 2.0 mm. Calculate its resistance.
R_A = 8 × R_B — wire A has 8 times the resistance of wire B.
Q1. What are the SI units of resistivity?
Q2. How does the resistivity of a metal change as its temperature increases?
Q3. A wire has resistance 8.0 Ω, diameter 1.5 mm, and length 2.0 m. Calculate its resistivity.
Q4. What is the transition temperature of a superconductor?
Q5. A wire is cut in half. How does its resistance change?
Challenge Q1. A cylinder of carbon (ρ = 3.5 × 10⁻⁵ Ω m) has diameter 4.0 mm and length 8.0 mm. Calculate its resistance. Then calculate what length of copper wire (ρ = 1.7 × 10⁻⁸ Ω m, diameter 0.5 mm) would have the same resistance.
Challenge Q2. Explain quantitatively why it is more efficient to transmit 10 MW of power at 400 kV rather than at 11 kV along a cable of resistance 5.0 Ω. Calculate power lost in each case.
🔬 Required Practical Measure the resistivity of a wire using a micrometer and metre rule
Aim
To determine the resistivity of a metal wire by measuring resistance for different lengths, and measuring the wire's diameter with a micrometer.
Equipment
Resistance wire (e.g. nichrome or constantan) on a metre rule
Micrometer screw gauge
Ohmmeter (or voltmeter and ammeter)
Crocodile clips / sliding contact
Power supply (low voltage), connecting leads
Method
Measure the diameter of the wire at multiple positions using the micrometer. Calculate the mean diameter d, then A = πd²/4.
Fix one crocodile clip at the zero end of the wire. Move the second clip to L = 0.10 m, 0.20 m, 0.30 m … 1.00 m.
At each length, measure the resistance R (using ohmmeter, or calculate R = V/I using ammeter and voltmeter).
Record L and R in a table.
Analysis
Plot R (y-axis) against L (x-axis). From R = ρL/A: the graph should be a straight line through the origin with gradient = ρ/A.
gradient = ρ / A
ρ = gradient × A
Calculate A from the mean diameter, then multiply by the gradient to find ρ.
Safety and Experimental Considerations
Keep current low to avoid heating the wire — this would change its resistivity. Avoid touching the wire during measurements. Ensure connections are clean to minimise contact resistance.
Use short connection time to prevent heating
Check the wire is straight and not kinked
Measure d at 5+ positions, including perpendicular orientations, to check for non-circular cross-section
Resistance of connecting leads — subtract or keep leads much shorter than wire
Results Table
Length L (m)
Resistance R (Ω)
0.10
0.20
0.40
0.60
0.80
1.00
Analysis Questions
1. Why should you not connect the circuit for a long time before taking each reading?
2. Suggest one improvement to reduce the uncertainty in your value of ρ.