Experiment 10Bench 10MZU-PE / FPP-10Ready

Electrical and Thermal Properties of Metals

Four-Point Probe & Thermal Bench

Sweep a metal rod through temperature on the four-point probe and test the Wiedemann-Franz law.

Duration: 3 hours
Kittel: Chapter 6
Topics: ResistivityMatthiessen's ruleWiedemann-Franz

Four-point probe & thermal bar

Copper rod · 4 mm² section

Running

Probe voltage

µV

Resistivity ρ

nΩ·m

Gradient ΔT

Conductivity κ

W/m·KOFF

Resistivity against temperature

Matthiessen's rule: ρ = ρ_residual + ρ_phonon(T)

As measuredWorking point

Thermal conductivity

κ(T) from the Wiedemann-Franz electronic term plus phonons

κ at working point

440.7W/m·K

Electronic share

98.9 %

Lorenz number

L = κρ/T tested against L₀ = 2.44 × 10⁻⁸ W Ω K⁻²

Measured LL₀ = 2.44

L / L₀

1.011

Room-temperature comparison

ρ at 300 K for every rod in the rack (logarithmic)

15.9

16.8

22.1

26.5

52.8

69.9

96.1

NiCr

1100.0

Values in nΩ·m. Nichrome is nearly seventy times more resistive than copper.

Bench meters

Constant-current source · nanovoltmeter · differential thermocouple

Resistivity

Temperature difference

Specimen

ρ at 300 K

16.8nΩ·m

Residual ρ₀

0.15nΩ·m

Θ_D

343K

Residual resistance ratio

112.0

Marginally more resistive than silver but far cheaper, which is why the world is wired with it.

Transport bench control

MZU-PE / FPP-10

Ready

Specimen rod

Furnace / cryostat

Rod temperature300 K

Four-point probe

Probe current100 mA

Cold work+0.0 nΩm

Cold work adds dislocations, raising the residual term without touching the phonon term.

Thermal bar

Heater power2.00 W

ONOFF

Heater

SourceHeater

Key Equations

\rho(T) = \rho_{\text{res}} + \rho_{\text{ph}}(T)

Matthiessen's rule. The residual term is impurity and defect scattering and does not depend on temperature; the phonon term follows the Bloch-Grüneisen form, $\rho_{\text{ph}} \propto T^{5}$ well below $\Theta_D$ and $\propto T$ well above it.

\frac{\kappa}{\sigma T} = L_0 = 2.44\times10^{-8}\ \mathrm{W\,\Omega\,K^{-2}}

The Wiedemann-Franz law. It holds when the same electrons carry both charge and heat, and both are scattered the same way.

Why four probes?

A two-terminal measurement includes the resistance of the leads and of the contacts, which on a good metal is far larger than the sample itself. Passing the current through the outer pair and measuring the voltage with the inner pair means no current flows through the voltage leads, so their resistance does not matter.

Procedure

0/4

Aρ against TLoad copper and take the rod from 20 K to 900 K, recording the probe voltage and resistivity at about ten temperatures. Identify the region where ρ is proportional to T and the region where it is not.
BCompare the rodsAt 300 K, record the resistivity of every rod in the rack. Rank them, and check your ranking against the comparison panel.
CTest Wiedemann-FranzFor copper, silver and nichrome at 300 K, record ρ, the heater power, ΔT and κ. Work out L = κρ/T for each and compare with L₀ = 2.44 × 10⁻⁸.
DMatthiessen's ruleWith copper at 300 K, wind the cold-work control up and watch the whole ρ(T) curve shift upward without changing slope. Then check what it does to nichrome, whose residual term is already enormous.

Table 10 — transport measurements

0 entries

Record the probe voltage as well as the derived resistivity, so you can show your working.

#	Rod	T/ K	I/ mA	V probe/ µV	ρ/ nΩ·m	Heater P/ W	ΔT/ K	κ/ W/m·K	L/ 10⁻⁸ WΩK⁻²	Cold work/ nΩ·m
No readings yet — set the controls, then press “Record reading”.

Analysis & Reflection

Analysis questions

From your Task A data, plot $\rho$ against $T$ above $\Theta_D$ . Is it linear? Now plot $\log\rho$ against $\log T$ below 40 K and find the exponent. What does the Bloch-Grüneisen theory predict?
Silver is a better conductor than copper at 300 K, yet copper is used for wiring. Explain the physics of the difference and then the engineering of the choice.
Nichrome barely changes resistivity when heated. Explain that in terms of Matthiessen's rule and the disorder in a substitutional alloy.
For the pure metals your measured $L$ should sit close to $L_0$ ; for nichrome it comes out higher. Which heat carrier is responsible for the excess?
The residual resistance ratio shown on the panel is used industrially as a purity specification for copper. Explain why, and what cold working does to it.

Physics problems

A copper rod of cross-section 4 mm² carries 100 mA and gives 5.0 µV across probes 20 mm apart. Calculate $\rho$ , then check it against the bench.
Use $\kappa = L_0T/\rho$ to predict the thermal conductivity of copper at 300 K, and compare with the measured value. Where does the difference come from?
Explain why metals feel colder than wood at the same temperature, using the thermal conductivities you measured.
Why does the thermal conductivity of a pure metal rise to a peak at low temperature and then fall again? Consider each term in Matthiessen's rule in turn.