Resistivity and conductionEdexcel A-Level Physics: Revision notes

The resistance of a uniform conductor depends on its material and shape:

R = ρl/A

where ρ is the resistivity (Ω m), l is the length and A the cross-sectional area. Resistivity is a property of the material (at a given temperature), whereas resistance also depends on the shape.

For a wire of diameter d, A = π(d/2)² = πd²/4, so R is proportional to 1/d². Doubling the diameter divides R by four.

Worked example: 2.0 m of copper (ρ = 1.7 × 10⁻⁸ Ω m) of diameter 0.50 mm has A = π(0.25 × 10⁻³)² = 2.0 × 10⁻⁷ m², so R = (1.7 × 10⁻⁸ × 2.0)/(2.0 × 10⁻⁷) = 0.17 Ω.

Current is the rate of flow of charge. In a conductor of area A with n charge carriers per unit volume, each carrying charge q, moving with mean drift velocity v, the charge passing a point in 1 s is the charge in a length v of the conductor:

I = nqvA

For a metal, q = e = 1.6 × 10⁻¹⁹ C. Because n is so large (about 10²⁸–10²⁹ m⁻³), v is tiny, typically fractions of a millimetre per second. A lamp comes on at once because conduction electrons are already present throughout the circuit, and all start to drift as soon as the field is set up.

For 3.0 A in copper of area 1.0 × 10⁻⁶ m², v = I/(nqA) = 2.2 × 10⁻⁴ m s⁻¹.

Resistivity ranges over about 20 orders of magnitude:

• Conductors (metals, about 10⁻⁸ Ω m): huge n, about 10²⁹ m⁻³ free electrons.
• Semiconductors (pure silicon, about 10³ Ω m): much smaller n, about 10¹⁶ m⁻³. n rises with temperature, so resistance falls.
• Insulators (glass, about 10¹² Ω m or more): electrons are held in bonds, so almost no free carriers.

From I = nqvA, for a given p.d. the current is proportional to n (q is fixed), so a smaller n means a smaller current, a greater resistance and a greater resistivity. The variation in n is far larger than any variation in drift velocity.

Method: measure the diameter of the wire with a micrometer at several places along it and in perpendicular directions, then take a mean. Fix the wire to a ruler and connect different lengths l into a circuit with an ammeter and voltmeter across the length. Measure V and I for each l, and calculate R = V/I.

Plot R against l. Since R = (ρ/A) l, the graph is a straight line through the origin with gradient ρ/A, so ρ = gradient × A.

Precautions: use a small current and switch off between readings so the wire does not heat (which increases R); repeat and average. Contact resistance at the clips gives a positive intercept but does not alter the gradient, which is another reason to plot a graph rather than use one length.