Resistivity and conductionEdexcel International A Level Physics: Revision notes

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

$R = \frac{\rho l}{A}$

where $\rho$ is the resistivity of the material (unit: ohm metre, Ω m), $l$ the length and $A$ the cross-sectional area. Resistance is proportional to length and inversely proportional to area. For a round wire $A = \pi d^2/4$, so doubling the diameter divides the resistance by 4. Resistivity is a property of the material (at a given temperature), whereas resistance also depends on the shape.

Measure the diameter $d$ with a micrometer at several points along the wire and in perpendicular directions, then take the mean (check for zero error). Connect the wire in series with an ammeter and a variable supply, with a voltmeter across the length being tested. For several lengths $l$, measured with a metre rule against a crocodile clip or contact, record $V$ and $I$ and calculate $R = V/I$. Use small currents, switching off between readings, so the wire does not heat.

Plot $R$ against $l$: the graph is a straight line through the origin with gradient $\rho/A$, so $\rho = \text{gradient} \times A$. Using a graph reduces the effect of random errors, and an intercept shows contact resistance or a systematic error.

In a conductor of cross-sectional area $A$ with $n$ free charge carriers per unit volume, each of charge $q$ and mean drift velocity $v$, the current is

$I = nqvA$

The charge passing a cross-section in time $\Delta t$ is $nAv\Delta t \times q$, giving this result. Drift velocities are very small (about $10^{-4}$ m s⁻¹ in copper) even though the signal in a circuit spreads almost instantly.

Using $I = nqvA$:

• Conductors (metals): very large $n$ (about $10^{28}$–$10^{29}$ m⁻³), so a small drift velocity gives a large current: low resistivity (copper about $1.7 \times 10^{-8}$ Ω m)
• Semiconductors: much smaller $n$ (and $n$ rises with temperature), so a moderate resistivity
• Insulators: almost no free charge carriers, tiny $n$, so a very high resistivity (can exceed $10^{12}$ Ω m)

A much smaller $n$ means for a given p.d. a much smaller current.

In a uniform wire carrying a current $I$, the resistance is proportional to length. The p.d. between one end and a point at distance $x$ is $V = IR_x$, with $R_x \propto x$, so the potential varies linearly with distance along the wire. If the whole supply p.d. $V_0$ is across a wire of length $L$, the p.d. at $x$ is $V_0 x/L$. A graph of potential against distance is a straight line. This is the basis of the slide-wire (potentiometer) arrangement.