Resistance & Resistivity

PHYCurrent, Drift & Resistance

This subtopic links the resistance of a sample to the material property resistivity through its geometry, $R=\rho l/A$ , and to temperature through the linear coefficient $\alpha$ . The key idea is that resistivity depends only on the material (and temperature), while resistance also depends on length and cross-section.

ISC numericals routinely ask you to find $\rho$ from measured $R$ , predict $R$ at a higher temperature, or rescale $R$ when a wire is stretched.

Resistance from resistivity

R = \dfrac{\rho \, l}{A}, \qquad \rho = \dfrac{R A}{l}

R

resistance (

\Omega

\rho

resistivity (

\Omega \cdot m

l

length (m),

A

cross-sectional area (

m^2

). For a round wire of diameter

d

A = \pi d^2/4

Variation of resistance with temperature

R_T = R_0 \left[ 1 + \alpha \, (T - T_0) \right]

R_0

resistance at reference temperature

T_0

R_T

at temperature

T

\alpha

temperature coefficient of resistance (

^{\circ}C^{-1}

). For metals

\alpha>0

Stretched wire (volume conserved)

R' = R \left( \dfrac{l'}{l} \right)^{2}

Stretching to

k

times the length at constant volume makes

A

drop by

k

and

R

rise to

k^2 R

; tripling the length gives

9R

Resistivity and the microscopic model

\rho = \dfrac{m}{n e^{2} \tau}

m

electron mass,

n

free-electron density,

e

electronic charge,

\tau

relaxation time. Resistivity rises with temperature because

\tau

falls.

Resistivity is intrinsic to the material; resistance is extrinsic and changes with shape, so a thicker or shorter wire of the same metal has lower $R$ but identical $\rho$ .
For a circular wire use $A=\pi d^2/4$ , not $\pi d^2$ or $\pi r^2$ with the diameter mistaken for the radius.
Convert all lengths and areas to SI (metres and $m^2$ ): $1\,\mu m^2 = 10^{-12}\,m^2$ and $0.5\,\text{mm} = 5\times10^{-4}\,\text{m}$ .
$\alpha$ for metals is positive (resistance rises with temperature); for semiconductors and insulators it is negative.
Stretching keeps volume constant, so $A l$ is fixed; combine $R=\rho l/A$ with $A=V/l$ to get $R \propto l^2$ .
Resistivity depends on temperature: $\rho_T = \rho_0[1+\alpha(T-T_0)]$ has the same form as the resistance relation.
The reference temperature in the temperature relation is whatever temperature $R_0$ was measured at — usually $20^{\circ}C$ in these problems, not $0^{\circ}C$ .

Where the marks go

Using diameter as radius in $A=\pi r^2$ , which makes the area (and resistivity) four times too large.
Forgetting SI conversions for $\mu m^2$ and mm, giving answers off by many orders of magnitude.
Taking $R \propto l$ for a stretched wire instead of $R \propto l^2$ at constant volume.
Plugging the temperature difference in the wrong direction or using $0^{\circ}C$ as $T_0$ when $R_0$ was actually given at $20^{\circ}C$ .

How the board asks it

Numericalresistivity from R, l, A
A copper wire of length $2\,\text{m}$ and diameter $0.5\,\text{mm}$ has a resistance of $0.174\,\Omega$ . Calculate the resistivity of the copper.
Numericalstretched wire, volume conserved
A wire of resistance $R$ is stretched uniformly until its length is doubled. Calculate its new resistance in terms of $R$ .
Numericalresistance variation with temperature
The resistance of a coil is $25\,\Omega$ at $20^{\circ}\text{C}$ . If $\alpha=4.0\times10^{-3}\,^{\circ}\text{C}^{-1}$ , calculate its resistance at $80^{\circ}\text{C}$ .
Give reasonsresistivity intrinsic vs resistance extrinsic
Give reasons: when a wire is cut into two equal halves and the halves are joined in parallel, the resistance changes but the resistivity of the material does not.
Distinguishsign of alpha, metal vs semiconductor
Distinguish between the variation of resistivity with temperature for a metallic conductor and for a semiconductor, with reference to the sign of $\alpha$ .
Define / statedefinition and SI unit of resistivity
Define resistivity of the material of a conductor and state its SI unit. On what factors does it depend?

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.