Resistivity and its Temperature Dependence: MDCAT Physics notes
Resistivity and its Temperature Dependence for MDCAT: R = ρL/A, conductivity, stretching wires, temperature coefficient α, metals versus semiconductors.
Factors affecting resistance
At a fixed temperature the resistance of a uniform wire is proportional to its length and inversely proportional to its cross-sectional area:
$$R = \rho\frac{L}{A}$$
$\rho$ is the resistivity (specific resistance) of the material. Resistance depends on length, area, material and temperature. It does not depend on the mass or colour of the wire as such.
Resistivity
- $\rho = RA/L$; SI unit ohm metre ($\Omega$ m).
- Resistivity is the resistance of a metre cube of the material, measured between opposite faces.
- It depends only on the material and its temperature. Cutting, stretching or bending a wire changes $R$ but not $\rho$.
- The reciprocal of resistivity is conductivity, $\sigma = 1/\rho$ (unit $\Omega^{-1}$ m$^{-1}$ or S m$^{-1}$).
- The reciprocal of resistance is conductance, unit siemens (S) = $\Omega^{-1}$.
Changing the shape of a wire
| Change | New resistance |
|---|---|
| Length × 3 (same area) | $3R$ |
| Length × 2, area × ½ | $4R$ |
| Length × ½, diameter × ½ (area ¼) | $2R$ |
| Length × 2, radius × 2 (area × 4) | $R/2$ |
| Stretched to $n$ times length (volume fixed) | $n^2R$ |
| Compressed so radius × 2 (volume fixed, length ¼) | $R/16$ |
When the volume is constant, $A$ goes up as $L$ goes down, so $R \propto L^2 \propto 1/A^2 \propto 1/r^4$.
Thin film example: a square film of side $s$ and thickness $t$, current across opposite edges: $L = s$, $A = st$, so $R = \rho/t$. With $\rho = 10^{-6}\ \Omega$ m and $t = 10^{-6}$ m, $R = 1\ \Omega$ whatever the side.
Temperature dependence
Over a moderate range the resistance of a metal varies linearly with temperature:
$$R_t = R_0(1 + \alpha\Delta T) \qquad \alpha = \frac{R_t - R_0}{R_0\,\Delta T}$$
$\alpha$ is the temperature coefficient of resistance: the fractional change in resistance per kelvin (increase in resistance per ohm of original resistance per degree rise). Unit: K$^{-1}$. The same form applies to resistivity, $\rho_t = \rho_0(1 + \alpha\Delta T)$.
| Material | Sign of $\alpha$ | Effect of heating | Reason |
|---|---|---|---|
| Metals (Cu, pure metals) | Positive | Resistance increases | Lattice ions vibrate more and scatter free electrons |
| Semiconductors (Si, Ge), carbon | Negative | Resistance decreases | More charge carriers are released |
| Alloys (constantan, manganin) | Nearly zero | Resistance nearly constant | Used in standard resistors |
A zero temperature coefficient means resistance does not change with temperature. Near absolute zero the resistance of a metal becomes very small, because lattice vibrations almost stop.
Key formulas
- $R = \rho L/A$
- $\sigma = 1/\rho$; $G = 1/R$
- $R_t = R_0(1 + \alpha\Delta T)$
Common MDCAT traps
- Doubling the length or cutting a wire changes $R$, never $\rho$.
- Resistivity is defined for a metre cube, not a metre or a metre square.
- Halving the diameter quarters the area; do not treat diameter like area.
- Semiconductors have a negative temperature coefficient; metals positive.
- Unit of $\rho$ is $\Omega$ m, not $\Omega$ m$^{-1}$.
Quick revision
- $R \propto L$ and $R \propto 1/A$.
- Conductivity is the reciprocal of resistivity.
- Siemens is the unit of conductance.
- Resistivity of a metal rises with temperature.
- Unit of $\alpha$ is K$^{-1}$.