Cambridge A-Level Physics 9702 – Resistance and Resistivity
Resistance and Resistivity
Key Concepts
Resistance (\$R\$) is the opposition to the flow of electric current.
Resistivity (\$\rho\$) is an intrinsic property of a material, independent of its shape.
The relationship between them is \$R = \dfrac{\rho L}{A}\$, where \$L\$ is the length and \$A\$ is the cross‑sectional area.
For many conductors, especially metals, resistivity increases with temperature.
Temperature Dependence of Resistance
For a metallic conductor the resistance at temperature \$T\$ can be approximated by
\$R = R0\,[1 + \alpha\,(T - T0)]\$
where:
\$R0\$ – resistance at a reference temperature \$T0\$ (usually \$20^\circ\text{C}\$),
\$\alpha\$ – temperature coefficient of resistance (typical values: \$3.9\times10^{-3}\,\text{K}^{-1}\$ for copper, \$4.5\times10^{-3}\,\text{K}^{-1}\$ for tungsten),
\$T\$ – actual temperature of the conductor.
Why a Filament Lamp’s Resistance Rises with Current
When a voltage is applied, an electric current \$I\$ flows through the filament.
The filament dissipates electrical power as heat: \$P = I^2 R.\$
Because the filament is made of a metal (usually tungsten) with a high \$\alpha\$, its temperature rises sharply as \$P\$ increases.
The rise in temperature raises the resistivity \$\rho\$, and consequently the resistance \$R\$ (see the temperature‑dependence equation above).
Thus, as the current increases, the filament becomes hotter, its resistivity grows, and the measured resistance increases.
Quantitative Example (Tungsten Filament)
Assume a tungsten filament of length \$L = 5\,\$mm and cross‑sectional area \$A = 0.02\,\$mm². At \$20^\circ\text{C}\$ the resistivity of tungsten is \$\rho_{20}=5.6\times10^{-8}\,\Omega\!\cdot\!m\$.
The resistance more than triples as the filament temperature rises from room temperature to typical operating temperatures (\overline{2500} °C).
Implications for Circuit Analysis
Filament lamps cannot be treated as ohmic devices; \$V\$ and \$I\$ are not linearly related.
When analysing circuits with lamps, use the \$V\$–\$I\$ characteristic curve or iterative methods to account for the changing resistance.
In AC circuits, the same temperature effect applies, but the instantaneous power is \$P(t)=i(t)^2R(t)\$, leading to a dynamic temperature response.
Suggested diagram: A schematic showing a filament lamp in a simple circuit, with arrows indicating heating, temperature rise, and the resulting increase in resistance.
Summary
The resistance of a filament lamp increases with current because the electrical power dissipated as heat raises the filament’s temperature. Since the resistivity of the filament material (tungsten) rises sharply with temperature, the overall resistance follows the relation \$R = R0[1+\alpha(T-T0)]\$. This non‑linear behaviour is a key consideration in both theoretical calculations and practical circuit design at A‑Level physics.