Resistance and Resistor Networks
This domain tests the physical origin of resistance, the effect of temperature, and the reduction of series–parallel networks to a single equivalent resistance.
Resistance from dimensions
R = ρl/A, where ρ is resistivity in ohm-metre, l the length in metres and A the cross-section in m². Resistance rises with length and falls with area; resistivity is a property of the material, resistance is a property of the specimen.
Temperature effect
For metals, R_t = R_0(1 + αt), where α is the temperature coefficient (positive for copper, aluminium, tungsten). Carbon, electrolytes and semiconductors have negative α, so their resistance falls on heating.
Networks
- Series: R_s = R1 + R2 + … ; the same current flows through each resistor.
- Parallel: 1/R_p = 1/R1 + 1/R2 + … ; the same p.d. appears across each.
- Two resistors in parallel: R_p = R1R2/(R1 + R2).
Worked example
A copper wire of length 100 m and area 1 mm² (1 × 10⁻⁶ m²) has ρ = 1.7 × 10⁻⁸ Ω m. R = ρl/A = (1.7 × 10⁻⁸ × 100)/(1 × 10⁻⁶) = 1.7 Ω. If a 6 Ω and a 3 Ω resistor are joined in parallel across this combination in series with 4 Ω, the parallel part is (6 × 3)/(6 + 3) = 2 Ω, total 6 Ω; at 12 V the supply current is 2 A and the p.d. across the parallel pair is 4 V.
Traps
- Adding parallel resistances directly instead of adding conductances.
- Forgetting to invert 1/R_p at the end of the calculation.
- Substituting mm² into the formula without converting to m².
- Believing ρ changes with length; it changes with material and temperature only.
- Ignoring the power rating of a resistor when it is used in a network.