Science (086) — AI-generated practice question

(i) Factors affecting resistance:

The resistance of a metallic conductor depends on:

1. Length (l): $R \propto l$
2. Area of cross-section (A): $R \propto \dfrac{1}{A}$
3. Nature of material

Combining: $R = \rho \dfrac{l}{A}$

Resistivity (ρ): It is the constant of proportionality in the relation $R = \rho \dfrac{l}{A}$. It is a characteristic property of the material of the conductor. SI unit: Ω m.

---

(ii) Calculation of length of copper wire:

Given: $d = 0.4 \text{ mm} = 4 \times 10^{-4}$ m, $\rho = 1.62 \times 10^{-8}$ Ω m, $R = 10$ Ω

$$A = \frac{\pi d^2}{4} = \frac{3.14 \times (4 \times 10^{-4})^2}{4} = \frac{3.14 \times 16 \times 10^{-8}}{4} = 1.256 \times 10^{-7} \text{ m}^2$$

From $R = \rho \dfrac{l}{A}$:

$$l = \frac{R \times A}{\rho} = \frac{10 \times 1.256 \times 10^{-7}}{1.62 \times 10^{-8}} = \frac{1.256 \times 10^{-6}}{1.62 \times 10^{-8}} \approx 77.5 \text{ m}$$

Length of copper wire required ≈ 77.5 m

---

(iii) Resistance of aluminium wire (same length and diameter):

$$R_{Al} = \rho_{Al} \frac{l}{A} = 2.63 \times 10^{-8} \times \frac{77.5}{1.256 \times 10^{-7}} \approx 16.2 \text{ Ω}$$

Since $\rho_{Al} > \rho_{Cu}$, the aluminium wire has higher resistance (~16.2 Ω) compared to copper (10 Ω) for the same dimensions.

Source: Chapter 11 – Electricity, Section 11.5 Factors on which the resistance of a conductor depends

---

• (i) Examiners expect the three factors named clearly, both proportionality statements written, the combined formula $R = \rho l/A$, and a one-line definition of resistivity with its SI unit. Don't skip the unit.
• (ii) The most common mistake is forgetting to convert diameter to radius or to metres. Use $A = \pi d^2/4$ directly (not $\pi r^2$, unless you halve the diameter first — both are equivalent). Show substitution clearly for full marks.
• (iii) Simply apply the same formula with $\rho_{Al}$ and compare. A clear statement that resistance increases because aluminium has higher resistivity is essential for the mark.