State, qualitatively, the relationship of the resistance of a metallic wire to its length and to its cross-sectional area
Resistance of a Metallic Wire – Cambridge IGCSE Physics (0625)
1. What is resistance?
- Resistance (R) is a property of a material that opposes the flow of electric charge.
- It is measured in ohms (Ω). For a linear (ohmic) resistor the relationship between the potential difference (V) and the current (I) is given by Ohm’s law:
V = I R
2. Why does a metallic wire have resistance?
- Free electrons in a metal drift under the influence of an electric field.
- Collisions with the lattice atoms impede this drift, producing a voltage drop – this is the resistance.
3. Factors that affect the resistance of a uniform metallic wire (material and temperature kept constant)
| Parameter | How it changes | Effect on resistance |
|---|---|---|
| Length (L) | Increase | Resistance increases – directly proportional (R ∝ L) |
| Length (L) | Decrease | Resistance decreases |
| Cross‑sectional area (A) | Increase | Resistance decreases – inversely proportional (R ∝ 1/A) |
| Cross‑sectional area (A) | Decrease | Resistance increases |
4. Quantitative relationship – the resistivity equation
R = ρ · L ⁄ A
- ρ (rho) – resistivity of the material (Ω·m). It is a constant for a given metal at a specified temperature.
- L – length of the wire (m).
- A – cross‑sectional area (m²). For a cylindrical wire, A = π r² or A = π (d/2)².
Exam tip: When a question asks only for the *relationship* (e.g. “state how R varies with L”), you can write “R ∝ L” and “R ∝ 1/A”.
5. Temperature dependence of resistance (metals)
For most metals the resistance varies approximately linearly with temperature over the range used in the IGCSE.
R = R₀ [1 + α (T – T₀)]
- R₀ – resistance at a reference temperature T₀ (usually 20 °C).
- α – temperature coefficient of resistance (typical values: copper 4.0 × 10⁻³ K⁻¹, aluminium 3.9 × 10⁻³ K⁻¹).
- α is positive for metals → resistance rises when the temperature rises.
6. Measuring resistance (IGCSE practical)
- V‑I method – Connect the unknown resistor in series with an ammeter, apply a known voltage, read V and I, then calculate R = V/I. Plotting a straight‑line V‑I graph also gives the slope = R.
- Ohmmeter / multimeter – Connect the leads across the resistor; the instrument displays the resistance directly. Important: the component must be disconnected from any circuit and from any power source.
7. Series and parallel combinations of resistors
| Configuration | Formula for total resistance Rtot |
|---|---|
| Series (end‑to‑end) | Rtot = R₁ + R₂ + R₃ + … |
| Parallel (same two nodes) | 1⁄Rtot = 1⁄R₁ + 1⁄R₂ + 1⁄R₃ + … |
Worked example (series + parallel)
R₁ = 2 Ω, R₂ = 3 Ω, R₃ = 6 Ω. R₂ and R₃ are in parallel; this combination is in series with R₁.
- Parallel part: 1⁄R₂₃ = 1⁄3 + 1⁄6 = ½ → R₂₃ = 2 Ω.
- Series total: Rtot = R₁ + R₂₃ = 2 Ω + 2 Ω = 4 Ω.
8. V‑I characteristic of an ohmic resistor
- Straight line through the origin → constant resistance (ohmic).
- Gradient (ΔV/ΔI) gives the value of R.
- If the graph curves, the device is non‑ohmic (e.g. a filament lamp).
9. Sample exam‑style questions (AO1–AO2)
- Qualitative – length
A copper wire of length 2.0 m has R = 0.5 Ω. If the length is doubled while the area stays the same, what is the new resistance?
Answer: R ∝ L, so R doubles → 1.0 Ω.
- Qualitative – area
The same wire is now cut so that its cross‑sectional area is halved (length unchanged). What happens to the resistance?
Answer: R ∝ 1/A, so R doubles → 1.0 Ω.
- Quantitative – temperature
A 5.0 Ω copper wire at 20 °C (α = 4.0 × 10⁻³ K⁻¹) is heated to 70 °C. Find its resistance.
Solution: R = 5.0[1 + 4.0 × 10⁻³ (70‑20)] = 5.0[1 + 0.20] = 6.0 Ω.
- Series/parallel calculation
Two identical 3 Ω resistors are connected in parallel, and this combination is placed in series with a 2 Ω resistor. Find Rtot.
Solution: Parallel part Rp = (3 · 3)/(3+3) = 1.5 Ω. Total R = 1.5 Ω + 2 Ω = 3.5 Ω.
- V‑I graph interpretation
A straight‑line graph passes through (2 V, 0.4 A) and (6 V, 1.2 A). Determine the resistance.
Solution: Gradient = ΔV/ΔI = (6‑2)/(1.2‑0.4) = 4/0.8 = 5 Ω.
10. Practical safety (key points for the IGCSE)
Safety box – Working with resistive elements
- Never touch a wire that has been carrying current – it may be hot.
- Use insulated leads and ensure all connections are tight to avoid sparking.
- Do not exceed the rated voltage of a resistor; overheating can cause burns or fire.
- When measuring resistance with a multimeter, disconnect the component from any power source.
- For high‑current circuits include a fuse or a suitable current‑limiting device.
11. Link to later topics (electromagnetism & power)
Resistance determines the heat produced in a conductor: P = I²R. In d.c. motors, transformers and heating elements, the wire’s resistance influences efficiency, temperature rise and power loss. Understanding R = ρ L/A therefore helps predict performance and design safe, reliable equipment.
12. Suggested diagram for the syllabus
