The resistance of a metallic wire increases when the wire is made longer. Think of a water pipe: the longer the pipe, the more friction the water experiences, so it takes more effort to push the water through. Similarly, electrons face more collisions as they travel a longer distance, which raises the resistance.
Mathematically, this is expressed as:
\$R \propto L\$
where L is the length of the wire.
A wire with a larger cross‑sectional area offers less resistance. Imagine a wide highway versus a narrow one: cars (electrons) can move more freely on the wide road. The resistance decreases as the area increases:
\$R \propto \frac{1}{A}\$
where A is the cross‑sectional area of the wire.
\$A = \pi r^2 = \pi (0.5\times10^{-3}\,\text{m})^2 = 7.85\times10^{-7}\,\text{m}^2\$.
\$R = 1.68\times10^{-8}\,\Omega\cdot\text{m} \times \dfrac{2.0\,\text{m}}{7.85\times10^{-7}\,\text{m}^2} \approx 0.043\,\Omega\$.
| Length (m) | Area (mm²) | Resistance (Ω) |
|---|---|---|
| 1.0 | 1.0 | 0.021 |
| 2.0 | 1.0 | 0.043 |
| 1.0 | 2.0 | 0.0105 |
Remember: In questions, always look for the formula \$R = \rho \dfrac{L}{A}\$ and check units.
• If the problem gives diameter, convert to radius first.
• If resistivity (ρ) is not provided, you may need to use a standard value for the material.
• Pay attention to the direction of the relationship: longer → higher, larger area → lower.
• Use the analogy of water flow or traffic to explain your reasoning in short answer questions.