A longer wire has more resistance
- Resistance arises because the free electrons collide with the vibrating positive ions of the metal lattice as they drift along the wire.
- A longer wire gives them more lattice to travel through, so there are more collisions and the resistance is greater.
- Doubling the length of a uniform wire doubles the resistance, so 2 m2\ \text{m}2 m of a wire has twice the resistance of 1 m1\ \text{m}1 m of the same wire.
- Written as a proportionality this is R∝lR \propto lR∝l, which plots as a straight line through the origin when RRR is graphed against lll.
- A variable resistor uses exactly this: sliding its contact changes the length of resistance wire in the circuit, and so changes the resistance.
A thicker wire has less resistance
- A thicker wire has a larger cross-sectional area, which gives the electrons more routes through and lets more of them drift past each second.
- Resistance therefore falls as the wire is made thicker, and a wire of twice the cross-sectional area has half the resistance.
- Written as a proportionality this is R∝1AR \propto \dfrac{1}{A}R∝A1, so plotting RRR against 1A\dfrac{1}{A}A1 gives a straight line through the origin.
- Be careful when a question gives the diameter: the area of a round wire is A=πr2A = \pi r^{2}A=πr2, so doubling the diameter multiplies the area by four and cuts the resistance to a quarter.
- This is why the thick copper cable feeding a cooker has a far lower resistance than the thin flex on a phone charger, and so wastes much less energy as heat.
Using the two relationships together
- Combining them, the resistance of a wire is proportional to lA\dfrac{l}{A}Al for a given metal at a fixed temperature.
- In a comparison question, deal with the two factors one at a time and combine the effects at the end.
- Both relationships hold only while the temperature stays constant, since a hot wire has a higher resistance than a cold one.
- They also assume the same material, because constantan and copper wires of identical size have very different resistances.
Common Mistake
- Do not treat diameter as if it were area, because doubling the diameter multiplies the area by four.
- Resistance goes down as a wire is made thicker, so check the direction of the change before writing your answer.
- These relationships apply to a metallic conductor at constant temperature, so they cannot be used on a filament lamp that is heating up.
- A graph of RRR against AAA is a curve, not a straight line; the straight line comes from plotting RRR against 1A\dfrac{1}{A}A1.
Exam technique
- State the proportionality before calculating, as in "R∝lR \propto lR∝l, so tripling the length triples the resistance".
- For a qualitative question two clean sentences are enough: longer means more resistance, thicker means less.
- If a graph is asked for, name the axes, since RRR against lll and RRR against 1A\dfrac{1}{A}A1 both give straight lines through the origin.
- Watch for a question that changes length and thickness at the same time, and show each step separately so partial credit is available.
Self review
- State how the resistance of a metallic wire depends on its length.
- State how it depends on the cross-sectional area.
- Explain, in terms of electrons and ions, why a longer wire has a greater resistance.
- A 1.0 m1.0\ \text{m}1.0 m wire has resistance 4.0 Ω4.0\ \Omega4.0 Ω. What is the resistance of 2.5 m2.5\ \text{m}2.5 m of the same wire?
- Why does doubling the diameter of a wire reduce its resistance to a quarter?
