Problem — long wires and thick wires

Problem

A coil of thin wire (high resistance) heats up more than a thick wire (low resistance), even though it carries the same current. Why? And why are mains power cables “thick”, while those of a toaster are thin? Answer in five lines.

Solution

Why the thin wire heats up more — for the same current, the power dissipated through Joule heating is $P = R\,i^2$ So, for fixed $i$, whoever has the greater resistance heats up more. Resistance depends on geometry through the second law of Ohm: $R = \rho\,\frac{L}{A}$ A thin wire (small cross-section $A$) and long (large $L$) has a high $R$; a thick wire has a large $A$ and hence a small $R$. With the same current, the thin wire dissipates far more and heats up.

Mains power cables (thick) — they must carry current without heating up or wasting energy: a small $R$ is desired, hence a large cross-section. The power dissipated $R\,i^2$ is pure waste (and a hazard), to be minimised.

Toaster wire (thin) — here heat is the purpose: the high resistance of the thin wire (in a high-resistivity alloy) is intentional, because it is precisely $R\,i^2$ that produces the heat that toasts the bread.

Same physics, opposite goals: minimise $R$ to transport, maximise it to heat.

$\ev{P = R\,i^2 \text{ and } R = \rho L/A:\ \text{thin wire} \Rightarrow R \text{ high} \Rightarrow \text{more heat for the same } i}$

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Topics: Circuiti elettrici Concepts: Legge di Ohm · Effetto Joule