Problem — copper with zero resistivity

Problem

Imagine copper had resistivity $\rho = 0$ at room temperature. What would happen to: (a) a circuit with a bulb and a battery; (b) an electric convector heater; (c) the coils of a motor? Discuss in five lines for each case.

Solution

With $\rho = 0$ the copper connecting wires would have resistance $R = \rho L/A = 0$: no voltage drop and no heat along the cables. But only copper becomes ideal; components made of other materials keep their resistance.

(a) Circuit with a bulb and a battery — the bulb’s filament is made of tungsten, not copper: it keeps its resistance. The bulb still works, indeed better, because no voltage is lost across the copper wires any more: the full e.m.f. reaches its terminals and it shines at maximum. Careful though: if the copper wires directly closed the battery circuit (without the bulb), we would have a perfect short circuit, with current limited only by the battery’s internal resistance.

(b) Electric convector heater — it works precisely thanks to the resistance of its heating element (a high-resistivity alloy) which dissipates $P = R\,i^2$. If the element were made of copper with $\rho = 0$, it would have no resistance and would produce no heat: the convector heater would no longer heat up. Made instead of resistive alloy, it would keep working, while the copper supply cables would waste no energy.

(c) Motor coils — the windings are made of copper: with $\rho = 0$ the Joule losses in the windings would disappear, making the motor almost perfect from a resistive standpoint (only magnetic and mechanical losses would remain). The flip side: at start-up, with no resistance to limit the current, there would be enormous inrush currents, hard to manage for the power supply (the inductance would still limit rapid changes, though).

$\ev{\text{bulb: shines (non-copper filament); copper convector heater: does not heat; motor: no losses but enormous inrush currents}}$

Collegamenti

Argomenti: Circuiti elettrici Concetti: Legge di Ohm Competenze: Analisi di casi limite e fantafisica