Joule Heating

When a current $i$ flows through a resistor, the product $\Delta V\cdot i$ represents the power dissipated as heat: this is the Joule effect, identified experimentally by James Prescott Joule in 1841. It is the phenomenon that heats the filament of a light bulb, the element of a heater, or the wire of a fuse.

Key formula — Joule heating

The power dissipated in a resistor $R$ carrying a current $i$, with voltage drop $\Delta V = Ri$, is $\ev{P_J = R\,i^2 = \Delta V\cdot i = \frac{(\Delta V)^2}{R}}$

The three expressions are equivalent: one moves from one to another by substituting the first law of Ohm $\Delta V = Ri$. The relationship between the three forms is thus a simple algebraic rewriting; which to use depends on which quantities are known in the problem.

Why is energy dissipated?

Microscopically, the conduction electrons, accelerated by the electric field in the wire, collide with the ions of the crystal lattice, losing kinetic energy: this energy becomes thermal vibrational energy of the lattice, that is, heat. The drift velocity of the electrons remains on average constant — it is a “terminal” regime, analogous to that of a raindrop falling at limiting speed through the atmosphere: the energy supplied by the electric field does not accumulate as kinetic energy, but is continuously transferred to the lattice. This is the classic example of electrical work becoming heat, and it finds its natural place within the first law of thermodynamics (Mazzoldi 2008).

Example — An electric heater

An electric heater of $P = 2000$ W connected to the mains at $\Delta V = 230$ V draws a current $i = \frac{P}{\Delta V} \approx 8{,}7\ \mathrm{A}$ and its element has resistance $R = \frac{(\Delta V)^2}{P} \approx 26\ \Omega$ To generate 10 MJ of heat (enough to warm a 50 m³ room of air by 5 °C) it takes $\ev{\Delta t = \frac{10^7}{2000} = 5000\ \mathrm{s} \approx 83\ \text{minutes}}$

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Topics: Circuiti elettrici Concepts: Effetto Joule · Potenza

Related exercises: Problema — resistenza riscaldante · Problema — riscaldatore sottoalimentato · Esercizio svolto — resistore usato come riscaldatore