Energy and power in circuits

Every time a current flows through a resistor, electrical energy is converted into heat: this is the Joule effect. The moving charges collide with the conductor’s lattice, giving up energy that appears as heating. Quantifying this dissipation — and more generally the flow of energy in a circuit — is essential for correctly sizing components and installations.

Power dissipated by a resistor

The power dissipated in a current-carrying resistor can be written in three equivalent forms, obtained by combining $P = \Delta V \cdot i$ with Ohm’s law $\Delta V = R\,i$:

Key formula — power in the resistor

$P_R = R\,i^2 = \Delta V \cdot i = \frac{(\Delta V)^2}{R}$

The three expressions are the same thing seen from different angles: the first is handy when the current is known, the third when the voltage across the resistor is known. All of this power leaves the circuit as heat.

Power balance

In a circuit in steady state, a conservation-of-energy principle holds in the form of power: the power the sources feed into the circuit must equal the power dissipated by the resistors.

Principle — Power balance

$\sum_{\text{gen}} \mathcal{E}\,i = \sum_{\text{res}} R\,i^2$

On the left, the power delivered by each source (electromotive force times the current flowing through it); on the right, the power dissipated by the Joule effect across all the resistors. This balance is often the quickest way to check the consistency of the results of a network analysis: if the two sides do not match, there is an error somewhere.

Beware of parallel connections

The power balance explains an important, and at first sight counter-intuitive, practical fact: adding appliances in parallel increases overall consumption, it does not divide it.

Overloading the installation

At constant voltage, the lower the equivalent resistance, the higher the current, and hence the higher the dissipated power. Connecting too many appliances in parallel makes the equivalent resistance plummet, the total current rise, and the installation can be overloaded — with the risk of the wires overheating and the protections tripping.

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Topics: Electric circuits Concepts: Power · Joule effect

Related exercises: Problem — heating resistor · Problem — underpowered heater · Worked exercise — resistor used as a heater