Law of conservation of energy

In steady state, energy does not accumulate anywhere in the circuit: everything the sources feed in every second is dissipated every second by the resistors. A global power balance follows.

Principle — Energy balance (steady state)

The power overall fed in by the sources equals the power overall dissipated by the resistors: $\ev{\sum_k \mathcal{E}_k\,i_k \;=\; \sum_j R_j\,i_j^2}$

Equivalently, the algebraic sum of all the powers is zero:

$\sum_k P_{\text{gen},k} \;-\; \sum_j R_j\,i_j^2 = 0$

This result is simply the work–energy theorem applied to the charges: what the source does on the circuit (positive work), the resistors convert entirely into heat. Nothing is lost, nothing is created.

Not an equation independent of Kirchhoff

The energy balance does not add new information: it is obtained by multiplying each loop equation by its corresponding current and summing. It is, however, a valuable checking tool, and often a quicker route in problems where powers are known instead of resistances.

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Topics: Electric circuits Concepts: Power Skills: Conservation of energy

Related exercises: Problem — resistance and power of an ohmic conductor · Problem — ranking the power of three light bulbs · Problem — current and power in a resistance