Kirchhoff’s Second Rule

Kirchhoff’s second rule (the loop rule) is an application of conservation of energy. The loop rule is stated in terms of potential, \(V\), rather than potential energy, but the two are related since \({\text{PE}}_{\text{elec}}=\text{qV}\). Recall that emf is the potential difference of a source when no current is flowing. In a closed loop, whatever energy is supplied by emf must be transferred into other forms by devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit. This figure illustrates the changes in potential in a simple series circuit loop.

Kirchhoff’s second rule requires \(\text{emf}-\text{Ir}-{\text{IR}}_{1}-{\text{IR}}_{2}=0\). Rearranged, this is \(\text{emf}=\text{Ir}+{\text{IR}}_{1}+{\text{IR}}_{2}\), which means the emf equals the sum of the \(\text{IR}\) (voltage) drops in the loop.

Part a shows a schematic of a simple circuit that has a voltage source in series with two load resistors. The voltage source has an e m f, labeled script E, of eighteen volts. The voltage drops are one volt across the internal resistance and twelve volts and five volts across the two load resistances. Part b is a perspective drawing corresponding to the circuit in part a. The charge is raised in potential by the e m f and lowered by the resistances.

The loop rule. An example of Kirchhoff’s second rule where the sum of the changes in potential around a closed loop must be zero. (a) In this standard schematic of a simple series circuit, the emf supplies 18 V, which is reduced to zero by the resistances, with 1 V across the internal resistance, and 12 V and 5 V across the two load resistances, for a total of 18 V. (b) This perspective view represents the potential as something like a roller coaster, where charge is raised in potential by the emf and lowered by the resistances. (Note that the script E stands for emf.)

This lesson is part of:

Circuits and DC Instruments

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