Learning objectives
By the end of this section, you will be able to:
 Analyze a complex circuit using Kirchhoff’s rules, applying the conventions for determining the correct signs of various terms.
The information presented in this section supports the following AP® learning objectives and science practices:

5.B.9.1 The student is able to construct or interpret a graph of the energy changes within an electrical circuit with only a single battery and resistors in series and/or in, at most, one parallel branch as an application of the conservation of energy (Kirchhoff’s loop rule).
(S.P. 1.1, 1.4)

5.B.9.2 The student is able to apply conservation of energy concepts to the design of an experiment that will demonstrate the validity of Kirchhoff’s loop rule in a circuit with only a battery and resistors either in series or in, at most, one pair of parallel branches.
(S.P. 4.2, 6.4, 7.2)

5.B.9.3 The student is able to apply conservation of energy (Kirchhoff’s loop rule) in calculations involving the total electric potential difference for complete circuit loops with only a single battery and resistors in series and/or in, at most, one parallel branch.
(S.P. 2.2, 6.4, 7.2)

5.B.9.4 The student is able to analyze experimental data including an analysis of experimental uncertainty that will demonstrate the validity of Kirchhoff’s loop rule.
(S.P. 5.1)

5.B.9.5 The student is able to use conservation of energy principles (Kirchhoff’s loop rule) to describe and make predictions regarding electrical potential difference, charge, and current in steadystate circuits composed of various combinations of resistors and capacitors.
(S.P. 6.4)

5.C.3.1 The student is able to apply conservation of electric charge (Kirchhoff’s junction rule) to the comparison of electric current in various segments of an electrical circuit with a single battery and resistors in series and in, at most, one parallel branch and predict how those values would change if configurations of the circuit are changed.
(S.P. 6.4, 7.2)

5.C.3.2 The student is able to design an investigation of an electrical circuit with one or more resistors in which evidence of conservation of electric charge can be collected and analyzed.
(S.P. 4.1, 4.2, 5.1)

5.C.3.3 The student is able to use a description or schematic diagram of an electrical circuit to calculate unknown values of current in various segments or branches of the circuit.
(S.P. 1.4, 2.2)

5.C.3.4 The student is able to predict or explain current values in series and parallel arrangements of resistors and other branching circuits using Kirchhoff’s junction rule and relate the rule to the law of charge conservation.
(S.P. 6.4, 7.2)

5.C.3.5 The student is able to determine missing values and direction of electric current in branches of a circuit with resistors and NO capacitors from values and directions of current in other branches of the circuit through appropriate selection of nodes and application of the junction rule.
(S.P. 1.4, 2.2)
Many complex circuits, such as the one in
[link] , cannot be analyzed with the seriesparallel techniques developed in
Resistors in Series and Parallel and
Electromotive Force: Terminal Voltage . There are, however, two circuit analysis rules that can be used to analyze any circuit, simple or complex. These rules are special cases of the laws of conservation of charge and conservation of energy. The rules are known as
Kirchhoff’s rules , after their inventor Gustav Kirchhoff (1824–1887).