This circuit cannot be reduced to a combination of series and parallel connections. Kirchhoff’s rules, special applications of the laws of conservation of charge and energy, can be used to analyze it. (Note: The script E in the figure represents electromotive force, emf.)
Kirchhoff’s rules
Kirchhoff’s first rule—the junction rule. The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
Kirchhoff’s second rule—the loop rule. The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.
Explanations of the two rules will now be given, followed by problem-solving hints for applying Kirchhoff’s rules, and a worked example that uses them.
Kirchhoff’s first rule
Kirchhoff’s first rule (the
junction rule ) is an application of the conservation of charge to a junction; it is illustrated in
[link] . Current is the flow of charge, and charge is conserved; thus, whatever charge flows into the junction must flow out. Kirchhoff’s first rule requires that
(see figure). Equations like this can and will be used to analyze circuits and to solve circuit problems.
Making connections: conservation laws
Kirchhoff’s rules for circuit analysis are applications of
conservation laws to circuits. The first rule is the application of conservation of charge, while the second rule is the application of conservation of energy. Conservation laws, even used in a specific application, such as circuit analysis, are so basic as to form the foundation of that application.
The junction rule. The diagram shows an example of Kirchhoff’s first rule where the sum of the currents into a junction equals the sum of the currents out of a junction. In this case, the current going into the junction splits and comes out as two currents, so that
. Here
must be 11 A, since
is 7 A and
is 4 A.
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,
, rather than potential energy, but the two are related since
. 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.
[link] illustrates the changes in potential in a simple series circuit loop.
Kirchhoff’s second rule requires
. Rearranged, this is
, which means the emf equals the sum of the
(voltage) drops in the loop.
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.)
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