Kirchhoff's Circuit Laws
The Current Law (Junction Rule)
The current law arises from the basic principle that charge is conserved. The law states
that the total current which enters a junction, must equal the total current which leaves the
junction.
In the circuit element above there are currents (conventional) entering and leaving the
junction. Applying the law one would obtain the relationship: I1+I2=I3 If this condition
were not true there would either be more less charge entering the junction compared to
the charge leaving the junction. This would suggest that it is not a steady state.
The Voltage Law
This law arises from the basic principle of conservation of energy. The law states that the
sum of all the voltage (potential difference) increases and decreases around a complete
path must add to zero. Since you have returned to the same position the change in electric
potential energy must equal zero.
There are certain steps which make it easier to obtain relationships. For an example of the
necessary steps, the example circuit below (fig a.) will be used. The circuit consists of
two batteries with emfs (voltage) ξ1 and ξ2 and internal resistances r1 and r2, connected to
two external resistances R1 and R2.
Fig. a
Fig. b
Steps: ( see Fig. b)
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Label the directions of conventional current. This is completely arbitrary.
o Currents I1, I2, and I3 have been labelled
One can label junctions and corners (with letters) to help indicate pathways taken.
Label the high potential (+) sides and low potential (-) sides of resistors and emf
sources.
o Conventional current always flows from high (+) to low potential (-)
through a resistor.
o The Long side of an emf source by definition is the positive side (+), and
the short side is the negative (-).
Follow a complete pathway in any direction and obtain a relationship. Note the
Ohm's law is used to obtain the potential difference across a resistor in terms of
the current and resistance.
o Example pathways
 Path AFEDCBA: +I2r2-ξ2-I1R1+ξ1-I1r1=0
 Path FCDEF: +I3R2+ξ2-I2r2=0
One of course could use the junction rule to obtain: I1+I2+I3=0
If a current is found to have a negative solution it means the actual conventional current
is the opposite direction. One would expect at least one of the currents in this example to
be negative. It would be impossible for all currents entering the junction and have none
leaving. Another possibility is that all the currents are zero.