LAB 4: KIRCHOFF’S CIRCUIT RULES
(Reference: Physics Laboratory Experiments - J. D. Wilson, DC Heath & Co.)
Objectives:
Distinguish between circuit branches and junctions
Apply Kirchoff’s rules to multiloop circuits
Explain how Kirchoff’s rules are related to the conservation of charge and energy
Theory:
A junction is a point in a circuit at which three or more wires are joined together, or a
point where the current divides or comes together in a circuit.
A branch is a path connecting two junctions, and it may contain one element or two or
more elements.
A loop is a closed path of two or more branches.
Kirchoff’s rules:
These rules do not represent any new physical principles. They embody two fundamental
conservation laws: conservation of electrical charge and conservation of electrical
energy.
A current flows in each branch of a circuit. In Figure 1 these are labeled I1, I2 and I3. At a
junction, by the conservation of electrical charge, the currents into a junction equal the
currents leaving the junction.
A
R1
2Ω
V1
I1
V2
12 V
B
I3
I2
C
6Ω
6V
R2 4 Ω
Loop1
Loop 2
R3
D
Figure 1
I1 = I2 + I3
By conservation of electrical charge, this means that charge cannot “pile up” or “vanish”
at a junction. This current equation may be written
I1 - I 2 - I 3 = 0
Of course, we do not generally know whether a particular current flows into or out of a
junction by looking at a multiloop circuit diagram. We simple assign labels and assume
the directions the branch currents flow at a particular junction.
If these assumptions are wrong, we will soon find out from the mathematics, as will be
shown in the following example. Notice that once the branch current directions are
assigned at one junction, the currents at a common branch junction are fixed: for example
in Figure 1, at junction D:
I2 + I3 = I1 (current in = current out)
(1)
Equation 1 may be written in mathematical notation as
Σ Ii = 0
which is a mathematical statement of Kirchoff’s first rule or junction theorem:
The algebraic sum of the currents at any junction is zero
In a simple single-loop circuit as shown in Figure 2, it is easy to see that by conservation
of energy the voltage drop across the resistor must be equal to the voltage rise of the
battery.
V battery = V resistor
where the voltage drop across the resistor is by Ohm’s law equal IR i.e. Vresistor = IR. By
the conservation of energy, this means that the energy (per charge) delivered by the
battery to the circuit is the same as that expended in the resistances. The conservation law
holds for any loop in a multiloop circuit, although there may sometimes be more than one
battery and more than one resistor in a particular loop.
Similar to the summation of the currents on the first rule, we may write for the voltages
Kirchoff’s second rule or loop theorem:
Σ Vi = 0
or
The algebraic sum of the voltage changes around a closed loop is zero.
Since one may go around a circuit loop either in a clockwise or counterclockwise
direction, it is important to establish a sign convention for voltage changes. For example,
if we went around a loop in one direction and crossed a resistor, this might be a voltage
drop (depending on the current flow). However, if we went around the loop in the
opposite direction, we would have a voltage rise in terms of potential.
Example:
Apply Kirchoff’s rules to the circuit shown in Figure 1 and find the value of the current
in each branch.
Solution:
By rule 1 we have
I1 = I2 + I3
(2)
with directions as seen in Figure 1.
Going around loop 1 as indicated in Figure 1 with Kirchoff’s second rule we have:
V1 – I1R1 – V2 – I2R2 = 0
or with known values
6 - I1(2) – 12 – I2(4) = 0
and
I1 + 2I2 = -3
(3)
Similarly, around loop 2, starting at battery 2,
V2 – I3R3 + I2R2 = 0
or with known values
12 – I3(6) + I2(4) = 0
and
3I3 – 2I2 = 6
(4)
Equations 2, 3 and 4 constitute a set of three equations with three unknowns I1, I2, I3 and
they can be solved to give:
I1 = -
3
A
11
I2 = -
15
A
11
I3 =
12
A
11
The negative values of I1 and I2 indicate that the current is flowing in the direction
opposite to that indicated in Figure 1.
Procedure:
A) Connect the circuit shown in Figure 2 and experimentally find the current I1, I2 and I3.
Then using Kirchoff’s rules calculate I1, I2 and I3 and compare the values. Complete
Table I.
R1
I2
V
R3
I1
R2
I3
R4
Table I
Measured Value
(
)
V1
I1
I2
I3
Theoretical value
(
)
Percent error
B) Connect the circuit shown in Figure 3 and experimentally find the current I1, I2 and I3.
Then using Kirchoff’s rules calculate I1, I2 and I3 and compare the values. Complete
Table II.
R1
I1
R3
I3
I2
R2
V
Table II
Measured value
( )
V1
I1
I2
I3
Theoretical value
( )
Percent error
Questions
1) Mathematically state Kirchoff’s loop theorem.
2) Mathematically state Kirchoff’s junction theorem.
3) What is a multimeter? What quantities can you measure with it?
4) How should a voltmeter be connected to measure the potential difference across a
resistor – in series or in parallel?
5) How should an ammeter be connected to measure the current through a resistor – in
series or in parallel?
6) Which of Kirchoff’s rule is based on conservation of charge?
7) Which of Kirchoff’s rule is based on conservation of energy?
8) Use Kirchoff’s rules to calculate the currents in the following circuit.
I1
V1
6V
R1
470 Ω
I3
R2
1000 Ω
V2
12 V
I2
R3
680 Ω