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, and. At a junction, by the conservation of electrical charge, the currents into a junction equal the currents leaving the junction. 6 V A 2 Ω V 2 12 V B C 6 Ω 4 Ω Loop1 Loop 2 D Figure 1 = + By conservation of electrical charge, this means that charge cannot pile up or vanish at a junction. This current equation may be written - - = 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: + = (current in = current out) (1) Equation 1 may be written in mathematical notation as Σ I i = 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. V resistor = 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 V i = 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 = + (2) with directions as seen in Figure 1. Going around loop 1 as indicated in Figure 1 with Kirchoff s second rule we have: V 2 = 0 or with known values 6 - (2) 12 (4) = 0 and + 2 = -3 (3) Similarly, around loop 2, starting at battery 2, V 2 + = 0 or with known values 1 2 (6) + (4) = 0 and 3 2 = 6 (4) Equations 2, 3 and 4 constitute a set of three equations with three unknowns,, and they can be solved to give: = - 11 3 A 15 = - A 11 12 = A 11 The negative values of and 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, and. Then using Kirchoff s rules calculate, and and compare the values. Complete Table I. V R 4 Table I Measured Value Theoretical value Percent error
B) Connect the circuit shown in Figure 3 and experimentally find the current, and. Then using Kirchoff s rules calculate, and and compare the values. Complete Table II. I1 V Table II Measured value 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. 6 V 470 Ω 1000 Ω V 2 12 V 680 Ω