Abstract
Kirchhoff’s law states that the total of the emf values in a closed loop equals the sum of the potential drops in that loop. Kirchhoff’s first and second laws are used to verify that in a closed loop, energy is neither created nor destroyed (Scherz, Paul, and Simon Monk 2013). Energy is converted from one form to another. Electric charge is given the necessary energy to flow in the circuit by the electromotive force source, preferable batteries into circuit elements. Electrical energy is converted into heat energy recorded as impedance in capacitors, resistors, or inductors. In the case of a lamp, electrical energy will be converted into light energy.
This experiment is aimed at verifying the validity of the Kirchhoff’s law of conservation of energy and charge. In the analysis, batteries are used as the sources of electromotive force whereas resistors are used as the circuit elements. Measurements of voltages and currents in the loop were taken. In the first experiment, current values were first assumed, in the second experiment, real values of current were measured by the use of a BK as an ammeter.
Introduction
In Kirchhoff’s law, terminologies such as a junction, branch, elements, and loop et al. are used. A junction in an electric circuit is a point where three or more connections (wires) join; in a junction, current divides or come together (Kosky 2013). A branch is termed as a path that connects two junctions. It may contain one, two or numerous elements such as resistors, capacitors or inductors. A loop is a closed path with two or more branches. In Kirchhoff’s laws, two fundamental conservative rules are embodied: conservation of electric charge and conservation of electric energy. Current flows in each branch in a circuit. As current flows through the wires, electric charge in the closed circuit is conserved. When current divides up, as in the junction containing elements in parallel, the total current through the branches must be equal to the original current. For instance, three elements in parallel, total current in the branches is given as; .This expression means that no charge is gained or lost when the current branches into parts in the circuit. The algebraic sum of currents (i) at a junction equals to zero; Kirchhoff’s Current Law.
Energy is conserved in a closed loop: as the current traverses through a circuit loop, all the potential energy changes caused by elements in the circuit is equal to the total electric potential energy available at the origin. or . There expressions show that no energy is gained or lost as one loops through the circuit. The algebraic sum of potential changes in a closed loop are equal to zero; Kirchhoff’s Voltage Law.
The figure below represents the schematic diagram of the circuit. Three resistors and two batteries are used.
Power supply for the circuit is provided by the batteries given in terms of EMFs. The difference between the rated voltage of the EMF supply and the actual operation value in the circuit, is quite small. Sometimes the difference is ignored and assumed that the batteries provide their rated voltage such that and respectively.
Aims and objectives
1. Apply the Kirchhoff’s rules to multiloop circuits.
2. Describe how the Kirchhoff’s rules are related to conservation of electric charge and electric energy.
Procedure
Using a large digital multimeter (DMM) as an ohmmeter, the preciseness of the three resistors was measured, and their values recorded on the DATA SHEET #1.
The circuit pictured above was constructed that had a separate post between the two EMF providers for R2 connection. This was meant for the later of convenience when the meter was attached. The connection was made such that the batteries were connected opposite to each other. Their two negative terminals were connected to each other. The DMM was used as a voltmeter to measure the potential differences across the terminals of the two batteries.
After the circuit was complete and current started flowing through the resistors, the unknown values of the currents were used, no direct measurements were done at this stage. Just a reasonable guess was about which way the current through each resistor is going. Arrows were drawn above each resistor, and appropriate labeling was done.
4. using the DDM as a voltmeter, the differences, and across the three resistors were measured. When measuring the voltages, the measurement was done according to the assigned directions for the currents, indicated by arrows drawn in the circuit. The lead for the positive input of the meter was connected to the side of resistor corresponding to a tail of the current arrow that of the COMmon input of the DMM was connected to the arrowhead of the current in assumption. This is because the conventional current flowed from the high potential to the lower potential. The sign got from for each voltage, and the current was consistent with the choice of current direction.
The batteries were disconnected from the circuit for later use in the experiment. Kirchhoff’s law was used to construct enough equations in DATA SHEET #2 and #3 to achieve the values of the unknown currents I1, I2, and I3.
The batteries were connected to measure the current directly using the BK handheld meter as the ammeter. The ammeter was connected in series with a resistor. The current reading of 200 mA scale was chosen, and readings with 2 mA setting were taken. The ammeter was placed in the circuit so that the current of assumption entered the meter at the positive mA input and left it through the COMmon terminal. The data was recorded on DATA SHEET #4. The same procedure was used to measure the other two assumed currents.
Data analysis
DATA SHEET #1
A. DIRECT MEASUREMENTS of resistors using the handheld BK meter. Uncertainties in the measurements were mainly from the instruments. An assumption of 0.15 for OHMMETER VALUES was assumed.
B. DIRECT MEASUREMENT OF POTENTIAL CHANGES (VOLTAGE) USING THE DMM ACROSS EACH RESISTOR SEPARATELY, ASSUME UNCERTAINTY OF 0.05%
C. CALCULATION OF CURRENTS USING OHM’S LAW
On DATA SHEET #2. Kirchhoff’s law was combined together with Ohm’s law, Vi=RiIi where subscript I represents resistors 1, 2, and 3. Two equation; for loop 1 and 2 respectively were deduced from this relationship. Kirchhoff’s law was used to write the third equation
The labeled loops are imaginary paths that were arbitrarily chosen through the circuit.
DATA SHEET #2
The two loop equations and one node evaluation were written on this sheet of paper. The figure below was used come up with the equations.
DATA SHEET #3
D. DIRECT MEASUREMENT OF CURRENTS. An uncertainty of 1.0 assumed.
FOR THE 200 mA setting
FOR THE 2 mA SETTING
The diagram below shows schematic circuit of the final correct directions
QUESTION AND ANSWERS
1. Was the initial assumption for directions of current in the circuit correctly represented? Answer-NO
2. Did the two different meter settings for the current measurements make any variation in the currents measured? WHY?
Answer-YES
Setting the ammeter to 200 mA recorded nothing on the scale since the setting was out of range, that is, the setting was very large compared to the amount of current to be measured in the loop.
At loop #1
The emf from the battery (Ea) is equal to the drop at resistors R1 and R2. Currents I1 and I2 are involved in the loop.
1
At node 1, the currents can be represented as
..2
At loop #2
The emf from the battery (Eb) is equal to the drop at resistors R3 and R2. Currents I3 and I2 are involved in the loop.
3
At node 2, the currents can be represented as
.4
Discussion
This lab experiment aims at verifying the certainty of conservation of energy using the Kirchhoff’s Law model. Circuits are applied to transfer electric energy from the EMF sources to the elements, resistors; devices that utilize the energy. The Kirchhoff’s first and second law (KCL) and (KVL) respectively are used to determine the real conservation of energy and charge in the loop (Kosky 2013).
In this experiment, the junction rule is based on the conservation of electric charge. This rule states that “the net electric charge in an isolated system is conserved.” The charge from the EMF sources flowed into the junction where the wires joined; no build up was noticed. All the charge coming into the junction was equal to the total charge leaving the node. Charge flowing per unit second is referred to as the current, and it is equal to the one out of the junction (Irwin, and Nelms 2011). The junction rule then verified the validity of energy conservation by stating that the amount of the total current directed into the junction was equal to the magnitude of the total current directed of the junction.
The experiment confirms the Kirchhoff’s rule by proving that in an electrical circuit, energy is neither created nor destroyed, it is only transformed from one form to another. The batteries, emf sources gave electric charges energy to move in the circuit. The resistors in the circuit recorded values of resistance: electrical energy is converted into heat energy in the resistors. Heat energy is the energy lost out of the circuit through the resistors. When the charge goes round the circuit for a complete round and arrives back at the origin, there is no net change in energy. The gains in the circuit are matched to the losses (Alexander, Charles, and Matthew, Sadiku 2009).
The Kirchhoff’s rule expresses the conservation of energy by stating that the sum of all drops equals the sum of all rises in a closed loop. Mathematically, the Kirchhoff’s rule was represented as a sum of the voltages and equated to zero; Therefore, Kirchoff's rule is just a simplification of the Faraday’s law of induction. It holds on the assumption that there is no fluctuating magnetic field in the closed loop.
Conclusion
As quoted in the aims of the experiment, in a closed loop, energy is conserved as it only converted from one form to another. The closed circuit is a straightforward model of how electrical energy and electrical charge are conserved. The voltage that is generated in the emf is equal to the voltage that is recorded in the resistors. Also, the current entering a node is equal to the current out of the node. Therefore, from the experiment, it is true to say that Kirchhoff’s law of conservation of energy and charge is correct.
Recommendation.
The rated voltage of an emf source is assumed as the voltage of operation, yet there is some voltage drop due to internal resistances. In future, the effect of internal resistance should be put into consideration.
Work cited
Alexander, Charles K, and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. Boston: McGraw-Hill, 2009. Print.
Irwin, J D, and R M. Nelms. Basic Engineering Circuit Analysis. Hoboken, N.J: John Wiley, 2011. Print.
Kosky, P G. Exploring Engineering: An Introduction to Engineering and Design. Oxford: Academic Press, 2013. Internet resource.
Scherz, Paul, and Simon Monk. Practical Electronics for Inventors. New York: McGraw-Hill, 2013. Internet resource.
