When a circuit is closed, the potential difference causes charges to move within the circuit thus constituting current passing through various appliances to perform designated functions. The electrical power generated from the power source is not all utilized as desired as some of the power is lost in the form of heat dissipated on the circuit. Power loss is attributed to resistance within the circuit. Different elements or circuit parts have different resistance. Ohm’s law correlates the current, voltage and resistance within a circuit. The effective resistance from the network within the circuit can be calculated and used to determine the value of any unknown resistance of any appliance or any circuit part. In this lab report, investigation of Ohm’s law was conducted with the aim of determining the effective resistance. Using two different methods, the results showed conformity to theoretical values with limited errors.
1 Introduction
1.1 Background information
The function of potential difference is to make charges move within the circuit and result in current. Current is required in a circuit system to facilitate the operation of various circuit appliances. Different appliances require different current size to be operated. For instance, a 100 Watt bulb may not operate when a limited current is supplied across its terminals (Barsoukov et al., 2018). However, the same current supply, may be sufficiently enough to make a 5 Watt bulb to produce light. Current supply to the appliance can be increased or decreased by varying the potential difference at the circuit power supply. In most current carrying conductors, the relation between current and voltage supplied across the terminals are directly correlated. This correlation constitutes Ohm’s law which states that the magnitude of potential difference or voltage source across the circuit is directly proportional to the current passing through the appliances connected in the circuit (Jin et al., 2015; Krause and Rädler, 2016 and Mitchell et al., 2018). Materials which concurs with the Ohm’s law and displays the said relationship between current and voltage are referred to as Ohmic materials. On the other hand, materials which do not conform to Ohm’s law are referred to as non-ohmic materials (Bain, 2017).
The resistance of a current currying conductor is dependent on a number of factors such as cross sectional area of the conductor, length of the conductor and materials resistivity. Cross sectional area is inversely proportional to the resistance of a conductor. Conductor with larger cross sectional area have relatively lower resistance to the flow of current (Carbonell-Ballestero et al., 2015). Larger cross sectional areas provide a greater current density with respect to number of electrons available to move, and hence reduce resistance or opposition of current flow. This is why high power transmission are anchored on larger cables to minimize the resistance to the current flow. On the other hand, resistance and length are directly proportional. The longer the conductor, the higher the resistance offered (Davies et al., 2015).
1.2 Experimental Aim
In this experiment, investigation of Ohm’s law was conducted with the aim of confirming the resistance law with the following objectives
i. To build an electrical circuit diagrams that can be used to measure various variables through varying of other variables
ii. To measure the small lengths measurements using micrometer screw gauge
iii. To estimate the value of resistivity of constantan
2 Theory
According to Ohm’s law, the correlation between current I, potential difference or voltage V as well as resistance R can be derived mathematically by;
(i)
Replacing the proportionality sign in equation (i) with an equal sign, a constant of proportionality, in this case R, must be added. Thus equation (i) can be rewritten as follows;
(ii)
From equation (ii), it can be deduced that current within a circuit is a function of resistance magnitude. The higher the resistance, the lower the current and vice versa for a constant voltage supply across the circuit terminals (Chou et al., 2014). In other words, high resistance compels the current size to reduce so that the right hand side of equation (ii) becomes equal to the left hand side.
Resistance of a conductor is also directly proportional to the length of the conductor, thus;
(iii)
In addition, resistance of a conductor is also inversely proportional to the cross sectional area of the conductor, thus;
(iv)
Equation (iii) and (iv) can be combined as follows;
(v)
Introducing equal sign in equation (v) requires a new constant to be added as follows;
(vi)
Where the constant of proportionality is known as resistivity of the material and is a function that describes the degree of resistance offered to flow of current. Higher resistivity implies that there will be corresponding higher resistance and vice versa (Turner and Taborisskiy, 2017).
Substituting equation (vi) into equation (ii) gives an equation that connects voltage supply and current size to the length and cross sectional area pf the conductor.
(vii)
Note that equation (vii) is only applicable to a case where the resistance is to be determined the conductor but it cannot be used in cases of resistors as an appliance. For resistors network with known resistance, equation (ii) is most preferred. Different materials have different magnitude of impedance offered to the flow of current within the circuit. Nonetheless, within a circuit, it is possible to determine the effective resistance from all the appliances and conductors within the circuit. The effective resistance or impedance depends on the way the resistors network are arranged within the circuit. There are two types connection for resistance network, namely; series connection and parallel connection.
In series connection, all resistors are connected in the same line and have same current passing through all of them. Figure 1 shows a series resistance network with three resistor connected in series.
Figure 1: Series connection resistor network for three resistors
Same current magnitude flows across all the resistors but the voltage drop differs depending on the magnitude of the resistor itself. The effective voltage drop can be calculated by adding the respective voltage drops across the series network. The effective voltage from the network in figure 1 is given by;
(viii)
Applying equation (ii) on equation (viii) gives;
(ix)
From the series connection, the effective current equals the current that passes through the implements. Thus, equation (ix) can be rewritten as;
(x)
From equation (x), current can be factored out resulting in the equation for calculating the effective resistance in series connection.
(xi)
Equation (xi) shows that the effective resistance is equal to the sum of the individual resistance offered by each resistor.
On the other hand, in parallel connection, all resistors are connected in a way that the voltage across their terminals equals the effective voltage across the entire network. Parallel connection network of resistors is anchored on the junction rule which states that the current getting into the junction must be equal to the current getting out of the junction (Augustine et al., 2015). Figure 2 shows a parallel resistance network with three resistor connected in parallel.
Figure 2: Parallel connection resistor network for three resistors
Using the junction rule, it can be confirmed that current getting into the network at the junction labeled a is equal to the current getting out of the junction labeled b. However, this does not imply that current passing through the respective resistors are equal.
Unlike in series connection, the parallel connection network of resistors has differing current magnitude flowing across different resistors. The effective current flowing through all the resistors within the network must therefore be equal to the total current supplied by the effective potential difference. The effective current from the network in figure 2 is given by;
(xii)
Making current in equation (ii) the subject of the formula and then substituting it on equation (xii) gives;
(xiii)
Equation (xiii) can further be reduced considering that fact that all the voltages are similar. That is effective voltage across the entire circuit equals the voltage across all the individual resistors within the parallel network connection. Thus, equation (xiii) can be rewritten as;
(xiv)
By factoring out effective voltage, the equation of determining the effective resistance in parallel connection can be arrived at as follows;
(xv)
Equation (xv) shows that the inverse of effective resistance is equal to the sum of the individual inverse of resistance offered by each resistor. The effective resistance is therefore less than individual resistance.
3 Experimental Methodology
3.1 Experimental Apparatus
The following apparatus were used in conducting this experiment.
i. A micrometer
ii. A set of resistors of 1kΩ, 10kΩ and 15kΩ to be used in the network.
iii. 2 Digital multimeters DMM for measurements.
iv. A blue circuit box and sufficient wires for connecting the circuit parts.
v. Variable Voltage for a direct current power supply.
vi. A variable resistor with graded length scale.
vii. Phywe wire box, collection of wires.
3.2 Experimental Precaution
Before connection, precaution was taken to ensure that the right port for a 20 mA was used. Multimeter has two such ports, one labeled -20 A and another labelled 20 mA hence attention was therefore necessary to eliminate possible measurement errors. In addition, it was also ensured that before connection to the port a resistor was connected in series to minimize the current across the network to a value less than 20 mA. This was purposely done to avoid blowing up the fuse when heavy current is allowed to flow through it.
3.3 Experimental procedure
3.3.1 Task 1: Series Connection Network
The set up was connected as shown in figure 3 with R1 of 4.7 kilo ohms and R2 of 0.33 kilo ohms connected in series. A small potential difference of 2 V was applied from the power source after which measurement and recording of the respective potential drop V1, V2 and V3
as well as current through the connection in A using Digital multimeters. One the Digital multimeters was used as an ammeter for measurement of current while the other one set as a voltmeter for measurement of voltage drop.
The effective resistance RTotal, R1 and R2 were calculated from the data recorded in table 1 and the results associated with the possible uncertainties.
Figure 3: Experimental setup for series connection network (task 1)
3.3.2 Task 2: Parallel Connection Network
The set up was connected as shown in figure 4 followed by application of a small potential difference of 2 V from the power source. The potential across each component was then measured and recorded in table 2. Similar to the first task, measurements were conducted using the two Digital multimeters with one the set as an ammeter for measurement of current while the other one set as a voltmeter for measurement of voltage drop. The effective resistance RTotal, R1
and R2 were calculated from the data recorded in table 2 and the results associated with the possible uncertainties.
Figure 4: Experimental setup for Parallel resistor network (task 2)
3.3.3 Task 3: Series Connection Network
In this task the resistance variation with length of a specified conductor, constantan, was investigate. Different pieces of constantan wire with differing lengths and diameters (corresponding to the variation in cross sectional area) were used. Four constantan wires, labeled wire 1, wire 2, wire 3 and wire 4 were first subjected to measurements of length and thickness using micrometer screw gauge. The resistance of the wires was then measured and recorded using Ohm’s law and using direct approach.
3.3.3 Task 4: Data analysis procedure
With the data from the two tables, table 1 and table 2, a graph of resistance was drawn and effective resistance calculated from the slope of the graphs. The value obtained was compared with the corresponding effective resistance calculated from theoretical equations derived in the theory section. The later value was considered as the theoretical value and it was used to evaluate the experimental value and attach possible explanation to any uncertainty encountered.
4 Result and analysis
4.1 Results for Series Resistors Network
Table 1: Tabulation of experimental data for task 1
Current I (A)
Potential difference (V)
Resistance R from Ohm's law in ohms
0.07 x 10^-3
0.39
5571.43
0.11 x 10^-3
0.59
5363.64
0.15 x 10^-3
0.8
5333.33
0.19 x 10^-3
0.99
5210.53
0.23 x 10^-3
1.21
5260.87
By applying equation (xi), the effective resistance from the two resistors was calculated as follows;
Effective Resistance Rtotal = R1 + R2
Effective Resistance Rtotal = 4.7 X 103 + 0.33 X 103
= 5.03 x 103 Ohms
4.2 Results for Parallel Resistors Network
Table 2: Tabulation of experimental data for task 2
Current I (A)
Potential difference (V)
Resistance R from Ohm's law
6.52 x 10^-3
2
306.75
9.79 x 10^-3
3.01
3.7.46
12.9 x 10^-3
3.97
307.75
16.27 x 10^-3
5.01
307.9
19.46 x 10^-3
6
308.32
From equation (xv), the effective resistance for the parallel network of resistors was calculated as follows;
The total resistance was then calculated from the inverse of the above equation as follows;
4.3 Results for task 4
4.3.1 A graph of potential difference versus current for series resistors network
Figure 5 shows a straight line graph of the variation of potential and corresponding current variation for the series connection of resistors.
Figure 5: A graph of variation of voltage against current for series network
From the graph, the slope of the of line was considered as the effective resistance in accordance with equation (ii) derived in the theory section.
Slope
= 5166.67
4.3.2 A graph of potential difference versus current for Parallel resistors network
Figure 6 shows a straight line graph of the variation of potential and corresponding current variation for the parallel connection of resistors.
Figure 6: A graph of variation of voltage against current for parallel network
Similarly, the effective resistance was calculated from the slope of the of line as follows;
Slope
= 309.12
5. Uncertainties
Arguing that the values calculated from equation (ii) represented the theoretical and that calculated from the slope represented the experimental values of the effective resistance. The percentage error was first calculated as follows;
Error in task 1;
= 2.7%
Error in task 2;
= 2.5%
6 Discussion
According to Ohm’s law, the ratio of potential difference to the current supply is called resistance (Wilks et al., 2017). Different materials have different resistance applicable for different functions. Materials with very high resistance to the flow of current are termed as resistors. Resistors conduct current at a very low rate as compared to good conductors. For instance, an electric heater coil is made using a material with very high resistance to impede the flow of current generated and hence generated the required heat.
The value of effective resistance obtained in data for series network was in accordance with the theoretical explanation. The effective resistance for a series network of resistors is usually greater than each individual resistance value. Similarly, the value of effective resistance obtained corresponding to the parallel network of resistors was also in accordance with the theoretical explanation. The effective resistance for parallel network of resistors is usually less than each individual resistance value.
From the results obtained in task 4 through the graphical calculation of the respective effective resistance, it can be concluded that indeed the experiment was conducted successfully since the values from the two different methods were almost equal. However, it is necessary to note that all the methods were based on the Ohm’s law statements. The experimental findings mirrored the findings in the research by Kvatinsky et al., (2015).
7 Conclusion
The error in both tasks were relatively low implying that the experimental accuracy was very high as expected. The error can be attributed to the fact that some resistance within the circuit were not considered in the theoretical method (Chua, 2014; Kvatinsky et al., 2015; Yuan et al., 2015 and Yoshida " Furuya, 2015). According to (Lu et al., 2014), connecting wires used in the experiment also had some level of resistance which can be significant enough when the length is higher. This explains why the experimental values was always greater than the theoretical respective values (Jovcic " Ahmed, 2015). Other than resistance of the other appliances in the circuit, the error can also be attributed to measurement and calculation errors.
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