Lab Report Exp 17

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Feb 20, 2024

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Report for Experiment #17 DC Circuits Sebastian Krauss Lab Partner: Hayley Ha TA: Dave Anderson 10/2/2023

Abstract In this experiment we investigate the qualitative values of several circuits arranged in varying manners using different circuit components. With the use of a DMM or Digital Multimeter, we are able to read the voltage, current, and resistance values of any circuit element across any point on the circuit. Through equations relating to these three values, and through our actual observations, we are able to assess the effect of real-world confounding variables. Including the measurement tools tendency to change that which it measures. We investigate the behavior of these values and circuits for both one, and two batteries, for both one and two resistors, and with one and two DMMs, across both series and parallel. Outside of ideal circumstances, these three values must be found through individual measurement and calculations. Introduction Many household devices operate via direct current (DC). Direct current circuits allow the flow of charged particles in only one direction. This is different from the alternating current (AC) circuits that allow for the charge to flip directions repeatedly. Because of these limitations, the number of extra components for a direct current circuit are greatly reduced. An example of a component that produces a DC circuit is the battery. A battery uses chemical reaction to create a potential difference in voltage across the positive and negative terminals of the battery, allowing this voltage to be applied and stored. The positive terminal will always direct current towards the negative terminal. This experiment examines DC circuits, along with the components and various circuit configurations that could arise. In doing so a deeper relationship between voltage, current, and resistance can be ascertained. The constant voltage that the battery maintains is called the electromotive force (emf) ε . ε is defined as the ratio of work ΔW done to charge Δq (i.e., the energy required per unit charge) in moving Δq from the negative to the positive. With units of Joule/Coulomb. One Joule of work is needed to raise one Coulomb of charge by one Volt. 𝜀 = ∆? ∆𝑞 (1) Current (I) is defined as the charge Q per unit time. It is measured in Amperes. Resistance (R) is the ratio of voltage supplied and current. Higher resistance means less current flow. Resistance is measured in Ohms ( Ω ) 𝑅 = ∆? 𝐼 (2) Power P (energy/time and work per unit time ΔW/Δt ) is measured by the Watt (W), which is the product ΔV × I of voltage drop ΔV across the element and current I. Power can be defined by the equation below. 𝑃 = ∆? ∆𝑡 = ∆𝑉𝐼 = ∆? 2 𝑅 = 𝐼 2 𝑅 (3)

There are two rules that govern the currents and voltages in circuits known as Kirchhoff’s Rules . They are as follows. The Loop Rule: The algebraic sum of the electrical potential differences around any closed circuit is zero. 𝜀 − Δ𝑉 𝑏 − Δ𝑉 𝑤 = 0 ( 4 ) This shows that the difference in voltages across the circuit is equal to the emf of the battery. This also show that the resistances sum to equal the total resistance, as the current will remain constant in a series. On the other hand, the second of Kirchoffs rules covers the behavior of current over a parallel circuit. Showing that rather than being constant, the total current is equal to the sum of the current of junctions. The Junction Rule: At any junction, the algebraic sum of the currents flowing into the junction must be zero. Additionally, internal resistance dictates that the voltage difference will not be exactly equal to the emf, due to the resistance of the battery itself. In this experiment the use of a Voltmeter, Ammeter, and Ohmmeter will be critical. The function is as follows. A voltmeter measures the potential difference between two points called digital voltmeters. Ammeters measure current by opening a leg in a circuit and inserting it into the break. When a resistor is connected to the two leads, the terminals of the Ohmmeter become output terminals for a small current that flows through the resistor. The meter measures that current, which depends on the magnitude of the resistor, and converts it to a resistance value. Based on these instruments, the experiment allowed for analysis of the voltage, current, and resistance across differing circuits. Results would be recorded in an excel spreadsheet. Investigation 1 — Electromotive Force of Battery Combinations In investigation 1 we examine various combinations of batteries and wires, without the use of resistors. Start by creating a table in excel. Examine the DMMs (Digital multimeter). Set the DMM to measure voltage by pressing the DC V button. Connect both red (positive) and black (negative) terminals to the corresponding terminals of the battery using a patch cable. Record the voltage shown. Continue and measure the other battery, then arrange the batteries in combination using the wires and DMM. The circuit diagram for the position of the meter, battery, and cables in figure 1 . Additionally in figures 2 and 3, we can see the circuits in series and parallel, respectively, using two batteries. The record values of voltage for these circuits are listed in table 1 below. Figure 1. Voltmeter attached to a single battery.

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Figure 2. Voltmeter attached to two batteries in series. Figure 3. Voltmeter attached to two batteries in parallel.

Table 1. The voltage recorded is shown in the table below, along with the voltage recorded with the lead connections reversed. This voltage is equivalent to the emf of the battery. Includes voltage measurements for configuration in series and parallel, with errors for all measurements. Investigation 1 volt error reverse error Battery 1 1.5592 0.00005 -1.5621 0.00005 Battery 2 1.5373 0.00005 -1.5393 0.00005 Series 3.0971 0.00005 Parallel 1.538 0.00005 Additionally in this table, we have the values for the two batteries when connected in series, ie, connected in a continuous line of current, with recorded error. And when connected in parallel, i.e., with the two positive ends and negative ends connected to each other, with recorded error. The voltages seen in these configurations align with each other based on expected voltage values. That being the voltage is summed in series, and constant in parallel. The practical usage of batteries in parallel are numerous, as they will not reduce in power over consecutive addition to the circuit. The lights in your house are in parallel and as such do not dim greatly. Series circuits on the other hand increase the voltage of the circuit with each addition, and are less prone to overheating. Investigation 2-Ohmic Resistance In investigation two, we introduce Ohmic resistors. Ohmic resistors are often color coded based on resistance and tolerance of nominal values. The Power rating refers to the amount of power that can be dissipated without significant heat buildup. This investigation includes a similar setup to investigation 1, and will use the same batteries, DMM and wires. Start by disconnecting the batteries and use the DMM as

an ohmmeter to measure the value of the 100 Ohm resistor in the provided circuit element box. The recorded value of 100 Ohms matches up perfectly with the nominal resistance and is within nominal tolerance. Then measure and record the value for the battery we will be using in this experiment. The recorded value is 1.5375 V. Proceed to construct a circuit to measure the current through the 100 Ohm resistor when connected in series using a battery. One of the DMMs will need to be set as an ammeter, (requires the use of two separate DMMS), Figure 4. Take care to use the correct input in order to not blow a fuse in the ammeter. Specifically the red 500 mA MAX input and black COMM input was used. The current can be seen in table 2. It is equal to 14.58 mA. Next, the same circuit will be used to measure the voltage across the lone resistor. The recorded value was 1.4813 V, and the circuit diagram can be seen below, figure 4. The resistor value can be calculated from these values using Ohms Law. The calculated resistance is 101.5980796 Ohms, and although there is a discrepancy between this value and the value from step 1, the numbers are nearly within tolerance of each other (1%). After this, the voltmeter was disconnected and used to measure the voltage across the current meter, Figure 5. This value corresponded to the voltage drop the DMM was responsible for. This value was equal to 41.48 mV, a very small number. From the variables we have ascertained we can find the internal resistance of the battery dividing the total voltage drops by the total current of the circuit. From this we found that the internal resistance is equal to 1.0096 Ohms Figure 4. Voltage drop across the lone resistor. Lone resistor circuit with ammeter attached in series.

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Figure 5. Voltage drop across the ammeter. Table 2. Corresponding values to steps 1 through 7. See text and diagrams for context

Investigation 2 1 100 Ohm Resistance. 2 1.5375 V Voltage found from battery. 3 14.58 mA Current measured from single resistor circuit. 4 1.4813 V Voltage drop across resistor. 5 101.5980796 Ohm Calculated resistance by dividing 4/(.001*5). 6 41.48 mV Voltage drop across Ammeter. 7 1.0096 Ohm Internal resistance of battery. Calculated by (2-4-6)/3 Investigation 3-Simple Resistor Networks This investigation follows a procedure similar to the other two, and involve the use of the same resistor board, and the same batteries and wire. However, in this investigation, the measured voltage and currents will be derived from circuits consisting of two resistors. The battery will be replaced by a DC power supply. This functions similarly to a battery, only more consistent and better suited for our study. The voltage and current can be controlled by knobs on the front of the supply. The power supply will be set to constant voltage mode, meaning the voltage drops remain the same regardless of current. To start, we calculate two resistors in a series. In order to organize the data better in Table 3, we will list the following steps in order, and with a corresponding number. Step One was to use the Ohmmeter to actually confirm the resistances of the 470 Ohm and 1000 Ohm resistors on the resistor board. These measured values have been used in our calculations, rather than the expected values. Step Two was to confirm the accuracy of the power supply by attaching it the the voltmeter while off, then turning it on and slowly adjusting the volts until reaching 5 V. This voltage setting was used for the rest of the investigation. Step three involved the actual creation of our circuit. Once attached to the power supply, every circuit element was examined using the voltmeter. Using a second voltmeter, the current was measured as well. A circuit diagram for this was created and is seen below in figure 6. Figure 6. Two resistors in series. With voltages being measure across each circuit element (Ammeter, resistor 1, resistor 2)

From these values we can compare them to the measured value of the power supply. Calculate the power dissipated in each resistor using equation 3 in the introduction as part of step 4. Additionally calculate the total resistance from the values of step one, with the values calculated through Ohms law, and the values from step 3. Based on these values we can see that the total resistance is equal to the sum of either resistor. Neglecting the burden voltage in these experiments would affect the final result, however due to the small magnitude of the burden voltage, it will barely affect the final result. The next step with recorded information is in step 7. Where the circuit is then rearranged to place the two resistors in parallel. The power supply remained at 5 volts, with one DMM connected to measure the total current, while the other measured the voltages across all three circuit elements, then measure the current passing through each resistor. A diagram of this can be seen below in figure 7. For step 8, proceed to compare the total resistance from the measurements taken in step 1 to the measurements taken in step 7, with the resistance calculated using Ohms law. The measurements agree with the rule of addition of two resistors in parallel. In step 9, compare the currents through the two resistors again the current recorded from the power supply, then find the power dissipated at each resistor. Then compare the power dissipation across the series and parallel circuits. Table 3. Investigation 3 1 468.57 Ohm 1.1008 kOhm 2 0 mV 4.9965 V 3 2.9871 mA 470 1.4864 V

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1000 3.498 V PS 4.992 V DMM 7.08 mV ideal actual 4 0.001502983 V^2/ohm 0.001407816 V^2/ohm 0.008323812 V^2/ohm 0.007796762 V^2/ohm 0.003395918 V^2/ohm 0.015879024 V^2/ohm 0.004816327 V^2/ohm 0.03194046 V^2/ohm 7 15.02 mA PS 4.953 V 470 4.931 V 1000 4.931 V 470 (I) 10.562 mA 1000(I) 4.1265 mA 8 466.8623367 1194.959409 9 0.015493326 V^2/ohm 0.015493326 V^2/ohm Conclusion This experiment consisted of three investigations. All 3 involving the use of DMMs in DC circuits, with the DMM allowing us to understand the circuits we explore using the resistor board and elements of simple DC Circuits. Through the measurements of two batteries connected to each other, we are able to see the relationship between voltage and the flow of current in a circuit, i.e. whether it is in series or in parallel. Through Ohmic resistors we can see how the circuit configuration affects the current values over each junction while being adjusted by said resistors. Through the use of two resistors, we are able to see how these values act in parallel, and we are able to dive into how each component affects the flow of current, ultimately causing dissipation of power. Additionally, we can see how each of our measurement tools actually affects the circuit we are measuring. In order to reproduce these results of this experiment in a more reliable/better way, we must address some of the potential confounding variables. These include the tolerance of the equipment used, specifically the batteries, which have been used multiple times and have not always perform consistently. The DMMs used only measured the values up to a certain decimal point, more accurate measurement devices could be used. Questions

A 6 V battery is connected in series with a 1.5 V flashlight cell. What possible terminal voltages are available? When you connect a 6V battery in series with a 1.5V flashlight cell, the total voltage depends on the orientation of the batteries. With Vtotal = 6V + 1.5V = 7.5V With Vtotal = 6V - 1.5V = 4.5V You are given two equal resistors. Will the total resistance be larger when they are in series or parallel? What will the new resistance be in each case? 2R for series 2/R for parallel. The series has a larger resistance. Does a 1.5 V battery have an internal resistance? If the maximum current the battery can supply is 200 mA, what is the value of its internal resistance? Using Ohms law, V = IxR we can calculate that R is equal to 7.5 Ohms What is the resistance of a 1.5 kW/110 V electric teapot? Using P=(V^2)/p we can calculate that the teapot has a resistance of 8.07 ohms. Determine the resistance between points A and B in Figs. 17.8a and 17.8b, where the dotted lines indicate that the visible circuit pattern is repeated infinitely to the right. For the series the resistance is infinite, for the parallel the resistance is infinitely close to zero. Acknowledgments I would like to thank my Lab Partner Hayley Ha. I would like to thank our TA David Anderson References [1] H. Young and R. Freedman, University Physics, p 111, Pearson Education, 14th edition. [2] Hyde, Batishchev, and Altunkaynak, Introductory Physics Laboratory, pp 189-201, Hayden-McNeil, 2017.

Lab Report Exp 17

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