Lecture Notes on DC Circuits - General Physics Laboratory II | PHY 222, Study notes of Physics

Download Lecture Notes on DC Circuits - General Physics Laboratory II | PHY 222 and more Study notes Physics in PDF only on Docsity! V. DC CIRCUITS=================================== f05.01 INTRODUCTION DC stands for “direct-current”. A direct current stays the same in magnitude and does not change its direction. Ohm’s Law and the rules for effective resistance of resistors con- nected in series and in parallel, which we studied in the previous experiment, can be used to understand only simple DC circuits. More general approach, which can be applied to analyze any DC circuit, is based on so called Kirchhoff’s Rules. PURPOSE • Experimental verification of Kirchhoff’s Rules. • Analysis of a complex DC circuit with Kirchhoff’s Rules. DC CIRCUITS V-1 PRE-LAB ASSIGNMENTS A. Readings: There are two Kirchhoff’s Rules: junction rule - The sum of the currents entering any junction must equal the sum of the currents leaving that junction. loop rule - The sum of the potential differences across each element around any closed circuit loop must be zero. The first rule reflects more general law of charge conservation. (Net charge cannot disappear or pop-out from nothing at the junction). The second rule stems from the law of energy conservation. Since the electric forces are conservative, energy done by electric forces on a test charge along any closed loop must be zero. Let us apply Kirchhoff’s Rules to the circuit shown in Fig. 1. In fact, we will study this circuit experimentally as well. This is an example of the circuit which cannot be understood from the rules for resistors in series and in parallel. 1R R + I I I ε ε 2R 3 1 2 3 21 + - - Figure 1. DC circuit analyzed in Experiment C. In the first step current direction in each loop needs to be assumed. It does not matter if we assume a wrong direction, since the sign of the current in the solution to the Kirchhoff’s Rules will tell us if we assumed the right direction (negative solution indicates the opposite current direction). Fig. 1 shows one possible direction of currents that we adopt here. There are two junctions in our circuit. In general, for N junctions it is enough to apply the junction rule to N − 1 junctions. Thus, we need to write the first rule only once: I1 + I2 = I3 (1) V-2 DC CIRCUITS REPORT SHEET V–1 Date Name Instructor PRE-LAB EXERCISES Exercise 1. Write junction rule for the junction shown below: 1I 4 I I I 2 3 Exercise 2. Assume that in the circuit shown in Fig. 1 all resistances are the same, R1 = R2 = R3 = R. Express I1 in terms of ²1, ²2 and R. You can start from equation (5). Show your algebra. DC CIRCUITS V-5 blank V-6 DC CIRCUITS LABORATORY ASSIGNMENTS Experiment A: Junction Rule The Task: To experimentally verify Kirchhoff’s Junction Rule. Materials Needed: • Resistors: 10kΩ, two 220Ω • Rheostat • Power Supply • Dual Channel Amplifier with two voltage probes • ULI computer interface box • 4.5V Battery (for apparatus tests only) • Voltmeter (for apparatus tests only) • Cables Procedures 1Ι ~30 V Ι 2 Rheostat Ι Ι2 Ι1 Junction 10 kΩ Figure 2. DC circuit used to verify Kirchhoff’s Junction Rule. DC CIRCUITS V-7 blank V-10 DC CIRCUITS Experiment B: Loop Rule The Task: To experimentally verify Kirchhoff’s Loop Rule. Materials Needed: • Resistor: 220Ω • Rheostat • Dual Channel Amplifier with two voltage probes • ULI computer interface box • 4.5V Battery • Cables Procedures 2 Rheostat R =220 R 21V V ~4.5 V 1 Figure 3. DC circuit used to verify Kirchhoff’s Loop Rule. B-1. The circuit used in this experiment is shown in Fig. 3. There are three elements in this loop: battery and two resistors (rheostat plays a role of one of them). Kirchhoff’s second rule written for this loop is: V1 + V2 − ² = 0 (7) where V1 is the potential difference across the first resistor (R1 = 220Ω), V2 is the potential difference across the second resistor (i.e. rheostat - R2 is variable) and ² is the potential difference generated by the battery (² ≈ 4.5V ). Since ² is constant, to DC CIRCUITS V-11 prove the second rule we need to demonstrate that V1 + V2 remains constant while V1 and V2 can vary individually. Go to “File” menu and “Open. . . ” to load “dc-rule2” set-up file. When the program asks you if you would like to save changes to the previous set-up click on “No”. Connect the circuit as shown in Fig. 3. Make sure that one connection to the rheostat is to the slider. Connect one voltage probe across 220Ω resistor and the second one across the rheostat. Make both voltage readings positive (reverse the order of the voltage lead connections if necessary). B-2. Click on “Collect” and move the slider of the rheostat in its full range. Copy V1 vs. V2 and V1 + V2 vs. Time graphs onto Report Sheet V–3. Did the loop rule work? B-3 Determine mean value of V1 + V2 from your data and compare it to the value of ² measured for the battery (Report Sheet V–3). V-12 DC CIRCUITS Experiment C: Complex DC Circuit The Task: To experimentally verify predictions based on the solution to Kirchhoff’s Junction and Loop Rules. Materials Needed: • Power Supply • 4.5V Battery • Voltage Divider Box • Three 100Ω Resistors • Dual Channel Amplifier with two voltage probes • ULI computer interface box • Cables Procedures We will now study circuit which served as an example in the theoretical introduction (Fig. 1). We will use R1 = R2 = R3 = R = 100Ω, 4.5V battery for ²2 and the power supply as a source of variable ²1. Using the formula (5) derived in the introduction we get: I1 = 2²1 − ²2 3R (8) I2 = 2²2 − ²1 3R (9) I3 = I1 + I2 = ²1 + ²2 3R (10) To verify these equations we will measure dependence of the current (in each branch) on ²1. C-1. Connect three 100Ω resistors as shown in Fig. 1. Connect 4.5V battery as ²2 (pay attention to the polarity). Connect the power supply across the terminals A and B of the voltage divider box (negative voltage to the terminal B). Connect the terminals B and C of the voltage divider box as ²1 in the circuit. Pay attention to the polarity of the power supply connection; the negative pole of the power supply (black) should be on the side of negative pole of the battery. To measure ²1 connect the voltage probe 1 across the terminals C and B of the box. It is easy to make a mistake when wiring this circuit. Make sure that the voltage probe is not connected directly to the power supply (terminals A and B). C-2 To verify formula (8) we will measure I1 by potential difference reading across the resistor R1. Connect the second voltage probe across this resistor (I1 = V2/100Ω). DC CIRCUITS V-15 Load the program set-up which was previously used to study Ohm’s Law (“File” menu → “Open. . . ” → file “ohms” in PHY222 subdirectory). With the power supply on and set to the maximum, V1 and V2 (or I ≡ I1) should be both positive. Switch the order of voltage probe lead connections to reverse the sign if needed. Start collecting the data and vary ²1 ≡ V1 by turning the knob of the power supply from its minimal to its maximal setting (you can also move back-and-fourth between the extreme settings). C-3 The formula (8) predicts that the current I1 should be zero (i.e. no current flowing through the first loop) for ²1 = ²2/2. From your I vs. V graph read-out V for which the current is zero. Is it half of the potential difference supplied by the battery? (see Report Sheet V–4). C-4 We can further verify formula (8) by fitting straight line to the Current vs. Potential data. Select this graph by clicking on it. Then go to “Analyze” menu and select “Linear Fit”. From the box superimposed on the graph read the slope and the intercept of the fitted line (y = mx + b, x = ²1, y = I1, m =slope, b =intercept). From formula (8) slope should be 2 3R and intercept − ²2 3R . Put into these formulae R = 100Ω and the battery voltage for ²2. To compare to the fitted slope and intercept multiply your theoretical predictions by 1000, since the current on your graph is displayed in units of mA rather than A. Report the measured and predicted values in Report Sheet V–4. C-5 Switch the voltage probe 2 to measure current I2. Collect data for varying ²1. Fit a line to your I vs. V graph. Compare the fitted slope and intercept to the expected ones from formula (9). Report the measured and predicted values in Report Sheet V–4. C-6 Switch the voltage probe 2 to measure current I3. Collect data to verify formula (10). V-16 DC CIRCUITS REPORT SHEET V–4 Date Name Instructor Partner(s) C-3. I1 = 0 corresponds to ²1 = Half of battery voltage ²2/2 = Slope Intercept Expected Measured Expected Measured Formula Value Value Formula Value Value (mA/V) (mA/V) (mA) (mA) C-4 I1 vs. ²1 2 3R − ²2 3R C-5 I2 vs. ²1 C-6 I3 vs. ²1 DC CIRCUITS V-17