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EXERCISE 6 Electricity β II Objectives: In the present laboratory exercise students will measure the equivalent resistance of two and three resistors connected in series and in parallel and they will compare the results with those predicted from the theoretical equations. The students will determine the unknown resistance of a resistor using the ammeter and voltmeter and they will use to more complicated circuits. Theoretical background Ohmβs law As a reminder from the previous lab, the Ohmβs law states that the current πΌ flows through a conductor, like a wire, is the ratio of the voltage π applied across the conductor over its resistance. πΌ= π π The current is measured in Amperés (π΄), the voltage in Volts (π) and the resistance in Ohms (πΊ). The Ohmβs law can also be applied for the whole circuit. In this case, the π in equation (1) represents the voltage of the battery and πΌ the current flows through the circuit. Then, the π represents the equivalent resistance π ππ of the circuit. π ππ = π πΌ Equivalent resistance of resistors connected in series Let us assume three resistors connected in series as shown in figure 1. We are going to calculate the equivalent resistance of such a circuit. Figure 1: Three resistors connected in series. A voltmeter is connected parallel to the battery to measure the voltage and an ammeter is connected in series to the resistors to measure the current through the circuit. 1 With the term equivalent resistance, π ππ , we mean a resistor of appropriate value of resistance which replaces the existing three resistors. As shown in figure 1, when the switch is closed the battery provides to the circuit a current πΌ. This current πΌ flows through all the resistors and the voltage π of the battery is applied to the circuit. The voltage of the battery π equals to the sum of the voltages across each individual resistor, ππ 1 , ππ 2 , and ππ 3 . We get, π = ππ 1 + ππ 2 + ππ 3 (1) We use the Ohmβs law for the circuit, π = π ππ πΌ, and for each resistor, ππ π = π π πΌ, where π = 1, 2, 3. Equation (1) is now written as π ππ πΌ = π 1 πΌ + π 2 πΌ + π 3 πΌ βΊ π ππ πΌ = (π 1 + π 2 + π 3 )πΌ βΊ π ππ = π 1 + π 2 + π 3 (2) The result in equation (2) can be generalized for more resistors connected in series. If in an electric circuit there are π resistors connected in series the equivalent resistance is given by the formula π ππ = π 1 + π 2 + π 3 + β― + π π (3) Therefore, the equivalent resistance of resistors connected in series equals to their sum. Equivalent resistance of resistors connected in parallel We connect two resistors in parallel and parallel to the battery, too, as shown in figure 2. Figure 2: Two resistors are connected parallel to the battery. The voltage across each resistor is identical to the batteryβs voltage. The current πΌ the battery provides to the circuit splits to two currents on the node A. Note, there is no current through the voltmeter because of its very high resistance. 2 When the switch S is closed, the battery provides to the circuit a current πΌ which is measured by the ammeter. When the current πΌ reaches the point A, where three wires coincide 1, it splits into two currents πΌ1 and πΌ2 . The current πΌ1 flows through the resistor π 1 and the current πΌ2 flows through the resistor π 2 . Due to the charge conservation principle, it holds, πΌ = πΌ1 + πΌ2 (4) We apply now the Ohmβs law for the whole circuit in figure 2 and we get πΌ = π βπ ππ , and for the two resistors and we get πΌ1 = ππ 1 βπ 1 and πΌ2 = ππ 2 βπ 2 respectively. ππ ππ π = 1+ 2 π ππ π 1 π 2 (5) Because the wires have no resistance 2, from figure 2 we get ππ 1 = ππ 2 = π. Equation (4) is now written as π π π 1 1 1 = + βΊ = + π ππ π 1 π 2 π ππ π 1 π 2 (6) If in an electric circuit there are π resistors connected in parallel, the equivalent resistance is calculated by the formula 1 1 1 1 1 = + + + β―+ π ππ π 1 π 2 π 3 π π (7) Experimental β Data analysis Measurement of an unknown resistance of a resistor Equipment needed β’ Three battery sockets β’ One switch β’ The resistor module β’ One voltmeter β’ One ammeter β’ Some wires Procedures Hook up the electric circuit shown in figure 3, according to the following steps: 1 The point A where three wires coincide is called node of the electric circuit. In general, any point in an electric circuit where three or more wires coincide is called electric node. 2 The resistance of the wires is very low and is considered zero. 3 Connect the positive terminal of the battery to one terminal of the switch. The switch is open. Connect the other terminal of the switch to the red terminal of the ammeter Connect the other terminal of the ammeter to one terminal of the unknown resistance Connect the other terminal of the resistor to the negative terminal of the battery. Connect also the negative terminal of the battery to the black terminal of the voltmeter Connect the positive terminal of the battery to the red terminal of the voltmeter, as shown in figure 3 Close the switch and register the readings from the voltmeter and ammeter on the worksheet. Determine the resistance of the resistor using the Ohmβs law and register the result in the appropriate cell of the worksheet. Figure 3: Measurement of the unknown resistance of a resistor using an ammeter and a voltmeter Measurement of the equivalent resistance of three resistors connected in series Equipment needed β’ Three battery sockets β’ One switch β’ The resistor module β’ One voltmeter β’ One ammeter β’ Some wires Procedures Hook up the electric circuit shown in figure 4, according to the following steps: β’ Connect the positive terminal of the battery to one terminal of the switch. The switch is open. β’ Connect the other terminal of the switch to the red terminal of the ammeter β’ Connect the black terminal of the ammeter to the red terminal of the 15 πΊ resistor 4 β’ Connect the black terminal of the 15 πΊ resistor to the black terminal of the 10 πΊ resistor β’ Connect also the red terminal of the 10 πΊ resistor to the red terminal of the 5 πΊ resistor β’ Connect the black terminal of the 5 πΊ resistor to the negative terminal of the battery, as shown in figure 4 β’ Connect the black terminal of the voltmeter to the negative terminal of the battery β’ Connect the red terminal of the voltmeter to the positive terminal of the battery β’ Close the switch and register the readings from the voltmeter and ammeter on the worksheet. β’ Determine the equivalent resistance of three resistors connected in series using the Ohmβs law and register the result in the appropriate cell of the worksheet. β’ Calculate the equivalent resistance according to the values given on the resistor module. Use formula (3). β’ Compare the experimental equivalent resistance with that given by equation (3). β’ Calculate the quantity |π ππ,ππππ β π ππ,ππππ |βπ ππ,ππππ β 100% Figure 4: Measurement of the equivalent resistance of three resistors connected in series Measurement of the equivalent resistance of two resistors connected in parallel Equipment needed β’ Three battery sockets β’ One switch β’ The resistor module β’ One voltmeter β’ One ammeter β’ Some wires 5 Procedures Hook up the electric circuit shown in figure 5, according to the following steps: β’ Connect the positive terminal of the battery to one terminal of the switch. The switch is open. β’ Connect the other terminal of the switch to the red terminal of the ammeter β’ Connect the black terminal of the ammeter to the red terminal of the 15 πΊ resistor β’ Connect the red terminal of the 15 πΊ resistor to the red terminal of the 10 πΊ resistor β’ Connect also the red terminal of the 10 πΊ resistor to the red terminal of the 5 πΊ resistor β’ Connect the black terminal of the 15 πΊ resistor to the black terminal of the 10 πΊ resistor β’ Connect the black terminal of the 10 πΊ resistor to the black terminal of the 5 πΊ resistor, as shown in figure 5 β’ Connect the black terminal of the 5 πΊ resistor to the negative terminal of the battery β’ Connect the black terminal of the voltmeter to the negative terminal of the battery β’ Connect the red terminal of the voltmeter to the positive terminal of the battery β’ Close the switch and register the readings from the voltmeter and ammeter on the worksheet. β’ Determine the equivalent resistance of three resistors connected in parallel using the Ohmβs law and register the result in the appropriate cell of the worksheet. β’ Calculate the equivalent resistance according to the values given on the resistor module. Use formula (7). β’ Compare the experimental equivalent resistance with that given by equation (7). β’ Calculate the quantity |π ππ,ππππ β π ππ,ππππ |βπ ππ,ππππ β 100% Figure 5: Measurement of the equivalent resistance of three resistors connected in parallel 6 Measurement of the equivalent resistance of a mixed combination of resistors Equipment needed β’ Three battery sockets β’ One switch β’ The resistor module β’ One voltmeter β’ One ammeter β’ Some wires Procedures Hook up the electric circuit shown in figure 6, according to the following steps: β’ Connect the positive terminal of the battery to red terminal of the resistor π π₯ . β’ Connect the black terminal of the resistor π π₯ to the black terminal of the switch β’ Connect the red terminal of the switch to the red terminal of the ammeter β’ Connect the black terminal of the ammeter to the red terminal of the 10 πΊ resistor β’ Connect the red terminal of the 10 πΊ resistor to the red terminal of the 15 πΊ resistor β’ Connect also the black terminal of the 10 πΊ resistor to the black terminal of the 15 πΊ resistor β’ Connect the black terminal of the 10 πΊ resistor to the negative terminal of the battery β’ Connect the black terminal of the battery to the black terminal of the voltmeter β’ Connect the positive terminal of the battery to the red terminal of the voltmeter β’ Close the switch and register the readings from the voltmeter and ammeter on the worksheet. Figure 6: Measurement of the equivalent resistance of three resistors connected in a mixed way 7 β’ Determine the equivalent resistance of the three resistors in the electric circuit using the Ohmβs law and register the result in the appropriate cell of the worksheet. β’ Calculate the equivalent resistance according to the values given on the resistor module. Use formulae (3) and (7) appropriately. β’ Compare the experimental equivalent resistance with that given by the combination of equations (3) and (7). β’ Calculate the quantity |π ππ,ππππ β π ππ,ππππ |βπ ππ,ππππ β 100% Discussion and general comments In the last section of your report, you describe briefly the task(s) of this laboratory session. Then you have to write comments on what you measured and how the theory is compared to the data. As an option, you may also write some thoughts for the improvement of the accuracy of the measurements. 8 Electricity II Studentβs worksheet β Lab 6 Studentβs name: ____________________________________________ Grade: _________ Date: __________ Lab partner(s): ____________________________________________________________________________ Experiment 1: Measurement of an unknown resistance Batteryβs voltage Current through the resistor Resistance π (π) πΌ (π΄) π (πΊ) Experiment 2: Measurement of the equivalent resistance of three resistors connected in series Batteryβs voltage Current through Equivalent Equivalent resistance % Percentage π (π) the circuit resistance π ππ,ππππ (πΊ) discrepancy πΌ (π΄) π ππ,ππππ (πΊ) Experiment 3: Measurement of the equivalent resistance of three resistors connected in parallel Batteryβs voltage Current through Equivalent Equivalent resistance % Percentage π (π) the circuit resistance π ππ,ππππ (πΊ) discrepancy πΌ (π΄) π ππ,ππππ (πΊ) Experiment 4: Measurement of the equivalent resistance of three resistors in an electric circuit Batteryβs voltage Current through Equivalent Equivalent resistance % Percentage π (π) the circuit resistance π ππ,ππππ (πΊ) discrepancy πΌ (π΄) π ππ,ππππ (πΊ) 9
