# Σ I in = Σ I out E = IR 1 + IR 2 FXA 2008 KIRCHHOFF S LAWS 1. Candidates should be able to : LAW 1 (K1)

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1 UNT G482 Module Series & Parallel Circuits Candidates should be able to : KRCHHOFF S LAWS 1 LAW 1 (K1) State Kirchhoff s second law and appreciate that it is a consequence of conservation of energy. The sum of the currents flowing into any point in a Circuit is equal to the sum of the currents flowing Out of the point. Apply Kirchhoff s first and second laws to circuits. i.e. Σ in = Σ out Select and use the equation for the total resistance of two or more resistors in series. Greek letter sigma, which means the sum of all. Recall and use the equation for the total resistance of two or more resistors in parallel. LAW 2 (K2) Consider the circuit shown opposite. Solve circuit problems involving series and parallel circuits with one or more sources of e.m.f. xplain that all sources of e.m.f have an internal resistance. xplain the meaning of the term terminal pd. As charge flows through the battery (of e.m.f. ), electrical energy is supplied to each coulomb. The charge then flows through the resistor of resistance (R 1 ) and through the filament lamp of resistance (R 2 ). n each of these R 1 R R 2 L components the electrical energy is converted to heat and heat and light energy respectively. Select and use the equations : = (R + r) And = + r nergy supplied per coulomb by = The sum of the energies converted The battery (i.e. the e.m.f.) per coulomb in each component (i.e. the sum of the pd s) = R + L And since the current () is the same at each point in a SRS circuit : = R 1 + R 2

2 UNT G482 Module Series & Parallel Circuits This equation expresses KRCHHOFF S SCOND LAW (K2) which states that : (2) Use Kirchoff s Laws to calculate the currents 1, 2 and 3 in the circuit shown opposite. A 12 B 2 n any closed loop in a circuit, the sum of the emf s is equal to the sum of the pd s around the loop. Mark in the currents and label the closed loops ABCDF as shown. Applying Kirchhoff 1 to point F 1 F 3 1 5Ω C 2 2 As we have seen from our energy analysis of a simple circuit, f a unit charge follows a closed path in a circuit : 3 = (1) Applying Kirchhoff 2 to loop FCDF 4 D Total energy supplied = Total energy dissipated to the charge. by the charge. i.e. KRCHOFF S SCOND LAW is a consequence Of the PRNCPL OF CONSRATON OF NRGY. XAMPLS OF TH US OF KRCHHOFF S LAWS N TH SOLUTON OF CRCUT PROBLMS 4 = 20 3 So 3 = 4/20 = 0.2 A Applying Kirchhoff 2 to loop ABCFA 12 = = (20 x 0.2) = = 8/5 = 1.6 A Substituting for 1 & 2 in equation (1) 0.2 = So 2 = = A The negative sign tells us that 2 flows in a direction opposite to that chosen. (1) Use Kirchhoff s Law 2 to determine the current () in the circuit shown opposite. Net emf = sum of the pd s 1-2 = R 1 + R = ( x 15) + ( x 30) 4.5 = 45 = 4.5 = 0.1 A 45 1=6 2=1.5 R 1=15Ω R 2=30Ω (3) The circuit diagram opposite shows a battery charger of emf 5.5 and internal resistance 1.0 Ω being used to recharge a cell of emf 1.2 and internal resistance 0.20 Ω. Use Kirchhoff s Law 2 to determine the charging current. Applying Kirchhoff 2 to the loop = ( x 1.0) + ( x 0.20) 4.3 = Ω = 4.3/1.2 = 3.58 A

3 UNT G482 Module Series & Parallel Circuits PRACTC QUSTONS (1) 1 The diagram opposite shows a junction in a circuit. Use Kirhhoff s First Law ( Σ in = Σ out ) to determine the size and direction of the current. 0.5 A 3.7 A 1.5 A 1.8 A 3.2 A 0.2 A 0.7 A 4 Use Kirchhoff s laws to determine the current readings shown on the ammeters (A 1, A 2 & A 3 ) in the circuit shown opposite A 2 A 3 A Use Kirchhoff s Second Law to Calculate : 12 COMBNATON OF RSSTORS (a) The pd across resistor R. (b) The value of R. 0.2 A R 48Ω RSSTORS N SRS 3 Use Kirchhoff s laws to determine the size and direction of the current at point P in the circuit shown opposite. 4 Consider three resistors of resistance R 1, R 2 and R 3 connected N SRS as shown opposite. According to Kirchhoff s Law 1 : The current () is the same in each resistor. R 1 R 2 R T P Since energy is conserved : Total pd across the combination = The sum of the pds across each resistor T = (And since = R) R T = R 1 + R 2 + R 3 From which : R T = R 1 + R 2 + R 3 12 For any number of resistors connected N SRS, the TOTAL RSSTANC (R T ) is given by : R T = R 1 + R 2 + R 3

4 UNT G482 Module Series & Parallel Circuits SPCAL CAS FOR RSSTORS N PARALLL 4 RSSTORS N PARALLL f (N) resistors having the same resistance (R) are connected N PARALLL, the TOTAL RSSTANC (R T ) is given by : Consider three resistors of resistance R 1, R 2 and R 3 connected N PARALLL as shown opposite. 1 R 1 R T = R N According to Kirchhoff s Law 1 = (1) 2 R 2 PRACTC QUSTONS (2) The pd across each resistor = 3 R 3 1 Calculate the total resistance of the resistor combinations shown below : From the definition of resistance : = /R (a) (b) 5Ω And applying this to equation (1) : 2Ω 4Ω = + + R T R 1 R 2 R 3 3Ω 15Ω From which : 1 = R T R 1 R 2 R 3 6Ω 30Ω For resistors connected N PARALLL : The LOWST value resistors carry the GRATST proportion of the current. For any number of resistors connected N PARALLL, the TOTAL RSSTANC (R T ) of the combination is given by : 1 = R T R 1 R 2 R 3 The TOTAL RSSTANC of the combination is LSS than the SMALLST resistance in the combination. 2 (a) What resistor value needs to be connected in parallel with a 20 Ω resistor to give a combined total resistance of 10 Ω? (b) You are provided with several 100 Ω resistors. How could you combine the minimum number of these resistors to give a total resistance of 250 Ω? 3 You are provided with a 400 Ω resistor and two 200 Ω resistors. Calculate the total resistances which may be obtained by connecting some or all of these resistors in various combinations.

5 UNT G482 Module Series & Parallel Circuits MF (), TRMNAL PD () AND NTRNAL RSSTANC (r) 5 4 For the resistor network shown calculate : (a) The total resistance of the combination. 5A 1 2 4Ω 5A All sources of emf have some NTRNAL RSSTANC (r), since they are made from materials (e.g. metal wires, electrodes, chemical electrolytes) which have some electrical resistance. (b) The pd () across each resistor. (c) The value of the currents ( 1, 2 & 3 ) through each of the resistors. 3 f a voltmeter is connected across the terminals of an electrical supply (e.g. a cell or battery), it indicates what is called the TRMNAL PD () of the cell or battery. cell or battery 5 For the circuit shown opposite, calculate : (a) The resistance (R AB ) of the parallel combination. (b) The total resistance (R T ) of the circuit. (c) The currents, 1 and 2. 40Ω A Ω 50Ω B f the cell or battery is not part of an external circuit and the voltmeter is ideal (i.e. it has infinite resistance), then zero current is drawn and : voltmeter reading = cell emf () r oltmeter reading = (d) The pd across the 40 Ω resistor. 6 n the circuit shown opposite the voltmeter gives a reading of 8. Calculate : (a) The pd s and Ω 1600Ω The emf () of a cell or battery can be defined as its TRMNAL PD when it is NOT supplying a current. (b) The pd across the 1600 Ω resistor when the voltmeter s disconnected. 1

6 UNT G482 Module Series & Parallel Circuits MASURNG MF () AND NTRNAL RSSTANC (r) 6 f an external circuit is connected to the cell or battery (or the voltmeter is not perfect and draws some current), the reading on the voltmeter drops to a value less than. r oltmeter reading = (<) R A good estimate of the emf () of a cell, battery or power supply may be obtained by simply measuring the terminal pd with a DGTAL OLTMTR (These have a high resistance and will therefore only draw a very small current). alue of given by the digital voltmeter = This is because when there is a current through the cell, some of its energy is converted into heat by the cell s internal resistance. The decrease in voltage is called the LOST OLTS of the cell and it is proportional to the current. The reading () which is < indicated by the voltmeter is the TRMNAL PD of the cell and also the pd across the resistor R. Applying Kirchhoff s Law 2 to the circuit : mf of the cell = terminal pd + pd across the internal resistance (= pd across R) = + r PROCDUR The circuit shown opposite is used to obtain a more accurate determination of the emf () of a cell as well as its internal resistance (r). Corresponding values of the current () in the cell and pd () across the cell are obtained by adjusting the variable resistor. The results are recorded in the table shown below. A r = R + r = (R + r) RSULTS PD ()/ = (R + r) Current ()/A

7 UNT G482 Module Series & Parallel Circuits SOM FFCTS OF NTRNAL RSSTANC 7 GRAPH When a graph of pd () is plotted, a best fit straight line should be obtained as shown opposite. NOT : n practice the graph is curved because (r) changes as the current () increases. PD ()/ Gradient of line = -r Current ()/A Low voltage sources from which large currents are drawn (e.g. car batteries) should have LOW internal resistance, otherwise their terminal PD ( = - r) would be very low. High voltage sources (e.g. 5 k supplies which are sometimes used in schools) have a HGH internal resistance so as to limit the current supplied should they be short-circuited accidentally. The headlamps on a car will dim if the vehicle is started while they are switched on. This is because the starter motor draws a large current and this causes the battery terminal PD ( = - r) to drop sharply. ANALYSS OF GRAPH The equation which links MF (), TRMNAL PD (), CURRNT() and NTRNAL RSSTANC (r) is : n order to get an efficient transfer of energy from a source of emf to an external component of resistance (R), the internal resistance (r) of the source << R. = + r PRACTC QUSTONS (3) Rearranging : Comparing with the equation For a straight line : t can be seen that : = -r + y = mx + c 1 A battery of emf 4.5 and internal resistance 1.5 Ω is connected to a 10 Ω fixed resistor. Draw a circuit diagram of the arrangement and calculate the current in the circuit. 2 A battery of emf 15 and internal resistance 1.2 Ω is connected to an 8 Ω resistor. Calculate : (a) The total resistance of the circuit. ntercept on the y-axis = = Gradient of the / graph = r = Ω (b) The current through the battery. (c) The lost volts (d) The battery s terminal pd.

8 UNT G482 Module Series & Parallel Circuits HOMWORK QUSTONS 8 3 A battery of emf () and internal resistance (r) was connected in series with a variable resistor of resistance (R) and an ammeter. f the ammeter reading was 2.0 A when R was set to 4.0 Ω and it dropped to 1.5 A when R was set to 6.0 Ω, calculate the values of and r. 4 The pd across the terminals of a battery is found to be 3.0 when it is measured using a very high resistance voltmeter. The battery is then connected to a 10 Ω resistor and its terminal pd drops to 2.8. Calculate the internal resistance of the battery. 1 (a) On which conservation laws are Kirchhoff s first and second laws based? (b) For the circuit shown opposite, calculate : (i) The pd across the 125 Ω resistor A Ω R 5 A high resistance voltmeter gives a reading of 1.5 when connected to a dry cell on open circuit. When the cell is connected to a lamp of resistance R, there is a current of 0.30 A and the voltmeter reading drops to 1.2. Calculate : (ii) The pd across resistor R. (iii) The resistance of resistor R. (a) The emf of the cell. 2 (a) State Kirchhoff s Second Law. 4.5 (b) The internal resistance of the cell. (c) The value of the resistance R. 6 A car battery has an emf of 12 and an internal resistance of Ω. The current drawn by the starter motor is 96 A. (b) Apply Kirchhoff s Second Law to the circuit shown opposite to determine the current at point X. X 15 Ω 40 Ω 10 (a) Calculate the terminal pd of the battery when the car is being started. (b) f the headlamps are rated at 12, 36 W, what is their resistance? (c) Calculate the value of their power output when the starter motor is in operation. 3 You are given three resistors of resistance 4 Ω, 6 Ω and 8 Ω. Using all the resistors draw the combination to give : (a) The largest resistance. (b) The smallest resistance. n each case calculate the total resistance of the combination.

9 UNT G482 Module Series & Parallel Circuits 4 (a) State Kirchhoff s First Law. (b) The diagram below shows part of an electrical circuit. 9 6 The diagram below shows a circuit diagram including three resistors. 5.0 X 31 ma 2 22 ma A 1 20 ma 3 A 10 Ω B C 30 Ω (i) Name the component marked X. (a) The variable resistor is set on its maximum resistance of 20 Ω. (ii) Determine the magnitude of the currents 1, 2 and 3. (OCR AS Physics - Module January 2004) Calculate the resistance between points : (i) B and C. (ii) A and C. 5 The results of an experiment to determine the emf () and internal resistance (r) of a power supply are given in the table below. / /A (b) n the circuit shown in the diagram above, the battery has negligible internal resistance and an emf of 5.0. The vaiable resistor is now set on its lowest resistance of 0 Ω. Calculate the reading on the ammeter. (OCR AS Physics - Module January 2006) Plot a suitable graph and use it to find and r.

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