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Series and Parallel Resistances and Kirchhoff’s Laws Objectives • Calculate the equivalent resistance for resistors connected in both series and parallel combinations. • Construct series and parallel circuits of lamps (resistors). • Observe and explain relative lamp brightness in series and parallel circuits. • Apply Kirchhoff’s first and second laws. • Calculate the current and voltage for resistor circuits connected in parallel. • Calculate the current and voltage for resistor circuits connected in series. Physics terms • series circuit • parallel circuit • equivalent resistance • junction • loop Equations Equivalent resistance for resistors connected in series Equivalent resistance for resistors connected in parallel Equations Kirchhoff’s laws: current law: voltage law: Series circuits A series circuit has only one path for the flow of electric current. I I I Parallel circuits In a parallel circuit the electric current can split apart and come back together again. V V Series circuits The battery and the resistors in these circuits are identical. What is different about these circuits? Series circuits The circuit with 3 resistors is 3 times harder for the current to flow through. I1 I2 I3 When you add resistors in series, equivalent resistance increases and current flow decreases: I3 < I2 < I1 Equivalent resistance What is “equivalent resistance”? • Equivalent resistance is the total combined resistance of a group of resistors. • Equivalent resistance has the value of a single resistor that could replace the multiple resistors in a circuit, while keeping total current through the circuit the same. Equivalent resistance These two circuits have the same total resistance . . . . . . so they will have the same total current, I. Equivalent resistance: series For resistors in series, you can find the equivalent resistance by simply adding up the individual resistances: Engaging with the concepts A 10 Ω resistor, a 15 Ω resistor, and a 5 Ω resistor are connected in series. What is the equivalent resistance of this arrangement? 10 15 5 Engaging with the concepts A 10 Ω resistor, a 15 Ω resistor, and a 5 Ω resistor are connected in series. What is the equivalent resistance of this arrangement? 30 10 Ω + 15 Ω + 5 Ω = 30 Ω 10 15 5 Engaging with the concepts Two strings of tree lights, each with a resistance of 150 Ω, are connected together in series. What is the Req? 150 150 0 Engaging with the concepts Two strings of tree lights, each with a resistance of 150 Ω, are connected together in series. What is the Req? 300 Ω When you add resistors in series does Req increase or decrease? Does total current increase or decrease? 300 150 150 0 Engaging with the concepts Two strings of tree lights, each with a resistance of 150 Ω, are connected together in series. What is the Req? 300 Ω When you add resistors in series does Req increase or decrease? Req increases Does total current increase or decrease? I decreases 450 150 150 150 What is the Req of each circuit? Req = _______ Req = _______ What is the Req of each circuit? Req = 10 Ω Req = 15 Ω How much current flows? Req = 10 Ω Req = 15 Ω How much current flows? 1A 0.67 A Parallel circuits A parallel circuit has more paths for electric current to flow. The circuit on the right lets twice as much current flow. What is the Req for the parallel circuit? R 2R ½R Parallel circuits In the circuit with parallel resistors: total current flow doubles because total resistance is halved. What is the Req for the parallel circuit? R 2R ½R Equivalent resistance: parallel When you add resistors in parallel, total resistance goes DOWN. To find the Req you must add the inverse of the resistances: Equivalent resistance: parallel If you have two 4 Ω resistors in parallel, what is the equivalent resistance? A. ½ Ω B. 2 Ω C. 4 Ω D. 8 Ω Equivalent resistance: parallel Don’t forget to flip the fraction at the end! If you have two 4 Ω resistors in parallel, what is the equivalent resistance? A. ½ Ω B. 2 Ω C. 4 Ω D. 8 Ω Engaging with the concepts A 10 Ω resistor and a 15 Ω resistor are connected in parallel. What is the Req of this arrangement? 10 Equivalent resistance 15 Engaging with the concepts A 10 Ω resistor and a 15 Ω resistor are connected in parallel. What is the Req of this arrangement? 6 ohms 6 Equivalent resistance When resistors are connected in parallel, total resistance decreases. 10 15 Engaging with the concepts Two strings of lights, each with a resistance of 150 Ω, are connected together. What is the Req when the strings are connected in parallel? 150 Equivalent resistance 150 Engaging with the concepts Two strings of lights, each with a resistance of 150 Ω, are connected together. What is the Req when the strings are connected in parallel? 75 Ω 75 150 150 Equivalent resistance The total resistance is always less than any of the individual resistors added in parallel! What is the Req of each circuit? These are all 10 Ω resistors. What is the Req in each case? Req = ____ Req = ____ Req = ____ What is the Req of each circuit? These are all 10 Ω resistors. What is the Req in each case? Req = 10 Ω Req = 5 Ω Req = 3.3 Ω Adding resistors in parallel makes the total resistance decrease. How much current flows? Apply Ohm’s law ( V = IR ) to find the total current through the battery. Req = 10 Ω I = ____ Req = 5 Ω I = ____ Req = 3.3 Ω I = ____ How much current flows? Apply Ohm’s law ( V = IR ) to find the total current through the battery. Req = 10 Ω I=3A Req = 5 Ω I=6A Req = 3.3 Ω I=9A Analogy for resistors in series What happens to the potential energy of the water as it passes through each water wheel? How is this analogous to the voltage drops across each resistor? Voltage drops: resistors in series Half as much current flows in the series circuit. What are the voltage drops across the two resistors in series? Voltage drops: resistors in series 10 V drop 10 V drop Single resistor Resistor in series Voltage drops: resistors in parallel Twice as much current flows in the parallel circuit. What are the voltage drops across the resistors in parallel? Voltage drops: resistors in parallel 4A 4A 20 V drop Single resistor Resistor in parallel Each resistor in parallel has the full 20 V drop. Assessment 1. Two resistors with resistances of 10 Ω and 30 Ω are connected in series with a 20 volt battery. a) What is the equivalent resistance of the circuit? b) What is the current flow through the circuit? Assessment 1. Two resistors with resistances of 10 Ω and 30 Ω are connected in series with a 20 volt battery. a) What is the equivalent resistance of the circuit? b) What is the current flow through the circuit? Assessment 1. Two resistors with resistances of 10 Ω and 30 Ω are connected in series with a 20 volt battery. a) What is the equivalent resistance of the circuit? b) What is the current flow through the circuit? Assessment 2. Two resistors with resistances of 10 Ω and 30 Ω are connected in parallel with a 20 volt battery. a) What is the equivalent resistance of the circuit? b) What is the current flow through the circuit? Assessment 2. Two resistors with resistances of 10 Ω and 30 Ω are connected in parallel with a 20 volt battery. a) What is the equivalent resistance of the circuit? 7.5 Ω b) What is the current flow through the circuit? Assessment 2. Two resistors with resistances of 10 Ω and 30 Ω are connected in parallel with a 20 volt battery. a) What is the equivalent resistance of the circuit? 7.5 Ω b) What is the current flow through the circuit? Assessment 3. If you connect these two identical resistors as shown, will they together draw more or less current than one of the resistors alone? Assessment 3. If you connect these two identical resistors as shown, will they together draw more or less current than one of the resistors alone? They will draw more current, because they are connected in parallel. There are now two different, but identical paths for electricity to flow, so the current doubles! Kirchhoff’s laws Two fundamental laws apply to ALL electric circuits. These are called Kirchhoff’s laws, in honor of German physicist Gustav Robert Kirchhoff (1824–1887). Kirchhoff’s current law Kirchhoff’s first law is the current law. •It is a rule about electric current. •It is always true for ALL circuits. Kirchhoff’s current law The current law is also known as the junction rule. A junction is a place where three or more wires come together. This figure shows an enlargement of the junction at the top of the circuit. Kirchhoff’s current law Current Io flows INTO the junction. Currents I1 and I2 flow OUT of the junction. What do you think the current law says about I, I1, and I2? Kirchhoff’s current law Kirchhoff’s current law: The current flowing INTO a junction always equals the current flowing OUT of the junction. Kirchhoff’s current law Example: Why is the current law true? Why is this law always true? Conservation of charge Why is this law always true? It is true because electric charge can never be created or destroyed. Charge is ALWAYS conserved. Applying the current law This series circuit has NO junctions. The current must be the same everywhere in the circuit. Current can only change at a junction. Applying the current law A 60 volt battery is connected to three identical 10 Ω resistors. 10 Ω What are the currents through the resistors? 60 V 10 Ω 10 Ω Applying the current law A 60 volt battery is connected to three identical 10 Ω resistors. 10 Ω What are the currents through the resistors? Req = 30 Ω 60 V I = 60 V/30 Ω = 2 amps through each resistor 10 Ω 10 Ω Applying the current law This series circuit has two junctions. Find the missing current. ? Applying the current law This series circuit has two junctions. Find the missing current. I2 = 2 A 2 amps Applying the current law How much current flows into the upper junction? Applying the current law How much current flows into the upper junction? I=4A 4 amps Kirchhoff’s voltage law Kirchhoff’s second law is the voltage law. •It’s a rule about voltage gains and drops. •It is always true for ALL circuits. Kirchhoff’s voltage law The voltage law is also known as the loop rule. A loop is any complete path around a circuit. This circuit has only ONE loop. Pick a starting place. There is only ONE possible way to go around the circuit and return to your starting place. Kirchhoff’s voltage law This circuit has more than one loop. Charges can flow up through the battery and back through R1. That’s one loop. Can you describe a second loop that charges might take? Kirchhoff’s voltage law This circuit has more than one loop. Charges can flow up through the battery and back through R1. That’s one loop. Can you describe a second loop that charges might take? Charges can flow up through the battery and back through R2. That’s another loop. Kirchhoff’s voltage law Kirchhoff’s voltage law says that sum of the voltage gains and drops around any closed loop must equal zero. Kirchhoff’s voltage law If this battery provides a 30 V gain, what is the voltage drop across each resistor? Assume the resistors are identical. 30 V Kirchhoff’s voltage law If this battery provides a 30 V gain, what is the voltage drop across each resistor? -10 V Assume the resistors are identical. +30 V -10 V -10 V 10 volts each! Applying Kirchhoff’s voltage law A 60 V battery is connected in series with three different resistors. Resistor R1 has a 10 volt drop. Resistor R2 has a 30 volt drop. -10 V 60 V -30 V What is the voltage across R3? ? Applying Kirchhoff’s voltage law A 60 V battery is connected in series with three different resistors. Resistor R1 has a 10 volt drop. Resistor R2 has a 30 volt drop. -10 V 60 V -30 V What is the voltage across R3? -20 V 20 volts Applying Kirchhoff’s voltage law What if a circuit has more than one loop? Treat each loop separately. The voltage gains and drops around EVERY closed loop must equal zero. Applying Kirchhoff’s voltage law A 30 V battery is connected in parallel with two resistors. What is the voltage across R1? 30 V Applying Kirchhoff’s voltage law A 30 V battery is connected in parallel with two resistors. What is the voltage across R1? R1 must have a 30 V drop. 30 V Applying Kirchhoff’s voltage law A 30 V battery is connected in parallel with two resistors. What is the voltage across R1? R1 must have a 30 V drop. What is the voltage across R2? 30 V Applying Kirchhoff’s voltage law A 30 V battery is connected in parallel with two resistors. What is the voltage across R1? R1 must have a 30 V drop. What is the voltage across R2? R2 also has a 30 V drop. 30 V Why is the voltage law true? Why is this law always true? Why is the voltage law true? Why is this law always true? This law is really conservation of energy for circuits. All the electric potential energy gained by the charges must equal the energy lost in one complete trip around a loop. Conservation of energy All the electrical energy gained by passing through the battery is lost as charges pass back through the resistors. All the gravitational potential energy gained by going up a mountain is lost by going back to your starting place. Assessment 1. A current I = 4.0 amps flows into a junction where three wires meet. I1 = 1.0 amp. What is I2? Assessment 1. A current I = 4.0 amps flows into a junction where three wires meet. I1 = 1.0 amp. What is I2? Use the junction rule: I2 = 3.0 amps Assessment 2. A 15 volt battery is connected in parallel to two identical resistors. a) What is the voltage across R1? b) If R1 and R2 have different resistances, will they have different voltages? Assessment 2. A 15 volt battery is connected in parallel to two identical resistors. a) What is the voltage across R1? 15 volts (use the loop rule) b) If R1 and R2 have different resistances, will they have different voltages? Assessment 2. A 15 volt battery is connected in parallel to two identical resistors. a) What is the voltage across R1? 15 volts (use the loop rule) b) If R1 and R2 have different resistances, will they have different voltages? They will still both have a 15 V drop. Assessment 3. Two 30 Ω resistors are connected in parallel with a 10 volt battery. a) What is the total resistance of the circuit? a) What is the voltage drop across each resistor? c) What is the current flow through each resistor? Assessment 3. Two 30 Ω resistors are connected in parallel with a 10 volt battery. a) What is the total resistance of the circuit? 15 ohms a) What is the voltage drop across each resistor? c) What is the current flow through each resistor? Assessment 3. Two 30 Ω resistors are connected in parallel with a 10 volt battery. a) What is the total resistance of the circuit? 15 ohms a) What is the voltage drop across each resistor? 10 volts Each resistor is in its own loop with the 10 V battery, so each resistor has a voltage drop of 10 V. c) What is the current flow through each resistor? Assessment 3. Two 30 Ω resistors are connected in parallel with a 10 volt battery. a) What is the total resistance of the circuit? 15 ohms a) What is the voltage drop across each resistor? 10 volts Each resistor is in its own loop with the 10 V battery, so each resistor has a voltage drop of 10 V. c) What is the current flow through each resistor? 0.33 amps Assessment 4. Two 5.0 Ω resistors are connected in series with a 30 volt battery. a) What is the total resistance of the circuit? a) What is the current flow through each resistor? c) What is the voltage drop across each resistor? Assessment 4. Two 5.0 Ω resistors are connected in series with a 30 volt battery. a) What is the total resistance of the circuit? 10 ohms a) What is the current flow through each resistor? c) What is the voltage drop across each resistor? Assessment 4. Two 5.0 Ω resistors are connected in series with a 30 volt battery. a) What is the total resistance of the circuit? 10 ohms a) What is the current flow through each resistor? 3.0 amps The circuit has only one branch, so current flow is the same everywhere in the circuit. c) What is the voltage drop across each resistor? Assessment 4. Two 5.0 Ω resistors are connected in series with a 30 volt battery. a) What is the total resistance of the circuit? 10 ohms a) What is the current flow through each resistor? 3.0 amps The circuit has only one branch, so current flow is the same everywhere in the circuit. c) What is the voltage drop across each resistor? 15 volts Use the loop rule: