# Parallel Resistors (32.6)

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1 Parallel Resistors (32.6) Resistors connected at both ends are called parallel resistors Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

2 Parallel Resistors (32.6) Resistors connected at both ends are called parallel resistors The important thing to note is that: the two left ends of the resistors are at the same potential. Also, the two right ends are at the same potential. Therefore, the V for each resistor is the same! Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

3 Parallel Resistors (32.6) Resistors connected at both ends are called parallel resistors The important thing to note is that: the two left ends of the resistors are at the same potential. Also, the two right ends are at the same potential. Therefore, the V for each resistor is the same! Kirchhoff s Junction Law means that I = I 1 + I 2 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

4 Parallel Resistors (32.6) Resistors connected at both ends are called parallel resistors The important thing to note is that: the two left ends of the resistors are at the same potential. Also, the two right ends are at the same potential. Therefore, the V for each resistor is the same! Kirchhoff s Junction Law means that I = I 1 + I 2 Using Ohm s Law I = V 1 R 1 + V 2 R 2 = V R 1 + V R 2 ( 1 = V + 1 ) R1 R2 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

5 Parallel Resistors (32.6) Starting with ( 1 I = V + 1 ) R1 R2 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

6 Parallel Resistors (32.6) Starting with ( 1 I = V + 1 ) R1 R2 and using Ohm s Law again: R = V I ( 1 = + 1 ) 1 R1 R2 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

7 Parallel Resistors (32.6) Starting with ( 1 I = V + 1 ) R1 R2 and using Ohm s Law again: R = V I ( 1 = + 1 ) 1 R1 R2 So, for many parallel resistors ( 1 R eq = R1 + 1 R R N ) 1 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

8 Example 32.9 Example 32.9 Three resistors are connected in parallel to a 9V battery. Find the potential difference across and the current through each resistor. (Note: assume an ideal battery and ideal wires) Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

9 Example 32.9 We replace the 3 parallel resistors with a single equivalent resistor: ( 1 R eq = 15Ω + 1 4Ω + 1 ) 1 = (0.4417Ω 1 ) 1 = 2.26Ω 8Ω Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

10 Example 32.9 We replace the 3 parallel resistors with a single equivalent resistor: ( 1 R eq = 15Ω + 1 4Ω + 1 ) 1 = (0.4417Ω 1 ) 1 = 2.26Ω 8Ω The equivalent resistance is LESS than any one resistor! Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

11 Example 32.9 We replace the 3 parallel resistors with a single equivalent resistor: ( 1 R eq = 15Ω + 1 4Ω + 1 ) 1 = (0.4417Ω 1 ) 1 = 2.26Ω 8Ω The equivalent resistance is LESS than any one resistor! We can then determine the current across the equivalent circuit I = E R eq = 9V 2.26Ω = 3.98A Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

12 Example 32.9 Now, we know that the current divides at each junction (not equally!!) and that the potential is the same across each resistor I 1 = 9V 15Ω = 0.6A, I 2 = 9V 4Ω = 2.25A, I 3 = 9V 8Ω = 1.13A Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

13 Example 32.9 Now, we know that the current divides at each junction (not equally!!) and that the potential is the same across each resistor I 1 = 9V 15Ω = 0.6A, I 2 = 9V 4Ω = 2.25A, I 3 = 9V 8Ω = 1.13A The sum of the currents is 3.98A as expected. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

14 Lightbulb Puzzle What happens to the brightness of A and B when the switch is closed? How does the brightness of C compare to that of A and B? Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

15 Lightbulb Puzzle When the switch is open, A and B are in series with R eq = 2R and current is I before = E 2R Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

16 Lightbulb Puzzle When the switch is open, A and B are in series with R eq = 2R and current is I before = E 2R Closing the switch puts B and C in parallel with R eq = R/2. They are in series with A giving total resistance of 3 2 R I after = E 3R/2 = 2 E 3 R > I before Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

17 Lightbulb Puzzle When the switch is open, A and B are in series with R eq = 2R and current is I before = E 2R Closing the switch puts B and C in parallel with R eq = R/2. They are in series with A giving total resistance of 3 2 R I after = E 3R/2 = 2 E 3 R > I before So, A gets brighter, B gets dimmer (E/3R) and C is equal to B. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

18 Voltmeters A voltmeter measures the potential difference across an element of a circuit. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

19 Voltmeters A voltmeter measures the potential difference across an element of a circuit. It must be connected in parallel so that it has the same V as the element. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

20 Voltmeters A voltmeter measures the potential difference across an element of a circuit. It must be connected in parallel so that it has the same V as the element. It should have nearly infinite resistance. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

21 Resistor Circuits (32.7) Example Find the current through and the potential difference across each of the four resistors in the circuit above. We need to develop a strategy for complex circuits. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

22 Resistor Circuits (32.7) Example Find the current through and the potential difference across each of the four resistors in the circuit above. We need to develop a strategy for complex circuits. We will break the circuit down into pieces and make an equivalent circuit, then build it back up piece by piece. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

23 Example Break it Down Add the 600 and 400 Ohm resistors in parallel = 240 Ohm Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

24 Example Break it Down Add the 600 and 400 Ohm resistors in parallel = 240 Ohm Add 240 and 560 in series = 800 Ohm Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

25 Example Break it Down Add the 600 and 400 Ohm resistors in parallel = 240 Ohm Add 240 and 560 in series = 800 Ohm Add 800 and 800 Ohm in parallel = 400 Ohm Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

26 Example Break it Down Add the 600 and 400 Ohm resistors in parallel = 240 Ohm Add 240 and 560 in series = 800 Ohm Add 800 and 800 Ohm in parallel = 400 Ohm The current is I = E R = 12V 400Ω = 30mA Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

27 Example Build it Up Two parallel 800Ω resistors split the current equally. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

28 Example Build it Up Two parallel 800Ω resistors split the current equally. Splitting the 800Ω resistor leaves the same current in two series resistors (15mA) and known resistances, so voltages can be calculated. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

29 Example Build it Up Two parallel 800Ω resistors split the current equally. Splitting the 800Ω resistor leaves the same current in two series resistors (15mA) and known resistances, so voltages can be calculated. Parallel resistors have the same potential difference and known resistances, so current can be calculated. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

30 Example A 2-Loop Circuit Example Find the current and the potential difference across the 100Ω resistor. None of these resistors are in series or parallel, you cannot simplify it. It is a 2-loop circuit. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

31 Example A 2-Loop Circuit Apply the loop law twice. Choose a direction for the center wire. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

32 Example A 2-Loop Circuit Apply the loop law twice. Choose a direction for the center wire. For the left loop 19V (300Ω)I 1 (100Ω)I 3 12V = 0 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

33 Example A 2-Loop Circuit Apply the loop law twice. Choose a direction for the center wire. For the left loop 19V (300Ω)I 1 (100Ω)I 3 12V = 0 For the right loop 12V + (100Ω)I 3 (200Ω)I 2 = 0 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

34 Example A 2-Loop Circuit Apply the loop law twice. Choose a direction for the center wire. For the left loop 19V (300Ω)I 1 (100Ω)I 3 12V = 0 For the right loop 12V + (100Ω)I 3 (200Ω)I 2 = 0 Using I 3 = I 1 I 2 gives 2 equations and 2 unknowns 400I 1 100I 2 = 7 100I I 2 = 12 and we can solve for I 1, I 2 and I 3 Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

35 Grounding Circuits (32.8) We often hear about circuits being grounded. Why? Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

36 Grounding Circuits (32.8) We often hear about circuits being grounded. Why? Within a single circuit you only care about relative potential differences. However, if two circuits are connected it suddenly becomes necessary to worry about a common reference point. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

37 Grounding Circuits (32.8) We often hear about circuits being grounded. Why? Within a single circuit you only care about relative potential differences. However, if two circuits are connected it suddenly becomes necessary to worry about a common reference point. A circuit connected to the earth by a single wire (not in a loop) is said to be grounded. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

38 Grounding Circuits (32.8) We often hear about circuits being grounded. Why? Within a single circuit you only care about relative potential differences. However, if two circuits are connected it suddenly becomes necessary to worry about a common reference point. A circuit connected to the earth by a single wire (not in a loop) is said to be grounded. The ground wire carries no current, so does not effect the circuit. However, it is now possible to specify absolute voltages, not just differences. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

39 Grounding Circuits (32.8) We often hear about circuits being grounded. Why? Within a single circuit you only care about relative potential differences. However, if two circuits are connected it suddenly becomes necessary to worry about a common reference point. A circuit connected to the earth by a single wire (not in a loop) is said to be grounded. The ground wire carries no current, so does not effect the circuit. However, it is now possible to specify absolute voltages, not just differences. Grounding is an important saftey feature, keep the case surrounding a circuit at V = 0V at all times. Neil Alberding (SFU Physics) Physics 121: Optics, Electricity & Magnetism Spring / 1

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