# Algebra Based Physics

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1 Page 1 of 105

2 Algebra Based Physics Electric Current & DC Circuits Page 2 of 105

3 Electric Current & DC Circuits Circuits Conductors Resistivity and Resistance Circuit Diagrams Measurement Page 3 of 105

4 Circuits Page 4 of 105

5 Electric Current Electric Current is the rate of flow of electric charges (charge carriers) through space. More specifically, it is defined as the amount of charge that flows past a location in a material per unit time. The letter "I" is the symbol for current. I = ΔQ Δt ΔQ is the amount of charge, and Δt is the time it flowed past the location. The current depends on the type of material and the Electric Potential difference (voltage) across it. Page 5 of 105

6 Electric Current A good analogy to help understand Electric Current is to consider water flow. The flow of water molecules is similar to the flow of electrons (the charge carriers) in a wire. Water flow depends on the pressure exerted on the molecules either by a pump or by a height difference, such as water falling off a cliff. Electric current depends on the "pressure" exerted by the Electric Potential difference - the greater the Electric Potential difference, the greater the Electric Current. Page 6 of 105

7 Electric Current The current, I = ΔQ Δt has the units Coulombs per second. The units can be rewritten as Amperes (A). Amperes are often called "amps". 1 A = 1 C/s Page 7 of 105

8 Electric Current We know that if an Electric Potential difference is applied to a wire, charges will flow from high to low potential - a current. However, due to a convention set by Benjamin Franklin, current in a wire is defined as the movement of positive charges (not the electrons which are really moving) and is called "conventional current." Ben didn't do this to confuse future generations of electrical engineers and students. It was already known that electrical phenomena came in two flavors - attractive and repulsive - Ben was the person who explained them as distinct positive and negative charges. Page 8 of 105

9 Electric Current He arbitrarily assigned a positive charge to a glass rod that had been rubbed with silk. He could just as easily called it negative - 50/50 chance. The glass rod was later found to have a shortage of electrons (they were transferred to the silk). So if the glass rod is grounded, the electrons will flow from the ground to the rod. The problem comes in how Electric Potential is defined - charge carriers will be driven from high to low potential - from positive to negative. For this to occur in the glass rod - ground system, the conventional current will flow from the rod to the ground - opposite the direction of the movement of electrons. Page 9 of 105

10 Electric Current To summarize - conventional Electric Current is defined as the movement of positive charge. In wires, it is opposite to the direction of the electron movement. However - in the case of a particle accelerator, where electrons are stripped off of an atom, resulting in a positively charged ion, which is then accelerated to strike a target - the direction of the conventional current is the same as the direction of the positive ions! Page 10 of 105

11 Circuits An electric circuit is an external path that charges can follow between two terminals using a conducting material. For charge to flow, the path must be complete and unbroken. An example of a conductor used to form a circuit is copper wire. Continuing the water analogy, one can think of a wire as a pipe for charge to move through. Page 11 of 105

12 1 12 C of charge passes a location in a circuit in 10 seconds. What is the current flowing past the point? Page 12 of 105

13 2 20 C of charge passes a location in a circuit in 30 seconds. What is the current flowing past the point? Page 13 of 105

14 3 A circuit has 3 A of current. How long does it take 45 C of charge to travel through the circuit? Page 14 of 105

15 4 A circuit has 10 A of current. How long does it take 20 C of charge to travel through the circuit? Page 15 of 105

16 5 A circuit has 10 A of current. How much charge travels through the circuit after 20s? Page 16 of 105

17 6 A circuit has 2.5 A of current. How much charge travels through the circuit after 4s? Page 17 of 105

18 Batteries Each battery has two terminals which are conductors. The terminals are used to connect an external circuit allowing the movement of charge. Positive Terminal Batteries convert chemical energy to electrical energy which maintains the potential difference. The chemical reaction acts like an escalator, carrying charge up to a higher voltage. Negative Terminal Click here for a Battery Voltage Simulation from PhET Page 18 of 105

19 Reviewing Basic Circuits The circuit cannot have gaps. The bulb had to be between the wire and the terminal. A voltage difference is needed to make the bulb light. The bulb still lights regardless of which side of the battery you place it on. As you watch the video,observations and the answers to the questions below. What is going on in the circuit? What is the role of the battery? How are the circuits similar? different? Click here for video using the circuit simulator from PhET Page 19 of 105

20 Batteries and Current The battery pushes current through the circuit. A battery acts like a pump, pushing charge through the circuit. It is the circuit's energy source. Charges do not experience an electrical force unless there is a difference in electrical potential (voltage).therefore, batteries have a potential difference between their terminals. The positive terminal is at a higher voltage than the negative terminal. click here for a video from Veritasium's Derek on current How will voltage affect current? Page 20 of 105

21 Conductors Page 21 of 105

22 Conductors Some conductors "conduct" better or worse than others. Reminder: conducting means a material allows for the free flow of electrons. The flow of electrons is just another name for current. Another way to look at it is that some conductors resist current to a greater or lesser extent. We call this resistance, R. Resistance is measured in ohms which is noted by the Greek symbol omega (Ω) How will resistance affect current? Click here to run another PhET simulation Page 22 of 105

23 Current vs Resistance & Voltage Raising resistance reduces current. Raising voltage increases current. We can combine these relationships in what we call "Ohm's Law". Another way to write this is that: R = V I ORV = IR I = V R You can see that one Ω = V A click here for a Veritasium music video on electricity Page 23 of 105

24 7 A flashlight has a resistance of 25 Ω and is connected by a wire to a 120 V source of voltage. What is the current in the flashlight? Page 24 of 105

25 8 A flashlight has a resistance of 30 Ω and is connected by a wire to a 90 V source of voltage. What is the current in the flashlight? Page 25 of 105

26 9 What is the current in a wire whose resistance is 3 Ω if 1.5 V is applied to it? Page 26 of 105

27 10How much voltage is needed in order to produce a 0.70 A current through a 490 Ω resistor? Page 27 of 105

28 11 How much voltage is needed in order to produce a 0.5 A current through a 150 Ω resistor? Page 28 of 105

29 12 What is the resistance of a rheostat coil, if 0.05 A of current flows through it when 6 V is applied across it? Page 29 of 105

30 13 What is the resistance of a rheostat coil, if 20 A of current flows through it when 1000 V is applied across it? Page 30 of 105

31 Electrical Power Power is defined as work per unit time P = W t if W = QV then substitute: P = QV t if I = Q t then substitute: P = IV What happens if the current is increased? What happens if the voltage is decreased? Page 31 of 105

32 Electrical Power Let's think about this another way... The water at the top has GPE & KE. As the water falls, it loses GPE and the wheel gets turned, doing work.when the water falls to the bottom it is now slower, having done work. Page 32 of 105

33 Electrical Power Electric circuits are similar. A charge falls from high voltage to low voltage. In the process of falling energy may be used (light bulb, run a motor, etc). What is the unit of Power? Page 33 of 105

34 Electrical Power How can we re-write electrical power by using Ohm's Law? (electrical power) P = IV (Ohm's Law) I = V R P = VV R P = V 2 R Page 34 of 105

35 Electrical Power Is there yet another way to rewrite this? I = V can be rewritten as V = IR. R (electrical power) P = IV (Ohm's Law) V = I R We can substitute this into Power P = I(IR) P = I 2 R Page 35 of 105

36 Batteries D, C, AA, & AAA have the same voltage, however they differ in the amount of power they deliver. D C AAA AA 1.5 V 9 V For instance, D batteries can deliver more current and therefore more power. Page 36 of 105

37 14 A toy car's electric motor has a resistance of 17 Ω; find the power delivered to it by a 6-V battery. Page 37 of 105

38 15 A toy car's electric motor has a resistance of 6 Ω; find the power delivered to it by a 7-V battery. Page 38 of 105

39 16 What is the power consumption of a flash light bulb that draws a current of 0.28 A when connected to a 6 V battery? Page 39 of 105

40 17 What is the power consumption of a flash light bulb that draws a current of 0.33 A when connected to a 100 V battery? Page 40 of 105

41 18 A 30Ω toaster consumes 560 W of power: how much current is flowing through the toaster? Page 41 of 105

42 19 A 50Ω toaster consumes 200 W of power: how much current is flowing through the toaster? Page 42 of 105

43 20 When 30 V is applied across a resistor it generates 600 W of heat: what is the magnitude of its resistance? Page 43 of 105

44 21 When 100 V is applied across a resistor it generates 200 W of heat: what is the magnitude of its resistance? Page 44 of 105

45 "Pipe" size How could the wire in the circuit affect the current? If wire is like a pipe, and current is like water that flows through the pipe... if there were pipes with water in them, what could we do to the pipes to change the speed of the water (the current)? Page 45 of 105

46 ** Resistivity and Resistance Page 46 of 105

47 ** Resistivity & Resisitance Every conductor "conducts" electric charge to a greater or lesser extent. The last example also applies to conductors like copper wire. Decreasing the length (L) or increasing the cross-sectional area (A) would increase conductivity. Also, the measure of a conductor's resistance to conduct is called its resistivity. Each material has a different resistivity. Resistivity is abbreviated using the Greek letter rho (ρ). Combining what we know about A, L, and ρ, we can find a conductor's total resistance. R = ρl A Page 47 of 105

48 ** Resistivity & Resisitance Resistance, R, is measured in Ohms (Ω). Ω is the Greek letter Omega. Cross-sectional area, A, is measured in m 2 Length, L, is measured in m R = ρl A Resistivity, ρ, is measured in Ωm How can we define A for a wire? Page 48 of 105

49 ** Resisitance ρ = RA L What is the resistance of a good conductor? Low; low resistance means that electric charges are free to move in a conductor. Click here for a PhET simulation about Resistance Page 49 of 105

50 ** Resistivities of Common Conductors Material Silver Copper Gold Aluminum Tungsten Iron Platinum Mercury Nichrome Resistivity (10-8 Ωm) Page 50 of 105

51 22 ** Rank the following materials in order of best conductor to worst conductor. A Iron, Copper, Platinum B Platinum, Iron, Copper C Copper, Iron, Platinum Material Resistivity (10-8 Ωm) Silver 1.59 Copper 1.68 Gold 2.44 Aluminum 2.65 Tungsten 5.60 Iron 9.71 Platinum 10.6 Mercury 98 Nichrome Page 51 of 105

52 23 What is the resistance of a 2 m long copper wire whose cross-sectional area of 0.2 mm 2? ** Page 52 of 105

53 24 ** An aluminum wire with a length of 900 m and a cross-sectional area of 10 mm 2 has a resistance of 2.5 Ω. What is the resistivity of the wire? Page 53 of 105

54 25 What diameter of 100 m long copper wire would ** have a resistance of 0.10 Ω? Page 54 of 105

55 26 ** What is the cross-sectional area of a 10Ω copper wire of length is meters? Page 55 of 105

56 27 ** What is the length of a 10 Ω copper wire whose diameter is 3.2 mm? Page 56 of 105

57 Circuit Diagrams Page 57 of 105

58 Circuit Diagrams Drawing realistic pictures of circuits can be very difficult. For this reason, we have common symbols to represent each piece. Resistor Battery Wire *Note: Circuit diagrams do not show where each part is physically located. Page 58 of 105

59 Circuit Diagrams Draw a simple circuit that has a 9 V battery with a 3 Ω resistor across its terminals. What is the magnitude and direction of the current? Conventional current flows from the positive terminal to the negative terminal. Page 59 of 105

60 Circuit Diagrams There are two ways to add a second resistor to the circuit. Series Parallel R 1 R 2 R 1 R 2 V All charges must move through both resistors to get to the negative terminal. V Charges pass through either R 1 or R 2 but not both. Page 60 of 105

61 Circuit Diagrams Are the following sets of resistors in series or parallel? R 1 R 1 V R 2 V R 2 Page 61 of 105

62 Equivalent Resistance Resistors and voltage from batteries determine the current. Circuits can be redrawn as if there were only a single resistor and battery.by reducing the circuit this way, the circuit becomes easier to study. The process of reducing the resistors in a circuit is called finding the equivalent resistance (R eq ). R 1 R 2 V Page 62 of 105

63 Series Circuits: Equivalent Resistance R 1 R 2 What happens to the current in the circuit to the right? V Page 63 of 105

64 Series Circuits: Equivalent Resistance R 1 R 2 What happens to the voltage as it moves around the circuit? V Page 64 of 105

65 Series Circuits: Equivalent Resistance If V = V 1 + V 2 + V substitute Ohm's Law solved for V is: V = IR IR = I 1 R 1 + I 2 R 2 + I 3 R 3 IR = IR 1 + IR 2 + IR 3 but since current (I) is the same everywhere in a series circuit, I = I 1 = I 2 = I 3 R eq = R 1 + R 2 + R Now divide by I To find the equivalent resistance (R eq ) of a series circuit, add the resistance of all the resistors.if you add more resistors to a series circuit, what happens to the resistance? Page 65 of 105

66 28 What is the equivalent resistance in this circuit? R 1 = 5Ω R 2 = 3Ω V = 9 V Page 66 of 105

67 29 What is the total current at any spot in the circuit? R 1 = 5Ω R 2 = 3Ω V = 9 V Page 67 of 105

68 30 What is the voltage drop across R 1? R 1 = 5Ω R 2 = 3Ω V = 9 V Page 68 of 105

69 31 What is the voltage drop across R 2? R 1 = 5Ω R 2 = 3Ω V = 9 V hint: A good way to check your work is to see if the voltage drop across all resistors equals the total voltage in the circuit. Page 69 of 105

70 32 How much power is used by R 1? R 1 = 5Ω R 2 = 3Ω V = 9 V Page 70 of 105

71 33 What is the equivalent resistance in this circuit? R 1 = 10Ω R 2 = 20Ω V = 9 V Page 71 of 105

72 34 What is the total current at any spot in the circuit? R 1 = 10Ω R 2 = 20Ω V = 9 V Page 72 of 105

73 35 What is the voltage drop across R 1? R 1 = 10Ω R 2 = 20Ω V = 9 V Page 73 of 105

74 36 What is the voltage drop across R 2? R 1 = 10Ω R 2 = 20Ω V = 9 V Page 74 of 105

75 37 How much power is used by R 1? R 1 = 10Ω R 2 = 20Ω V = 9 V Page 75 of 105

76 38 How much power is used by R 2? R 1 = 10Ω R 2 = 20Ω V = 9 V Page 76 of 105

77 Parallel Circuits: Equivalent Resistance R 1 What happens to the current in the circuit to the right? R 2 V Page 77 of 105

78 Parallel Circuits: Equivalent Resistance R 1 What happens to the voltage as it moves around the circuit? R 2 V Page 78 of 105

79 Parallel Circuits: Equivalent Resistance If I = I 1 + I 2 + I 3 Rewrite Ohm's Law for I and substitute for each resistor V R = V 1 R 1 + V 2 R 2 + V 3 R 3 Also, since V = V 1 = V 2 = V 3 so we can substitute V for any other voltage V R = V R 1 + V R 2 + V R 3 Voltage is a common factor, so factor it out! V R = V( 1 R R = + + R eq R 2 R 1 R 3 1 R 3 ( Divide by V to eliminate voltage from the equation. If you add more resistors in parallel, what will happen the to resistance of the circuit? Page 79 of 105

80 39 What is the equivalent resistance in the circuit? R 1 = 3Ω R 2 = 6Ω V = 18V Page 80 of 105

81 40 What is the voltage at any spot in the circuit? R 1 = 3Ω R 2 = 6Ω V = 18V Page 81 of 105

82 41 What is the current through R 1? R 1 = 3Ω R 2 = 6Ω V = 18V Page 82 of 105

83 42 What is the current through R 2? R 1 = 3Ω R 2 = 6Ω V = 18V Page 83 of 105

84 43 What is the power used by R 1? R 1 = 3Ω R 2 = 6Ω V = 18V Page 84 of 105

85 44 What is the power used by R 2? R 1 = 3Ω R 2 = 6Ω V = 18V Page 85 of 105

86 45 What is the equivalent resistance in the circuit? Page 86 of 105

87 46 What is the voltage at any spot in the circuit? Page 87 of 105

88 47 What is the current through R 1? Page 88 of 105

89 48 What is the current through R 2? Page 89 of 105

90 49 What is the power used by R 1? Page 90 of 105

91 50 What is the power used by R 1? Page 91 of 105

92 51 What is the power used by R 2? Page 92 of 105

93 Measurement Page 93 of 105

94 Voltmeter Voltage is measured with a voltmeter. Voltmeters are connected in parallel and measure the difference in potential between two points. Since circuits in parallel have the same voltage, and a voltmeter has very high resistance, very little current passes through it. This means that it has little effect on the circuit. Page 94 of 105

95 Ammeter Current is measured using an ammeter. Ammeters are placed in series with a circuit. In order to not interfere with the current, the ammeter has a very low resistance. Page 95 of 105

96 Multimeter Although there are separate items to measure current and voltage, there are devices that can measure both (one at a time). These devices are called multimeters.multimeters can also measure resistance. Click here for a PhET simulation on circuits Page 96 of 105

97 52 A group of students prepare an experiment with electric circuits. Which of the following diagrams can be used to measure both current and voltage? L A B C D E Page 97 of 105

98 * Electromotive Force R eq A battery is a source of voltage AND a resistor. + r E _ Each battery has a source of electromotive force and internal resistance. Electromotive force (EMF) is the process that carries charge from low to high voltage. Another way to think about it is that EMF is the voltage you measure when no resistance is connected to the circuit. Page 98 of 105

99 * Electromotive Force R eq + r E _ Terminal voltage (V T ) is the voltage measured when a voltmeter is across its terminals. If there is no circuit attached, no current flows, and the measurement will equal the EMF. If however a circuit is attached, the internal resistance will result in a voltage drop, and a smaller terminal voltage. (E - Ir) Page 99 of 105

100 * Terminal Voltage R eq We say that the terminal voltage is: + r E _ V T = E - Ir Maximum current will occur when there is zero external current. When solving for equivalent resistance in a circuit, the internal resistance of the battery is considered a series resistor. R EQ = R int + R ext Page 100 of 105

101 53* When the switch in the circuit below is open, the voltmeter reading is referred to as: A EMF B Current C Power D Terminal Voltage E Restivity Page 101 of 105

102 54* When the switch in the circuit below is closed, the voltmeter reading is referred to as: A Terminal Voltage B EMF C Current D Resistance E Power Page 102 of 105

103 55* A 6V battery, whose internal resistance 1.5 Ω is connected in series to a light bulb with a resistance of 6.8 Ω. What is the current in the circuit? Page 103 of 105

104 56* A 6V battery, whose internal resistance 1.5Ω is connected in series to a light bulb with a resistance of 6.8Ω. What is the terminal voltage of the battery? Page 104 of 105

105 57* A 25 Ω resistor is connected across the terminals of a battery whose internal resistance is 0.6 Ω. What is the EMF of the battery if the current in the circuit is 0.75 A? Page 105 of 105

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