Slide 1 / 127 Slide 2 / 127 Electric Current & DC Circuits www.njctl.org Slide 3 / 127 How to Use this File Slide 4 / 127 Electric Current & DC Circuits Each topic is composed of brief direct instruction There are formative assessment questions after every topic denoted by black text and a number in the upper left. > Students work in groups to solve these problems but use student responders to enter their own answers. > Designed for SMART Response PE student response systems. > Use only as many questions as necessary for a sufficient number of students to learn a topic. Full information on how to teach with NJCTL courses can be found at njctl.org/courses/teaching methods Circuits Conductors Resistivity and Resistance Circuit Diagrams Measurement EMF & Terminal oltage Kirchhoff's Rules Capacitors RC Circuits Click on the topic to go to that section Slide 5 / 127 Slide 6 / 127 Electric Current Circuits 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. Return to Table of Contents The current depends on the type of material and the Electric Potential difference (voltage) across it.
Slide 7 / 127 Electric Current Slide 8 / 127 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. The current, I = ΔQ Δt has the units Coulombs per second. 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. The units can be rewritten as Amperes (A). Amperes are often called "amps". 1 A = 1 C/s Slide 9 / 127 Electric Current Slide 10 / 127 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." Benjamin Franklin 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 - Franklin was the person who explained them as distinct positive and negative charges. 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. Slide 11 / 127 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! 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. Slide 12 / 127 Circuits 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.
Slide 13 / 127 1 12 C of charge passes a location in a circuit in 10 seconds. What is the current flowing past the point? Slide 13 () / 127 1 12 C of charge passes a location in a circuit in 10 seconds. What is the current flowing past the point? Slide 14 / 127 2 A circuit has 3 A of current. How long does it take 45 C of charge to travel through the circuit? Slide 14 () / 127 2 A circuit has 3 A of current. How long does it take 45 C of charge to travel through the circuit? Slide 15 / 127 3 A circuit has 2.5 A of current. How much charge travels through the circuit after 4s? Slide 15 () / 127 3 A circuit has 2.5 A of current. How much charge travels through the circuit after 4s?
Each battery has two terminals which are conductors. The terminals are used to connect an external circuit allowing the movement of charge. 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. Slide 16 / 127 Batteries Click here for a Battery oltage Simulation from PhET Positive Terminal Negative Terminal Slide 17 / 127 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 Slide 18 / 127 Slide 19 / 127 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. Conductors 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 eritasium's Derek on current How will voltage affect current? Return to Table of Contents Slide 20 / 127 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) Slide 21 / 127 Current vs Resistance & oltage Raising resistance reduces current. Raising voltage increases current. We can combine these relationships in what we call "Ohm's Law". I = R Another way to write this is that: R = I OR = IR How will resistance affect current? Click here to run another PhET simulation You can see that one # = A click here for a eritasium music video on electricity
Slide 22 / 127 Current vs Resistance & oltage Raising resistance reduces current and raising voltage increases current. However, this relationship is linear in only what we call Ohmic resistors. If the relationship is not linear, it is a non-ohmic resistor. Slide 23 / 127 4 A flashlight has a resistance of 25 # and is connected by a wire to a 120 source of voltage. What is the current in the flashlight? oltage Non-Ohmic Ohmic Current Slide 23 () / 127 4 A flashlight has a resistance of 25 # and is connected by a wire to a 120 source of voltage. What is the current in the flashlight? Slide 24 / 127 5 How much voltage is needed in order to produce a 0.70 A current through a 490 # resistor? Slide 24 () / 127 Slide 25 / 127 6 What is the resistance of a rheostat coil, if 0.05 A of current flows through it when 6 is applied across it?
Slide 25 () / 127 6 What is the resistance of a rheostat coil, if 0.05 A of current flows through it when 6 is applied across it? Slide 26 / 127 Electrical Power Power is defined as work per unit time P = W t if W = Q then substitute: if I = Q then substitute: t P = Q t P = I What happens if the current is increased? What happens if the voltage is decreased? Slide 27 / 127 Electrical Power Slide 28 / 127 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. 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? Slide 29 / 127 Electrical Power How can we re-write electrical power by using Ohm's Law? (electrical power) P = I P = R P = 2 R (Ohm's Law) I = R Slide 30 / 127 Is there yet another way to rewrite this? (electrical power) P = I Electrical Power I = can be rewritten as = IR. R P = I(IR) P = I 2 R (Ohm's Law) = I R We can substitute this into Power
Slide 31 / 127 Batteries D, C, AA, & AAA have the same voltage, however they differ in the amount of power they deliver. Slide 32 / 127 7 A toy car's electric motor has a resistance of 17 # ; find the power delivered to it by a 6- battery. D C AAA AA 1.5 9 For instance, D batteries can deliver more current and therefore more power. Slide 32 () / 127 7 A toy car's electric motor has a resistance of 17 # ; find the power delivered to it by a 6- battery. Slide 33 / 127 8 What is the power consumption of a flash light bulb that draws a current of 0.28 A when connected to a 6 battery? Slide 33 () / 127 8 What is the power consumption of a flash light bulb that draws a current of 0.28 A when connected to a 6 battery? Slide 34 / 127 9 A 30Ω toaster consumes 560 W of power: how much current is flowing through the toaster?
Slide 34 () / 127 Slide 35 / 127 10 When 30 is applied across a resistor it generates 600 W of heat: what is the magnitude of its resistance? Slide 35 () / 127 Slide 36 / 127 10 When 30 is applied across a resistor it generates 600 W of heat: what is the magnitude of its resistance? Resistivity and Resistance Return to Table of Contents Slide 37 / 127 "Pipe" size Slide 37 () / 127 "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)? How could the wire in the circuit affect the current? If wire is like a pipe, and current is like water that flows change through the cross-sectional area of the pipe... the pipe making it bigger will allow more water to flow if there were pipes with water in them, what could we do to the change the length of the pipe pipes to change the speed of the increasing the length will water (the current)? increase the time it takes for the water to get to the end of its trip
Slide 38 / 127 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 (#). Combining what we know about A, L, and ρ, we can find a conductor's total resistance. Slide 39 / 127 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 = #L A Resistivity, ρ, is measured in Ωm How can we define A for a wire? R = #L A 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 Slide 40 / 127 Resisitance # = RA L Slide 42 / 127 11 Rank the following materials in order of best conductor to worst conductor. Slide 41 / 127 Resistivities of Common Conductors Material Silver Copper Gold Aluminum Tungsten Iron Platinum Mercury Nichrome Resistivity (10-8 Ωm) 1.59 1.68 2.44 2.65 5.60 9.71 10.6 98 100 Slide 42 () / 127 11 Rank the following materials in order of best conductor to worst conductor. A Iron, Copper, Platinum A Iron, Copper, Platinum B Platinum, Iron, Copper Material Resistivity (10-8 Ωm) B Platinum, Iron, Copper Material Resistivity (10-8 Ωm) C Copper, Iron, Platinum Silver Copper 1.59 1.68 C Copper, Iron, Platinum Silver Copper 1.59 1.68 Gold 2.44 Gold 2.44 Aluminum 2.65 Aluminum 2.65 Tungsten Iron Platinum Mercury 5.60 9.71 10.6 98 Tungsten Iron Platinum Mercury C 5.60 9.71 10.6 98 Nichrome 100 Nichrome 100
Slide 43 / 127 Slide 43 () / 127 12 What is the resistance of a 2 m long copper wire whose cross-sectional area of 0.2 mm 2? 2r Slide 44 / 127 13 The following resistors are made of the same material. Rank them from greatest resistance to least resistance. L 2r 2L R w R x R y R z r L r 2L 2r Slide 44 () / 127 13 The following resistors are made of the same material. Rank them from greatest resistance to least resistance. L 2r 2L R w R x R y R z r L r 2L A R x > R w > R z > R y B R z > R y > R x > R w C Ry > R z > R w > R x A R x > R w > R z > R y B R z > R y > R x > R w C Ry > R z > R w > R x B D R y > R w > R z > R x D R y > R w > R z > R x Slide 45 / 127 14 What diameter of 100 m long copper wire would have a resistance of 0.10 #? Slide 45 () / 127 14 What diameter of 100 m long copper wire would have a resistance of 0.10 #?
Slide 46 / 127 Slide 46 () / 127 15 The length of a copper wire is cut to half. By what factor does the resistance change? 15 The length of a copper wire is cut to half. By what factor does the resistance change? A 1/4 B 1/2 C 2 D 4 A 1/4 B 1/2 C 2 D 4 B Slide 47 / 127 Slide 47 () / 127 16 The radius of a copper wire is doubled. By what factor does the resistivity change? 16 The radius of a copper wire is doubled. By what factor does the resistivity change? A 1/4 B 1/2 C 1 D 2 A 1/4 B 1/2 C 1 D 2 C Slide 48 / 127 Slide 49 / 127 Circuit Diagrams 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 Return to Table of Contents *Note: Circuit diagrams do not show where each part is physically located.
Slide 50 / 127 Circuit Diagrams Draw a simple circuit that has a 9 battery with a 3 Ω resistor across its terminals. What is the magnitude and direction of the current? Slide 50 () / 127 Circuit Diagrams Draw a simple circuit that has a 9 battery with a 3 Ω resistor across its terminals. What is the magnitude and direction of the current? R = 3# I = 3A I = 9 Conventional current flows from the positive terminal to the negative terminal. Slide 51 / 127 Circuit Diagrams There are two ways to add a second resistor to the circuit. Conventional current flows from the positive terminal to the negative terminal. Slide 52 / 127 Circuit Diagrams Are the following sets of resistors in series or parallel? Series Parallel R 1 R 1 R 1 R 1 R 2 R 2 R 2 R 2 All charges must move through both resistors to get to the negative terminal. Charges pass through either R 1 or R 2 but not both. Slide 52 () / 127 Circuit Diagrams Are the following sets of resistors in series or parallel? R R 1 1 Slide 53 / 127 Equivalent Resistance Resistors and voltage from batteries determine the current. R 2 R 2 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). R1 R2 Series Parallel The "test" is to trace the shortest route around the circuit. The resistors found on the same route are in series; those not found on the same route are in parallel to those that were.
Slide 54 / 127 Series Circuits: Equivalent Resistance Slide 54 () / 127 Series Circuits: Equivalent Resistance R1 R2 R1 R2 What happens to the current in the circuit to the right? What happens to the current in the circuit to the right? The current passing through all parts of a series circuit is the same. For example: I = I 1 = I 2 Slide 55 / 127 Series Circuits: Equivalent Resistance Slide 55 () / 127 Series Circuits: Equivalent Resistance R1 R2 R1 R2 What happens to the voltage as it moves around the circuit? What happens to the voltage as it moves around the circuit? The sum of the voltage drops across each of the resistors in a series circuit equals the voltage of the battery. For example: = 1 2 Slide 56 / 127 Series Circuits: Equivalent Resistance Slide 57 / 127 17 What is the equivalent resistance in this circuit? If = 1 2 3... IR = I 1R 1 I 2R 2 I 3R 3 IR = IR 1 IR 2 IR 3 substitute Ohm's Law solved for is: = IR but since current (I) is the same everywhere in a series circuit, I = I 1 = I 2 = I 3 R 1 = 5# R2 = 3# R eq = R 1 R 2 R 3... Now divide by I = 9 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?
Slide 57 () / 127 17 What is the equivalent resistance in this circuit? Slide 58 / 127 18 What is the total current at any spot in the circuit? R 1 = 5# R2 = 3# R 1 = 5# R 2 = 3# Resistors in series: = 9 = 9 Slide 58 () / 127 18 What is the total current at any spot in the circuit? Slide 59 / 127 19 What is the voltage drop across R 1? R 1 = 5# R 2 = 3# R 1 = 5# R 2 = 3# Resistors in series: = 9 = 9 Slide 59 () / 127 19 What is the voltage drop across R 1? Slide 60 / 127 20 What is the voltage drop across R 2? R 1 = 5# R 2 = 3# Resistors in series: R 1 = 5# R 2 = 3# = 9 Net Current/equal everywhere: oltage Drop across R 1: = 9 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.
Slide 60 () / 127 20 What is the voltage drop across R 2? Slide 61 / 127 21 How much power is used by R 1? R 1 = 5# R 2 = 3# Net Current/equal everywhere: Resistors in series: R 1 = 5# R 2 = 3# = 9 oltage Drop across R 2: = 9 hint: A good way to check your work [This is object to see is a if pull the tab] voltage drop across all resistors equals the total voltage in the circuit. Slide 61 () / 127 Slide 62 / 127 Parallel Circuits: Equivalent Resistance R 1 What happens to the current in the circuit to the right? R 2 Slide 62 () / 127 Parallel Circuits: Equivalent Resistance Slide 63 / 127 Parallel Circuits: Equivalent Resistance R 1 R 1 What happens to the current in the circuit to the right? R 2 What happens to the voltage as it moves around the circuit? R 2 The sum of the currents through each of the resistors in a parallel circuit equals the current of the battery. For example: I = I 1 I 2
Slide 63 () / 127 Parallel Circuits: Equivalent Resistance Slide 64 / 127 Parallel Circuits: Equivalent Resistance R 1 If I = I1 I2 I3 Rewrite Ohm's Law for I and substitute for each resistor What happens to the voltage as it moves around the circuit? R 2 R = 1 R1 2 R2 3 R3 Also, since = 1 = 2 = 3 so we can substitute for any other voltage The voltage across all the resistors in a parallel circuit is the same. For example: = 1 = 2 R R 1 = Req = = ( 1 R1 R1 1 R1 R2 1 R2 1 R2 1 R3 R3 1 R3 ( oltage is a common factor, so factor it out! Divide by to eliminate voltage from the equation. Slide 65 / 127 If you add more resistors in parallel, what will happen the to resistance of the circuit? Slide 65 () / 127 22 What is the equivalent resistance in the circuit? 22 What is the equivalent resistance in the circuit? = 18 = 18 Slide 66 / 127 23 What is the voltage at any spot in the circuit? Slide 66 () / 127 23 What is the voltage at any spot in the circuit? 18 = 18 = 18
Slide 67 / 127 24 What is the current through R 1? Slide 67 () / 127 24 What is the current through R 1? = 18 = 18 Slide 68 / 127 25 What is the current through R 2? Slide 68 () / 127 25 What is the current through R 2? = 18 = 18 Slide 69 / 127 26 What is the power used by R 1? Slide 69 () / 127 26 What is the power used by R 1? Power used by R 1: = 18 = 18
Slide 70 / 127 27 What is the power used by R 2? Slide 70 () / 127 27 What is the power used by R 2? Power used by R 2: = 18 = 18 Slide 71 / 127 28 What is the total current in this circuit? Slide 71 () / 127 28 What is the total current in this circuit? R 3 = 4# R 3 = 4# = 18 = 18 Slide 72 / 127 29 What is the voltage drop across the third resistor? Slide 72 () / 127 29 What is the voltage drop across the third resistor? R 3 = 4# R 3 = 4# = 18 = 18
Slide 73 / 127 30 What is the current though the first resistor? Slide 73 () / 127 30 What is the current though the first resistor? R 3 = 4# R 3 = 4# = 18 = 18 Slide 74 / 127 Slide 74 () / 127 31 Which two of the following sets of resistors have the same equivalent resistance? Select two answers. 31 Which two of the following sets of resistors have the same equivalent resistance? Select two answers. A 3Ω B 4Ω A 3Ω B 4Ω 6Ω 4Ω 6Ω 4Ω A, D C 3Ω 6Ω D 1Ω 1Ω C 3Ω 6Ω D 1Ω 1Ω Slide 75 / 127 Slide 76 / 127 oltmeter Measurement oltage is measured with a voltmeter. oltmeters are connected in parallel and measure the difference in potential between two points. Return to Table of Contents 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.
Slide 77 / 127 Ammeter Slide 78 / 127 Multimeter 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. 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 Slide 79 / 127 Slide 79 () / 127 32 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 32 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 A B C D C D E E E Slide 80 / 127 Slide 81 / 127 Electromotive Force EMF & Terminal oltage Return to Table of Contents r R eq E _ A battery is a source of voltage AND a resistor. 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.
Slide 82 / 127 Electromotive Force Slide 83 / 127 Terminal oltage r R eq E _ Terminal voltage ( 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. r R eq E _ We say that the terminal voltage is: T = E - Ir Maximum current will occur when there is zero external current. If however a circuit is attached, the internal resistance will result in a voltage drop, and a smaller terminal voltage. (E - Ir) 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 Slide 84 / 127 33 When the switch in the circuit below is open, the voltmeter reading is referred to as: Slide 84 () / 127 33 When the switch in the circuit below is open, the voltmeter reading is referred to as: A EMF B Current C Power D Terminal oltage E Restivity A EMF B Current C Power D Terminal oltage E Restivity A Slide 85 / 127 34 When the switch in the circuit below is closed, the voltmeter reading is referred to as: Slide 85 () / 127 34 When the switch in the circuit below is closed, the voltmeter reading is referred to as: A Terminal oltage B EMF C Current D Resistance E Power A Terminal oltage B EMF C Current D Resistance E Power A
Slide 86 / 127 Slide 86 () / 127 35 A 6 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? Slide 87 / 127 Slide 87 () / 127 36 A 6 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? Slide 88 / 127 Slide 88 () / 127 37 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?
Slide 89 / 127 Slide 90 / 127 Kirchhoff's Rules Kirchhoff's Rules Up until this point we have been analyzing simple circuits by combining resistors in series and parallel and using Ohm's law. This works for simple circuits but in order analyze more complex circuits we need to use Kirchhoff's Rules which are based on the laws of conservation of charge and energy. Return to Table of Contents Slide 91 / 127 Slide 92 / 127 Kirchhoff's First rule, or junction rule is based on the law of conservation of charge. It states: At any junction point, the sum of all currents entering the junction point must equal the sum of all the currents exiting the junction. For example, I 1 I 2 = I 3 Kirchhoff's Rules I 3 I 1 I 2 Kirchhoff's Second rule, or loop rule is based on the law of conservation of energy. It states: The sum of all changes in potential around any closed path must equal zero. For example, Kirchhoff's Rules The sum of the voltage drops is equal to the voltage across the battery. 1 2-1 - 2 =0 OR = 1 2 1. Label and - for each battery. Slide 93 / 127 Problem Solving with Kirchhoff's Rules 2. Label the current in each branch with a symbol and an arrow. (Don't worry about the direction of the arrow. It it's incorrect the solution will be negative.) 3. Apply the junction rule to each junction. You need as many equations as there are unknowns. (You can also use Ohm's Law to reduce the number of unknowns.) 4. Apply the loop rule for each loop. (Pay attention to signs. For a resistor, the potential difference is negative. For a battery the potential difference is positive.) Slide 94 / 127 Problem Solving with Kirchhoff's Rules Find the unknowns in the following circuit: I2 = 2.6 A R1 = 5# R2 = 3# R3 =? R4 = 2# 3 = 1.87 = 12 5. Solve the equations algebraically.
Slide 95 / 127 Problem Solving with Kirchhoff's Rules First, label and - for each battery. I2 = 2.6 A Slide 96 / 127 Problem Solving with Kirchhoff's Rules Next, label the current in each branch with a symbol and an arrow. I1 I2 = 2.6 A R1 = 5# R2 = 3# R3 =? R4 = 2# 3 = 1.87 _ = 12 I3 R1 = 5# R2 = 3# I4 R3 =? R4 = 2# 3 = 1.87 _ = 12 I Slide 97 / 127 Slide 97 () / 127 Problem Solving with Kirchhoff's Rules Next, apply the junction rule to each junction. Problem Solving with Kirchhoff's Rules Next, apply the junction rule to each junction. I1 I2 = 2.6 A I1 I2 = 2.6 A R1 = 5# R2 = 3# R1 = 5# R2 = 3# I3 I4 R3 =? R4 = 2# 3 = 1.87 _ = 12 I I3 I4 R3 =? = R4 = I 2# = I 3 I 4 I 2 3 = 1.87 I 1 = I 3 _ = 12 I Slide 98 / 127 Slide 98 () / 127 Problem Solving with Kirchhoff's Rules Next, apply the loop rule to each loop. Problem Solving with Kirchhoff's Rules Next, apply the loop rule to each loop. I1 I2 = 2.6 A I1 I2 = 2.6 A R1 = 5# R2 = 3# R1 = 5# R2 = 3# I3 I4 R3 =? R4 = 2# 3 = 1.87 _ = 12 I I3 I4 = 3 1 2 R3 =? R4 = 2# 3 = 1.87 = 4 2 1 3 _ = 4 = 12 I I = I 3 I 4 I = I 2 I 1 = I 3 I = I 3 I 4 I = I 2 I 1 = I 3
Slide 99 / 127 Problem Solving with Kirchhoff's Rules Slide 100 / 127 Problem Solving with Kirchhoff's Rules List the givens and use ohm's law to solve for the unknowns. Find the unknowns in the following circuit: R3 = 2 # Current (Amps) oltage (olts) Resistance (Ohms) R1 5 R2 2.6 3 R3 1.87 R4 2 Total 12 R1 = 10?# I1 = 7 A 1 = 70? R2 = 23?# I2 = 3 A 2 = 70? I3 = 10 A 3 = 20? R4 = 3 # I4 = 10? A 4 = 30? I = I 3 I 4 I = I 2 I 1 = I 3 = 3 1 2 = 4 2 1 3 = 4 = 120 Slide 101 / 127 Capacitors Slide 102 / 127 Capacitance When a battery is connected to a capacitor, charge moves between them. Every electron that moves to the negative plate leaves a positive nucleus behind. As the plates charge, the potential difference between the places increases. The current through the circuit decreases until the capacitor becomes fully charged. C - - Return to Table of Contents The plates have equal magnitudes of charges, but one is positive, the other negative. L What does the lamp look like while the capacitor is charging? Slide 102 () / 127 Slide 103 / 127 Capacitance Capacitance When a battery is connected to a capacitor, charge moves between them. Every electron that moves to the negative plate leaves a positive nucleus behind. As the While plates the capacitor charge, the is charging, potential its difference resistance between is increasing the places so the increases. current in the circuit is decreasing. The Therefore current through the lamp the gets circuit dimmer decreases until until the the capacitor current becomes stops and fully the charged. lamp goes out The plates have equal magnitudes of charges, but one is positive, the other negative. C - - L If the battery is now disconnected and current is allowed to flow, the capacitor discharges. The current decreases until the capacitor is fully discharged. The potential difference decreases until the capacitor is discharged. What happens to the lamp? L C - What does the lamp look like while the capacitor is charging?
Slide 103 () / 127 Capacitance If the battery is now disconnected and current is allowed to flow, the capacitor discharges. Slide 104 / 127 38 In which of the following circuits will the capacitor store charge if the battery is disconnected? Select two answers. A B The current decreases until the capacitor is fully discharged. As the capacitor decreases, the potential difference also decreases The potential difference causing decreases the lamp until to start out bright the capacitor is discharged. but get dimmer until the capacitor is fully discharged and the lamp goes out. What happens to the lamp? C - C D L Slide 104 () / 127 Slide 105 / 127 38 In which of the following circuits will the capacitor store charge if the battery is disconnected? Select two answers. Equivalent Capacitance A B Circuits can be redrawn as if there were only a single capacitor and battery. By reducing the circuit this way, the circuit becomes easier to study. C D B, C The process of reducing the resistors in a circuit is called finding the equivalent Capacitance for capacitors in series (C S) and parallel (C P). Slide 106 / 127 Slide 107 / 127 Parallel Circuits: Equivalent Capacitance Parallel Circuits: Equivalent Capacitance What is the voltage across each capacitor? C2 The voltage across each capacitor is the same. C2 C1 = 1 = 2 C1 What is the charge on each capacitor? The total charge is the sum of the charge on all the capacitors. Q = Q 1 Q 2
Slide 108 / 127 Parallel Circuits: Equivalent Capacitance Slide 109 / 127 Series Circuits: Equivalent Capacitance Since = 1 = 2, Q = Q 1 Q 2 and Q = C C = C 1 1 C 2 2 Replace 1 and 2 with. C2 C1 The sum of the voltage drops across each of the resistors in a series circuit equals the voltage of the battery. C1 C2 C = C 1 C 2 Divide both sides by. = 1 2 The charge on each capacitor is the same. C = C 1 C 2 Q = Q 1 = Q 2 So, for capacitors in parallel C P = ΣC i Slide 110 / 127 Slide 111 / 127 Series Circuits: Equivalent Capacitance Equivalent Capacitance Since = 1 2, Q = Q 1 = Q 2 and = Q/C C1 C2 Finding equivalent capacitance is like finding equivalent resistance, only reversed. Capacitors in series are added like resistors in parallel. Q/C = Q 1/C 1 Q 2/C 2 Replace Q 1 and Q 2 with Q. Q/C = Q/C 1 Q/C 2 Divide both sides by Q. Capacitors in parallel are added like resistors in series. 1/C = 1/C 1 1/C 2 So, for capacitors in series 1/C S = Σ1/C i Slide 112 / 127 Slide 112 () / 127 39 What is the equivalent capacitance (in mf) if C 1 is 4mF and C 2 is 6mF? C2 C1
Slide 113 / 127 Slide 113 () / 127 40 What is the equivalent capacitance (in mf) if C 1 is 4mF and C 2 is 6mF? 40 What is the equivalent capacitance (in mf) if C 1 is 4mF and C 2 is 6mF? C1 C2 C1 C2 Slide 114 / 127 Slide 114 () / 127 41 What is the equivalent capacitance (in μf) if C 1 is 5μF and C 2 is 11μF? C2 C1 Slide 115 / 127 Slide 115 () / 127 42 What is the equivalent capacitance (in nf) if C 1 is 9nF and C 2 is 3nF? 42 What is the equivalent capacitance (in nf) if C 1 is 9nF and C 2 is 3nF? C1 C2 C1 C2
Slide 116 / 127 Slide 117 / 127 RC Circuits RC Circuits A resistor-capacitor circuit is a circuit composed of resistors and capacitors that are connected to voltage source. Return to Table of Contents Slide 118 / 127 RC Circuits When the switch is open, there is no current in the top branch with the capacitor and this circuit is simply two light bulbs in series. The current through the battery current would be /2R. Slide 119 / 127 RC Circuits Immediately after the switch is closed, there is no resistance in the branch with the capacitor so all of the current goes through that branch and bypasses the second light bulb. Therefore the current through the battery increases and is /R. Slide 120 / 127 RC Circuits A long time after the switch is closed, the capacitor is fully charged and there is no more current in the top branch. The rest of the circuit behaves as if the capacitor was disconnected. Therefore the current through the battery is back to /2R. Slide 121 / 127 43 Four identical light bulbs are connected in a circuit with a capacitor as shown. The switch is open. Rank the voltage from greatest to least. A 3 > 1 > 2 > 4 B 1 = 2 = 3 > 4 1 2 3 C 4 > 3 > 1 = 2 D 4 = 3 > 1 = 2 4
Slide 121 () / 127 Slide 122 / 127 43 Four identical light bulbs are connected in a circuit with a capacitor as shown. The switch is open. Rank the voltage from greatest to least. 44 Four identical light bulbs are connected in a circuit with a capacitor as shown. Immediately after the switch is closed, rank current from greatest to least. A 3 > 1 > 2 > 4 1 2 A I 4 = I 3 > I 1 > I 2 1 2 B 1 = 2 = 3 > 4 3 B I 1 = I 2 = I 4 > I 3 3 C 4 > 3 > 1 = 2 D 4 = 3 > 1 = 2 4 C C I 4 > I 3 > I 1 = I 2 D I 4 > I 3 = I 1 = I 2 4 Slide 122 () / 127 Slide 123 / 127 44 Four identical light bulbs are connected in a circuit with a capacitor as shown. Immediately after the switch is closed, rank current from greatest to least. 45 Four identical light bulbs are connected in a circuit with a capacitor as shown. The switch has been closed for a long time. Rank the light bulbs in order of brightness. A I 4 = I 3 > I 1 > I 2 1 2 A 4 > 3 > 1 = 2 1 2 B I 1 = I 2 = I 4 > I 3 C I 4 > I 3 > I 1 = I 2 D I 4 > I 3 = I 1 = I 2 4 3 D B 1 = 2 = 3 = 4 C 1 = 2 > 3 > 4 D 3 = 4 > 1 = 2 4 3 Slide 123 () / 127 Slide 124 / 127 45 Four identical light bulbs are connected in a circuit with a capacitor as shown. The switch has been closed for a long time. Rank the light bulbs in order of brightness. Energy Stored in RC Circuits Remember from the previous chapter that capacitors can store potential energy. A 4 > 3 > 1 = 2 B 1 = 2 = 3 = 4 C 1 = 2 > 3 > 4 D 3 = 4 > 1 = 2 4 1 2 A 3 The energy stored in a capacitor is given by: Using, we can also substitute for Q and and get the following: and
Slide 125 / 127 Slide 125 () / 127 46 A 3 mf capacitor is connected to a resistor with a resistance of 2Ω and battery with a potential difference of 6. How much energy, in Joules, is stored in the capacitor? 46 A 3 mf capacitor is connected to a resistor with a resistance of 2Ω and battery with a potential difference of 6. How much energy, in Joules, is stored in the capacitor? 2Ω 3mF 2Ω 3mF 6 6 Slide 126 / 127 Slide 126 () / 127 47 Now another resistor with a resistance of 1 Ω and a switch are connected to the capacitor in parallel. How much energy, in Joules, is stored in the capacitor? 47 Now another resistor with a resistance of 1 Ω and a switch are connected to the capacitor in parallel. How much energy, in Joules, is stored in the capacitor? 1Ω 1Ω 2Ω 3mF 2Ω Since the switch is open it is the same 3mFas it was without the resistor. 6 6 Slide 127 / 127 Slide 127 () / 127 48 Now the switch has been closed for a long time. How much energy, in Joules, is stored in the capacitor? 48 Now the switch has been closed for a long time. How much energy, in Joules, is stored in the capacitor? 1Ω 1Ω 2Ω 3mF 2Ω 3mF 6 6