2/13/07184 Lecture 202AnnouncementsAnnouncements We hope to open the correction set tonight You will have a week to complete the problems You can re-do all the problems from the exam You will receive 30% credit for the problems you missed To get credit, you must do all the problems in Corrections Set 1, not just the ones you missed Homework Set 5 is due on Tuesday, February 20 at 8 am
2/13/07184 Lecture 203 Kirchhoff’s Law, Multi-loop Circuits One can create multi-loop circuits that cannot be resolved into simple circuits containing parallel or series resistors. To handle these types of circuits, we must apply Kirchhoff’s Rules. Kirchhoff’s Rules can be stated as Kirchhoff’s Junction Rule The sum of the currents entering a junction must equal the sum of the currents leaving a junction Kirchhoff’s Loop Rule The sum of voltage drops around a complete circuit loop must sum to zero.
2/13/07184 Lecture 204 Circuit Analysis Conventions i is the magnitude of the assumed current
2/13/07184 Lecture 205 Multi-Loop Circuits To analyze multi-loop circuits, we must apply both the Loop Rule and the Junction Rule. To analyze a multi-loop circuit, identify complete loops and junction points in the circuit and apply Kirchhoff’s Rules to these parts of the circuit separately. At each junction in a multi-loop circuit, the current flowing into the junction must equal the current flowing out of the circuit.
2/13/07184 Lecture 206 Multi-Loop Circuits (2) Assume we have a junction point a We define a current i 1 entering junction a and two currents i 2 and i 3 leaving junction a Kirchhoff’s Junction Rule tells us that
2/13/07184 Lecture 207 Multi-Loop Circuits (3) By analyzing the single loops in a multi-loop circuit with Kirchhoff’s Loop Rule and the junctions with Kirchhoff’s Junction Rule, we can obtain a system of coupled equations in several unknown variables. These coupled equations can be solved in several ways Solution with matrices and determinants Direct substitution Next: Example of a multi-loop circuit solved with Kirchhoff’s Rules
2/13/07184 Lecture 208 The circuit here has three resistors, R 1, R 2, and R 3 and two sources of emf, V emf,1 and V emf,2 This circuit cannot be resolved into simple series or parallel structures To analyze this circuit, we need to assign currents flowing through the resistors. We can choose the directions of these currents arbitrarily. Example - Kirchhoff’s Rules
2/13/07184 Lecture 209 At junction b the incoming current must equal the outgoing current At junction a we again equate the incoming current and the outgoing current But this equation gives us the same information as the previous equation! We need more information to determine the three currents – 2 more independent equations Example - Kirchhoff’s Laws (2)
2/13/07184 Lecture 2010 To get the other equations we must apply Kirchhoff’s Loop Rule. This circuit has three loops. Left R 1, R 2, V emf,1 Right R 2, R 3, V emf,2 Outer R 1, R 3, V emf,1, V emf,2 Example - Kirchhoff’s Laws (3)
2/13/07184 Lecture 2011 Going around the left loop counterclockwise starting at point b we get Going around the right loop clockwise starting at point b we get Going around the outer loop clockwise starting at point b we get But this equation gives us no new information! Example - Kirchhoff’s Laws (4)
2/13/07184 Lecture 2012 We now have three equations And we have three unknowns i 1, i 2, and i 3 We can solve these three equations in a variety of ways Example - Kirchhoff’s Laws (5)
2/13/07184 Lecture 2013 Given is the multi-loop circuit on the right. Which of the following statements cannot be true: A) B) C) D) Clicker Question
2/13/07184 Lecture 2014 Given is the multi-loop circuit on the right. Which of the following statements cannot be true: A) B) C) D) Clicker Question Junction rule Not a loop! Upper right loop Left loop
2/13/07184 Lecture 2015 Ammeter and Voltmeters A device used to measure current is called an ammeter A device used to measure voltage is called a voltmeter To measure the current, the ammeter must be placed in the circuit in series To measure the voltage, the voltmeter must be wired in parallel with the component across which the voltage is to be measured Voltmeter in parallel High resistance Ammeter in series Low resistance
2/13/07184 Lecture 2016 RC Circuits So far we have dealt with circuits containing sources of emf and resistors. The currents in these circuits did not vary in time. Now we will study circuits that contain capacitors as well as sources of emf and resistors. These circuits have currents that vary with time. Consider a circuit with a source of emf, V emf, a resistor R, a capacitor C
2/13/07184 Lecture 2017 RC Circuits (2) We then close the switch, and current begins to flow in the circuit, charging the capacitor. The current is provided by the source of emf, which maintains a constant voltage. When the capacitor is fully charged, no more current flows in the circuit. When the capacitor is fully charged, the voltage across the plates will be equal to the voltage provided by the source of emf and the total charge q tot on the capacitor will be q tot = CV emf.
2/13/07184 Lecture 2018 Capacitor Charging Going around the circuit in a counterclockwise direction we can write We can rewrite this equation remembering that i = dq/dt The solution of this differential equation is … where q 0 = CV emf and = RC The term V c is negative since the top plate of the capacitor is connected to the positive - higher potential - terminal of the battery. Thus analyzing counter-clockwise leads to a drop in voltage across the capacitor!
2/13/07184 Lecture 2019 Capacitor Charging (2) We can get the current flowing in the circuit by differentiating the charge with respect to time The charge and current as a function of time are shown here ( = RC) Math Reminder:
2/13/07184 Lecture 2020 Now let’s take a resistor R and a fully charged capacitor C with charge q 0 and connect them together by moving the switch from position 1 to position 2 In this case current will flow in the circuit until the capacitor is completely discharged. While the capacitor is discharging we can apply the Loop Rule around the circuit and obtain Capacitor Discharging
2/13/07184 Lecture 2021 Capacitor Discharging (2) The solution of this differential equation for the charge is Differentiating charge we get the current The equations describing the time dependence of the charging and discharging of capacitors all involve the exponential factor e -t/RC The product of the resistance times the capacitance is defined as the time constant of a RC circuit. We can characterize an RC circuit by specifying the time constant of the circuit.
2/13/07184 Lecture 2022 Example: Time to Charge a Capacitor Consider a circuit consisting of a 12.0 V battery, a 50.0 resistor, and a 100.0 F capacitor wired in series. The capacitor is initially uncharged. Question: How long will it take to charge the capacitor in this circuit to 90% of its maximum charge? Answer: The charge on the capacitor as a function of time is
2/13/07184 Lecture 2023 Example: Time to Charge a Capacitor (2) We need to know the time corresponding to We can rearrange the equation for the charge on the capacitor as a function of time to get Math Reminder: ln(e x )=x
2/13/07184 Lecture 2024 Example: More RC Circuits A 15.0 k resistor and a capacitor are connected in series and a 12V battery is suddenly applied. The potential difference across the capacitor rises to 5V in 1.3 s. What is the time constant of the circuit? Answer: 2.41 s
2/13/07184 Lecture 2025 Clicker Question A 15.0 k resistor and a capacitor are connected in series and a 12V battery is suddenly applied. The potential difference across the capacitor rises to 5V in 1.3 s. What is the capacitance C of the capacitor? A) 161 pF B) 6.5 pF C) 0 D) 49 pF