PHY 184

1. PHY 184 Spring 2007 Lecture 20 Title: 184 Lecture 20

2. Announcements • 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 184 Lecture 20

3. 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. 184 Lecture 20

4. Circuit Analysis Conventions i is the magnitude of the assumed current 184 Lecture 20

5. 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. 184 Lecture 20

6. Multi-Loop Circuits (2) • Assume we have a junction point a • We define a current i1 entering junction a and two currents i2 and i3 leaving junction a • Kirchhoff’s Junction Rule tells us that 184 Lecture 20

7. 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 184 Lecture 20

8. Example - Kirchhoff’s Rules • The circuit here has three resistors, R1, R2, and R3 and two sources of emf, Vemf,1 and Vemf,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. 184 Lecture 20

9. Example - Kirchhoff’s Laws (2) • 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 thesame information as theprevious equation! • We need more informationto determine the three currents – 2 more independent equations 184 Lecture 20

10. Example - Kirchhoff’s Laws (3) • To get the other equations we must apply Kirchhoff’s Loop Rule. • This circuit has three loops. • Left • R1, R2, Vemf,1 • Right • R2, R3, Vemf,2 • Outer • R1, R3, Vemf,1, Vemf,2 184 Lecture 20

11. Example - Kirchhoff’s Laws (4) • 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 loopclockwise startingat point b we get • But this equation gives us no newinformation! 184 Lecture 20

12. Example - Kirchhoff’s Laws (5) • We now have three equations • And we have three unknowns i1, i2, and i3 • We can solve these three equations in a variety of ways 184 Lecture 20

13. Clicker Question • Given is the multi-loop circuit on the right. Which of the following statements cannot be true: • A) • B) • C) • D) 184 Lecture 20

14. Clicker Question • Given is the multi-loop circuit on the right. Which of the following statements cannot be true: • A) • B) • C) • D) Junction rule Not a loop! Upper right loop Left loop 184 Lecture 20

15. 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 184 Lecture 20

16. 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, Vemf, • a resistor R, • a capacitor C 184 Lecture 20

17. RC Circuits (2) • We then close the switch, and current begins to flow in the circuit, charging the capacitor. • The current is provided by thesource of emf, which maintainsa 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 qtot on the capacitor will be qtot = CVemf. 184 Lecture 20

18. Capacitor Charging • Going around the circuit in a counterclockwise direction we can write • We can rewrite this equationremembering that i = dq/dt • The solution of this differential equation is • … where q0 = CVemf and  = RC The term Vc 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! 184 Lecture 20

19. Capacitor Charging (2) Math Reminder: • 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) 184 Lecture 20

20. Capacitor Discharging • Now let’s take a resistor R and a fully charged capacitor C with charge q0 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 184 Lecture 20

21. 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. 184 Lecture 20

22. 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 184 Lecture 20

23. 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(ex)=x 184 Lecture 20

24. 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 ms 184 Lecture 20

25. 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 184 Lecture 20