My Chapter 18 Lecture Outline

1. My Chapter 18 Lecture Outline

2. Chapter 18: Electric Current and Circuits • Electric current • EMF • Current & Drift Velocity • Resistance & Resistivity • Kirchhoff’s Rules • Series & Parallel Circuit Elements • Applications of Kichhoff’s Rules • Power & Energy • Ammeters & Voltmeters • RC Circuits

3. e- e- e- e- e- e- e- e- §18.1 Electric Current A metal wire. Assume electrons flow to the right. Current is a measure of the amount of charge that passes though an area perpendicular to the flow of charge. Current units: 1C/sec = 1 amp

4. A current will flow until there is no potential difference. The direction of current flow in a wire is opposite the flow of the electrons. (In the previous drawing the current is to the left.)

5. Example: If a current of 80.0 mA exists in a metal wire, how many electrons flow past a given cross-section of the wire in 10.0 minutes?

6. §18.2 EMF and Circuits An ideal battery maintains a constant potential difference. This potential difference is called the battery’s EMF(). The work done by an ideal battery in pumping a charge q is W = q.

7. At high potential +  At low potential The circuit symbol for a battery (EMF source) is Batteries do work by converting chemical energy into electrical energy. A battery dies when it can no longer sustain its chemical reactions and so can do no more work to move charges.

8. §18.3 Microscopic View of Current in a Metal Electrons in a metal might have a speed of ~106 m/s, but since the direction of travel is random, an electron has vdrift = 0.

9. Only when the ends of a wire are at different potentials (E  0) will there be a net flow of electrons along the wire (vdrift  0). Typically, vdrift < 1 mm/sec.

10. Calculate the number of charges (Ne) that pass through the shaded region in a time t: l The current in the wire is:

11. Example (text problem 18.19): A copper wire of cross-sectional area 1.00 mm2 has a constant current of 2.0 A flowing along its length. What is the drift speed of the conduction electrons? Copper has 1.101029 electrons/m3.

12. §18.4 Resistance and Resistivity A material is considered ohmic if VI, where The proportionality constant R is called resistance and is measured in ohms (; and 1  = 1 V/A).

13. The resistance of a conductor is: where  is the resistivity of the material, L is the length of the conductor, andA is its cross sectional area. With R a material is considered a conductor if  is “small” and an insulator if  is “large”.

14. The resistivity of a material depends on its temperature: where 0 is the resistivity at the temperature T0, and  is the temperature coefficient of resistivity. A material is called a superconductor if  = 0.

15. Example (text problem 18.28): The resistance of a conductor is 19.8  at 15.0 C and 25.0  at 85.0 C. What is the temperature coefficient of resistivity? Values of R are given at different temperatures, not values of . But the two quantities are related. Multiply both sides of equation (2) by L/A and use equation (1) to get:

16. Example continued: Solve equation (3) for  and evaluate using the given quantities:

17. §18.5 Kirchhoff’s Rules Junction rule: The current that flows into a junction is the same as the current that flows out. (Charge is conserved) A junction is a place where two or more wires (or other components) meet. Loop rule: The sum of the voltage dropped around a closed loop is zero. (Energy is conserved.)

18. A B For a resistor: If you cross a resistor in the direction of the current flow, the voltage drops by an amount IR (write as IR). There is a voltage rise if you cross the other way (write as +IR). If the current flows from A to B, then the potential decreases from A to B. The potential difference between A and B is < 0 (V = IR). I

19. At high potential +  At low potential  For batteries (or other sources of EMF): If you move from the positive to the negative terminal the potential drops by  (write as ). The potential rises if you cross in the other direction (write as +).

20. A current will only flow around a closed loop. B A VAB is the terminal voltage. Applying the loop rule:

21. In a circuit, if the current always flows in the same direction it is called a direct current (DC) circuit.

22. §18.6 Series and Parallel Circuits Resistors: The current through the two resistors is the same. It is not “used up” as it flows around the circuit! These resistors are in series. Apply Kirchhoff’s loop rule:

23. The pair of resistors R1 and R2 can be replaced with a single equivalent resistor provided that Req = R1 + R2. In general, for resistors in series

24. Current only flows around closed loops. When the current reaches point A it splits into two currents. R1 and R2 do not have the same current through them, they are in parallel. Apply Kirchhoff’s loop rule: The potential drop across each resistor is the same.

25. Applying the junction rule at A: I = I1+ I2. From the loop rules: Substituting for I1 and I2 in the junction rule:

26. The pair of resistors R1 and R2 can be replaced with a single equivalent resistor provided that In general, for resistors in parallel

27. R1 = 15 A R2 = 12  R3 = 24  B Example (text problem 18.40): In the given circuit, what is the total resistance between points A and B? R2 and R3 are in parallel. Replace with an equivalent resistor R23.

28. R1 = 15 A R23 = 8  A R123 = 23  B B Example continued: The circuit can now be redrawn: The resistors R23 and R1 are in series: Is the equivalent circuit and the total resistance is 23 .

29. C1 C2  Capacitors: For capacitors in series the charge on the plates is the same. Apply Kirchhoff’s loop rule:

30. The pair of capacitors C1 and C2 can be replaced with a single equivalent capacitor provided that In general, for capacitors in series

31. C2 C1  Apply Kirchhoff’s loop rule: For capacitors in parallel the charge on the plates may be different. Here

32. The pair of capacitors C1 and C2 can be replaced with a single equivalent capacitor provided that Ceq= C1 + C2. In general, for capacitors in parallel

33. C1 A C3 C2 B Example (text problem 18.49): Find the value of a single capacitor that replaces the three in the circuit below if C1 = C2 = C3 = 12 F. C2 and C3 are in parallel

34. C1 A C23 B A C123 B Example continued: The circuit can be redrawn: The remaining two capacitors are in series. Is the final, equivalent circuit.

35. §18.8 Power and Energy in Circuits The energy dissipation rate is: For an EMF source: For a resistor:

36. Example: Use the results of the example starting on slide 35 to determine the power dissipated by the three resistors in that circuit.

37. R1 R2 A1 A3 A2  §18.9 Measuring Currents and Voltages Current is measured with an ammeter. An ammeter is placed in series with a circuit component. A1 measures the current through R1. An ammeter has a low internal resistance. A2 measures the current through R2. A3 measures the current drawn from the EMF.

38. V R1 R2  A voltmeter is used to measure the potential drop across a circuit element. It is placed in parallel with the component. A voltmeter has a large internal resistance. The voltmeter measures the voltage drop across R1.

39. Switch R  C + +   §18.10 RC Circuits Close the switch at t = 0 to start the flow of current. The capacitor is being charged. Apply Kirchhoff’s loop rule:

40. The current I(t) that satisfies Kirchhoff’s loop rule is: where  is the RC time constant and is a measure of the charge (and discharge) rate of a capacitor.

41. The voltage drop across the capacitor is: The voltage drop across the resistor is: The charge on the capacitor is: Note: Kirchhoff’s loop rule must be satisfied for all times.

42. Plots of the voltage drop across the (charging) capacitor and current in the circuit.

43. I S1 R S2 C +   While the capacitor is charging S2 is open. After the capacitor is fully charged S1 is opened at the same time S2 is closed: this removes the battery from the circuit. Current will now flow in the right hand loop only, discharging the capacitor. Apply Kirchhoff’s loop rule: The current in the circuit is But the voltage drop across the capacitor is now

44. The voltage drop across the discharging capacitor:

45. Example (text problem 18.85): A capacitor is charged to an initial voltage of V0 = 9.0 volts. The capacitor is then discharged through a resistor. The current is measured and is shown in the figure.

46. Example continued: (a) Find C, R, and the total energy dissipated in the resistor. Use the graph to determine . I0 = 100 mA; the current is I0/e = 36.8 mA at t = 13 msec. Since  = V0 = 9.0 volts, R = 90  and C = 144 F. All of the energy stored in the capacitor is eventually dissipated by the resistor.

47. Example continued: (b) At what time is the energy in the capacitor half of the initial value? Want:

48. Example continued: Solve for t:

49. Summary • Current & Drift Velocity • Resistance & Resistivity • Ohm’s Law • Kirchhoff’s Rules • Series/Parallel Resistors/Capacitors • Power • Voltmeters & Ammeters • RC Circuits