Lecture 6

1. Lecture 6 • Capacitance • Electric Current • Circuits • Resistance and Ohms law

2. Capacitors in Series • When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery • As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge • All of the right plates gain charges of –Q and all the left plates have charges of +Q

3. More About Capacitors in Series • An equivalent capacitor can be found that performs the same function as the series combination • The potential differences add up to the battery voltage

4. Fig. 16-19, p.551

5. Fig. 16-20, p.552

6. Capacitors in Series, cont • The equivalent capacitance of a series combination is always less than any individual capacitor in the combination • Demo

7. Fig. P16-34, p.564

8. Fig. P16-35, p.564

9. Problem-Solving Strategy • Be careful with the choice of units • Combine capacitors following the formulas • When two or more unequal capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same • The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor

10. Problem-Solving Strategy, cont • Combining capacitors • When two or more capacitors are connected in parallel, the potential differences across them are the same • The charge on each capacitor is proportional to its capacitance • The capacitors add directly to give the equivalent capacitance

11. Problem-Solving Strategy, final • Repeat the process until there is only one single equivalent capacitor • A complicated circuit can often be reduced to one equivalent capacitor • Replace capacitors in series or parallel with their equivalent • Redraw the circuit and continue • To find the charge on, or the potential difference across, one of the capacitors, start with your final equivalent capacitor and work back through the circuit reductions

12. Problem-Solving Strategy, Equation Summary • Use the following equations when working through the circuit diagrams: • Capacitance equation: C = Q / DV • Capacitors in parallel: Ceq = C1 + C2 + … • Capacitors in parallel all have the same voltage differences as does the equivalent capacitance • Capacitors in series: 1/Ceq = 1/C1 + 1/C2 + … • Capacitors in series all have the same charge, Q, as does their equivalent capacitance

13. Fig. 16-21, p.553

14. Fig. P16-57, p.566

15. Energy Stored in a Capacitor • Energy stored = ½ Q ΔV • From the definition of capacitance, this can be rewritten in different forms

16. Fig. 16-22, p.554

17. Applications • Defibrillators • When fibrillation occurs, the heart produces a rapid, irregular pattern of beats • A fast discharge of electrical energy through the heart can return the organ to its normal beat pattern • In general, capacitors act as energy reservoirs that can slowly charged and then discharged quickly to provide large amounts of energy in a short pulse

18. Capacitors with Dielectrics • A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance • Dielectrics include rubber, plastic, or waxed paper • C = κCo = κεo(A/d) • The capacitance is multiplied by the factor κ when the dielectric completely fills the region between the plates

19. Capacitors with Dielectrics

20. Dielectric Strength • For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct • This maximum electric field is called the dielectric strength

21. An Atomic Description of Dielectrics • Polarization occurs when there is a separation between the “centers of gravity” of its negative charge and its positive charge • In a capacitor, the dielectric becomes polarized because it is in an electric field that exists between the plates

22. More Atomic Description • The presence of the positive charge on the dielectric effectively reduces some of the negative charge on the metal • This allows more negative charge on the plates for a given applied voltage • The capacitance increases

23. Fig. 16-30, p.560

24. Table 16-1, p.557

25. Fig. 16-1, p.532

26. Fig. 16-23, p.557

27. Fig. 16-26, p.558

28. Fig. 16-28, p.560

29. Fig. 16-29a, p.560

30. Fig. 16-29b, p.560

32. Electric Current • Whenever electric charges of like signs move, an electric current is said to exist • The current is the rate at which the charge flows through this surface • Look at the charges flowing perpendicularly to a surface of area A • The SI unit of current is Ampere (A) • 1 A = 1 C/s

33. Electric Current, cont • The direction of the current is the direction positive charge would flow • This is known as conventional current direction • In a common conductor, such as copper, the current is due to the motion of the negatively charged electrons • It is common to refer to a moving charge as a mobile charge carrier • A charge carrier can be positive or negative

34. Current and Drift Speed • Charged particles move through a conductor of cross-sectional area A • n is the number of charge carriers per unit volume • n A Δx is the total number of charge carriers

35. Current and Drift Speed, cont • The total charge is the number of carriers times the charge per carrier, q • ΔQ = (n A Δx) q • The drift speed, vd, is the speed at which the carriers move • vd = Δx/ Δt • Rewritten: ΔQ = (n A vd Δt) q • Finally, current, I = ΔQ/Δt = nqvdA

36. Current and Drift Speed, final • If the conductor is isolated, the electrons undergo random motion • When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current

37. Charge Carrier Motion in a Conductor • The zig-zag black line represents the motion of charge carrier in a conductor • The net drift speed is small • The sharp changes in direction are due to collisions • The net motion of electrons is opposite the direction of the electric field Demo

38. Electrons in a Circuit • The drift speed is much smaller than the average speed between collisions • When a circuit is completed, the electric field travels with a speed close to the speed of light • Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s • c = 3 x 108 m/s

39. Meters in a Circuit – Ammeter • An ammeter is used to measure current • In line with the bulb, all the charge passing through the bulb also must pass through the meter

40. p.578

41. Fig. A17-1, p.591

42. Meters in a Circuit – Voltmeter • A voltmeter is used to measure voltage (potential difference) • Connects to the two ends of the bulb

43. Resistance • In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor • The constant of proportionality is the resistance of the conductor

44. Fig. 17-CO, p.568

45. Resistance, cont • Units of resistance are ohms (Ω) • 1 Ω = 1 V / A • Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor

46. Georg Simon Ohm • 1787 – 1854 • Formulated the concept of resistance • Discovered the proportionality between current and voltages

47. Ohm’s Law • Experiments show that for many materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents • This statement has become known as Ohm’s Law • ΔV = I R • Ohm’s Law is an empirical relationship that is valid only for certain materials • Materials that obey Ohm’s Law are said to be ohmic