1. Resistance

2. Review of Resistors The resistance is an intrinsic property of a material which impedes the flow of charge requiring a pd to be applied so that there can be current flow.

3. Review of Resistors The resistance is an intrinsic property of a material which impedes the flow of charge requiring a pd to be applied so that there can be current flow. From ohm’s law, the resistance of a device is the ratio of the potential difference across it to the current flowing through it.

4. The unit of the resistor is the ohm ( ).

5. RC Circuits

6. The current in the previous circuits are time independent once the emf of the source is time independent.

7. RC Circuits The current in the previous circuits are time independent once the emf of the source is time independent. However we may have circuits which are time dependent. An example is an RC circuit.

8. A RC circuit consists of a resistor R connected in series with a capacitor C.

9. The following circuit can be use the test the charging and discharging of the capacitor through the resistor.

10. Consider charging:

11. Initially the capacitor is uncharged.

12. Consider charging: Initially the capacitor is uncharged. When in the charging position current flows and the capacitor charges. From Kirchoff’s law:

13. Which can be written as:

14. Since We can rewrite the equation as,

15. Which can be written as: Since We can rewrite the equation as, Doing some algebra,

16. Which can be written as: Since We can rewrite the equation as, Doing some algebra, We must separate the variables so that we can integrate and find the final charge on the capacitor.

17. Separating variables,

18. Integrating,

19. Separating variables, Integrating,

20. Separating variables, Integrating, Which gives,

21. Taking the antilog and simplifying we get,

22. V bat Cq(t) t

23. The product RC in the previous equation is called the time constant. Has units of time. Time taken for the charge to increase from zero to 63% of its final value.

24. The pd across the capacitor Which gives V bat VcVc t

25. The current for the charging Which gives V bat /RI(t) t

26. Consider discharging:

27. For the discharge position, the battery is no longer in the circuit.

29. Since We can write that

30. Since We can write that Separating variables,

31. Since We can write that Separating variables, Which in separated form is,

32. Integrating,

33. We get Which after simplification is,

34. This can be written as,, noting that the initial charge is CV bat.

35. Differentiating gives the current, The voltage across the capacitor is,

36. Limiting conditions: 1.At t=0, q= CV bat. 2.At t=inf, q= 0. CV bat q t

37. V bat I(t) t t

38. Power, Energy

39. Power The net rate of energy transfer from the source (battery) P is given by, Power is in watts(W) or joules/second The rate at which energy is dissipated through through the resistor is, The energy lost is in the form of thermal energy. The power supplied to the capacitor is,

40. Energy The total energy supplied by the battery in a time t is given by, The total energy dissipated in a time t, The total energy supplied to the capacitor in time t,

41. Energy From the conservation of energy,

42. Resistance in Series and Parallel

43. Series:

44. From the conservation of energy,

45. where,

46. From the conservation of energy, where,

47. From the conservation of energy, where,

48. In general,

49. Parallel:

50. From the conservation of charge,

51. where,

52. From the conservation of charge, where,

53. From the conservation of charge, where,

54. From the conservation of charge, where,

55. In general,