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Resistance

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Resistance

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  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. RC Circuits • 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. Consider charging: • 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. Which can be written as: • 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. Separating variables, • Integrating,

  19. Separating variables, • Integrating,

  20. Separating variables, • Integrating, • Which gives,

  21. Taking the antilog and simplifying we get,

  22. q(t) VbatC t • Taking the antilog and simplifying we get,

  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. Vc Vbat t • The pd across the capacitor • Which gives

  25. The current for the charging • Which gives I(t) Vbat/R t

  26. Consider discharging:

  27. Consider discharging: • For the discharge position, the battery is no longer in the circuit.

  28. Since • We can write that

  29. Since • We can write that • Separating variables,

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

  31. Integrating,

  32. Integrating, • We get • Which after simplification is,

  33. This can be written as, , noting that the initial charge is CVbat.

  34. This can be written as, , noting that the initial charge is CVbat. • Differentiating gives the current, • The voltage across the capacitor is,

  35. Limiting conditions: • At t=0, q= CVbat. • At t=inf, q= 0. q CVbat t

  36. t I(t) Vbat t

  37. Power, Energy

  38. 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,

  39. 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,

  40. Energy • From the conservation of energy,

  41. Resistance in Series and Parallel

  42. Series:

  43. From the conservation of energy,

  44. From the conservation of energy, • where,

  45. From the conservation of energy, • where,

  46. From the conservation of energy, • where,

  47. In general,

  48. Parallel:

  49. From the conservation of charge,