Current and Resistance

1. Current and Resistance February 18, 2008

2. February 18, 2008 • Headlines: • Today we will finish capacitors and begin a study of current, resistors and DC circuits. • Watch for new WebAssign • Quiz on Friday • Regular Problem sessions on Wednesday and Thursday

3. ++++++++++++ q V0 ------------------ -q A Closer Look at this stuff.. Consider this virgin capacitor. No dielectric experience. Applied Voltage via a battery. C0

4. ++++++++++++ q V0 ------------------ -q Remove the Battery The Voltage across the capacitor remains V0 q remains the same as well. The capacitor is fat (charged), dumb and happy.

5. ++++++++++++ q - - - - - - - - -q’ +q’ V0 + + + + + + ------------------ -q Slip in a DielectricAlmost, but not quite, filling the space Gaussian Surface E E’ from induced charges E0

6. A little sheet from the past.. -q’ +q’ - - - +++ -q q 0 2xEsheet 0

7. Some more sheet…

8. + - Vo Another look

9. Original Structure Disconnect Battery Slip in Dielectric Vo + - + - + - Add Dielectric to Capacitor V0 Note: Charge on plate does not change!

10. so so si + si - + - What happens? Potential Difference is REDUCED by insertion of dielectric. Charge on plate is Unchanged! Capacitance increases by a factor of k as we showed previously

11. SUMMARY OF RESULTS

12. New Topic:Current and Resistance

13. Physical Resistors

14. - - What Happens? “+” “+” REMEMBER, THE ELECTRONS ARE ACTUALLY MOVING THEOTHER WAY! “+” “+”

15. What’s Moving? Charge!

16. What is making the charged move?? Battery

17. KEEP IN MIND • A wire is a conductor • We will assume that the conductor is essentially an equi-potential • It really isn’t. • Electrons are moving in a conductor if a current is flowing. • This means that there must be an electric field in the conductor. • This implies a difference in potential since E=DV/d • We assume that the difference in potential is small and that it can often be neglected. • In this chapter, we will consider this difference and what causes it.

18. Conducting material DQ,Dt DEFINITION • Current is the motion of POSITIVE CHARGE through a circuit. Physically, it is electrons that move but …

19. Conducting material DQ,Dt

20. UNITS • A current of one coulomb per second is defined as ONE AMPERE.

21. A charged belt, 30 cm wide, travels at 40 m/s between a source of charge and a sphere. The belt carries charge into the sphere at a rate corresponding to 100 µA. Compute the surface charge density on the belt. [8.33e-06] C/m2

22. Comment on Current Flow

23. Question ….. A small sphere that carries a charge q is whirled in a circle at the end of an insulating string. The angular frequency of rotation is ω. What average current does this rotating charge represent?

24. An electric current is given by the expression I(t) = 100 sin(120πt), where I is in amperes and t is in seconds. What is the total charge carried by the current from t = 0 to t = (1/240) s?

25. ANOTHER DEFINITION (Average) Current Density

26. The figure represents a section of a circular conductor of non-uniform diameter carrying a current of 5.00 A. The radius of cross section A1 is 0.400 cm. (a) What is the magnitude of the current density across A1? (b) If the current density across A2 is one-fourth the value across A1, what is the radius of the conductor at A2?

27. Ohm • A particular object will resist the flow of current. • It is found that for any conducting object, the current is proportional to the applied voltage. • STATEMENT: DV=IR • R is called the resistance of the object. • An object that allows a current flow of one ampere when one volt is applied to it has a resistance of one OHM.

28. Ohm’s Law

29. Graph

30. A DIODE Resistance Varies with Applied Voltage (actually with current)

31. Let’s look at the atomic level .. • Conduction is via electrons. • They are weak and small and don’t exercise much. • Positive charge is big and strong and doesn’t intimidate easily. • It’s an ugly situation … something like ……

32. + -

34. The Current • Electrons are going the opposite way from the current. (WHY?) • They probably follow a path like … Average “drift” speed - vd IN OUT

35. Notation • vd average drift velocity of the electron • n number of electrons (mobile) per unit volume. • Dt interval of time • Dx average distance the electron moves in time Dt. • Q total amount of CHARGE that goes through a surface of the conductor in time Dt.

36. The Diagram Often a Vector

37. We return to the diagram ….. • Consider an electron. • Assume that whenever it “bumps” into something it loses its momentum and comes to rest. • It’s velocity therefore starts at zero, the electric field accelerates it until it has another debilitating collision with something else. • During the time it accelerates, its velocity increases linearly . • The average distance that the electron travels between collisions is called the “mean free path”.

38. Starting when the electron is at rest: We showed two slides ago:: Let n= number of charge carriers per unit volume (mobile electrons)

39. Finally

40. Reference The average drift velocity of an electron is about 10-4 m/s SLOW !

41. Ponder How can a current go through a resistor and generate heat (Power) without decreasing the current itself?

42. Loses Energy Gets it back Exit

43. Conductivity In metals, the bigger the electric field at a point, the bigger the current density. s is the conductivity of the material. r=(1/s) is the resistivity of the material

45. A conductor of uniform radius 1.20 cm carries a current of 3.00 A produced by an electric field of 120 V/m. What is the resistivity of the material?

46. The rod in the figure is made of two materials. The figure is not drawn to scale. Each conductor has a square cross section 3.00 mm on a side. The first material has a resistivity of 4.00 × 10–3 Ω · m and is 25.0 cm long, while the second material has a resistivity of 6.00 × 10–3 Ω · m and is 40.0 cm long. What is the resistance between the ends of the rod?

47. Going to the usual limit …

48. Example • A cylindrical conductor of radius R has a current density given by • J0 (constant) • gr • Find the total current in each case.

49. Range of r and s

50. Ye old RESISTANCE Ohm's Law .. but now we know resistivity!