1. A Little Capacitance and Current and Resistance
February 19, 2006

2. Notes Complete Capacitance (Very Brief)
New topic today – Current and Resistance
Quiz on Friday
Friday 7:30 review session.
There is a new WebAssign.

3. DIELECTRIC RESULTS

4. APPLICATION OF GAUSS’ LAW

5. New Gauss for Dielectrics

6. New Topic: Current and Resistance

7. Physical Resistors

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

9. What’s Moving?
Charge!

10. What is making the charged move??
Battery

11. 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.

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

13. Conducting material
DQ,Dt

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

15. 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

16. Comment on Current Flow

17. 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?

18. 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?

19. (Average) Current Density
ANOTHER DEFINITION
(Average) Current Density

20. 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 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?

21. 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.

22. Ohm’s Law

23. Graph

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

25. 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 ……

26. + -

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

29. 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.

30. The Diagram
Often a Vector

31. 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”.

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

33. Finally

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

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

36. Loses Energy
Gets it back
Exit

37. Conductivity s is the conductivity of the material.
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

39. A conductor of uniform radius 1. 20 cm carries a current of 3
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?

40. The rod in the figure is made of two materials
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?

41. Going to the usual limit …

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

43. Range of r and s

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

45. REMEMBER

46. Temperature Effect Dr DT

47. A closed circuit

48. Power