1. CAPACITORS
February 11, 2008

2. Calendar of the Week Exams have been returned.
If you did badly in the exam you need to have a plan to succeed. Let me know if you want any help with this.
Regular problem solving sessions on Wednesday and Friday. Room :39 AM.
Quiz on Friday – Capacitors.
WebAssign should appear shortly if it hasn’t done so already.

3. Capacitors Part I

4. A simple Capacitor TWO PLATES WIRES WIRES Battery Remove the battery
Charge Remains on the plates.
The battery did WORK to charge the plates
That work can be recovered in the form of electrical energy – Potential Difference
WIRES
WIRES
Battery

5. INSIDE THE DEVICE

6. Two Charged Plates (Neglect Fringing Fields)
Air or Vacuum
Area A
- Q Q E
V=Potential Difference
Symbol
ADDED CHARGE

7. Where is the charge? + + + + + - AREA=A s=Q/A + - Q +Q d Air or Vacuum
V=Potential Difference

8. The capacitor therefore stores energy!
One Way to Charge:
Start with two isolated uncharged plates.
Take electrons and move them from the + to the – plate through the region between.
As the charge builds up, an electric field forms between the plates.
You therefore have to do work against the field as you continue to move charge from one plate to another.
The capacitor therefore stores energy!

9. Capacitor
Demo

10. More on Capacitors - Q +Q d Air or Vacuum Area A E
V=Potential Difference
Gaussian
Surface
Same result from other plate!

11. DEFINITION - Capacity
The Potential Difference is APPLIED by a battery or a circuit.
The charge q on the capacitor is found to be proportional to the applied voltage.
The proportionality constant is C and is referred to as the CAPACITANCE of the device.

12. UNITS
A capacitor which acquires a charge of 1 coulomb on each plate with the application of one volt is defined to have a capacitance of 1 FARAD
One Farad is one Coulomb/Volt

13. The two metal objects in the figure have net charges of +79 pC and -79 pC, which result in a 10 V potential difference between them.
(a) What is the capacitance of the system? [7.9] pF (b) If the charges are changed to +222 pC and -222 pC, what does the capacitance become? [7.9] pF (c) What does the potential difference become? [28.1] V

14. NOTE
Work to move a charge from one side of a capacitor to the other is qEd.
Work to move a charge from one side of a capacitor to the other is qV
Thus qV=qEd
E=V/d As before

15. Continuing…
The capacitance of a parallel plate capacitor depends only on the Area and separation between the plates.
C is dependent only on the geometry of the device!

16. Units of e0
pico

17. Simple Capacitor Circuits
Batteries
Apply potential differences
Capacitors
Wires
Wires are METALS.
Continuous strands of wire are all at the same potential.
Separate strands of wire connected to circuit elements may be at DIFFERENT potentials.

18. Size Matters!
A Random Access Memory stores information on small capacitors which are either charged (bit=1) or uncharged (bit=0).
Voltage across one of these capacitors ie either zero or the power source voltage (5.3 volts in this example).
Typical capacitance is 55 fF (femto=10-15)
Question: How many electrons are stored on one of these capacitors in the +1 state?

19. Small is better in the IC world!

20. TWO Types of Connections
SERIES
PARALLEL

21. Parallel Connection V
CEquivalent=CE

22. Series Connection The charge on each capacitor is the same ! q -q q -qV
C C2
q q q q
The charge on each
capacitor is the same !

23. Series Connection ContinuedV
C C2
q q q q

24. More General

25. Example C1 C2 series (12+5.3)pf (12+5.3)pf V C3 C1=12.0 uf C2= 5.3 uf
C3= 4.5 ud
C C2
series
(12+5.3)pf
(12+5.3)pf V C3

26. More on the Big C +q -q
E=e0A/d
+dq
We move a charge dq from the (-) plate to the (+) one.
The (-) plate becomes more (-)
The (+) plate becomes more (+).
dW=Fd=dq x E x d

27. So….

28. Not All Capacitors are Created Equal
Parallel Plate
Cylindrical
Spherical

29. Spherical Capacitor

30. Calculate Potential Difference V
(-) sign because E and ds are in OPPOSITE directions.

31. Continuing…
Lost (-) sign due to switch of limits.

32. Capacitor Circuits

33. A Thunker
If a drop of liquid has capacitance 1.00 pF, what is its radius?
STEPS
Assume a charge on the drop.
Calculate the potential
See what happens

34. Anudder Thunker
Find the equivalent capacitance between points a and b in the combination of capacitors shown in the figure.
V(ab) same across each

35. Thunk some more … C1 C2 (12+5.3)pf V C3 C1=12.0 uf C2= 5.3 uf
C3= 4.5 ud
C C2
(12+5.3)pf V C3

36. More on the Big C +q -q
E=e0A/d
+dq
We move a charge dq from the (-) plate to the (+) one.
The (-) plate becomes more (-)
The (+) plate becomes more (+).
dW=Fd=dq x E x d

37. So….
Sorta like (1/2)mv2

38. What's Happening?
DIELECTRIC

39. Polar Materials (Water)

40. Apply an Electric Field
Some LOCAL ordering Larger Scale Ordering

41. Adding things up..
Net effect REDUCES the field

42. Non-Polar Material

43. Non-Polar Material
Effective Charge is
REDUCED

44. We can measure the C of a capacitor (later)
C0 = Vacuum or air Value
C = With dielectric in place
C=kC0
(we show this later)

45. How to Check This Charge to V0 and then disconnect from The battery.
Connect the two together V
C0 will lose some charge to the capacitor with the dielectric.
We can measure V with a voltmeter (later).

46. Checking the idea.. V
Note: When two Capacitors are the same (No dielectric), then V=V0/2.

48. Messing with Capacitors
The battery means that the
potential difference across
the capacitor remains constant.
For this case, we insert the dielectric but hold the voltage constant,
q=CV
since C  kC0
qk kC0V
THE EXTRA CHARGE COMES FROM THE BATTERY! + V - + - + V -
Remember – We hold V constant with the battery.

49. Another Case We charge the capacitor to a voltage V0.
We disconnect the battery.
We slip a dielectric in between the two plates.
We look at the voltage across the capacitor to see what happens.

50. No Battery q0 q0 =C0Vo When the dielectric is inserted, no charge+ -
q0 =C0Vo
When the dielectric is inserted, no charge
is added so the charge must be the same. V0 V qk

51. Another Way to Think About This
There is an original charge q on the capacitor.
If you slide the dielectric into the capacitor, you are adding no additional STORED charge. Just moving some charge around in the dielectric material.
If you short the capacitors with your fingers, only the original charge on the capacitor can burn your fingers to a crisp!
The charge in q=CV must therefore be the free charge on the metal plates of the capacitor.

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

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

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

55. A little sheet from the past..
-q’ q’ -
+ + + -q q
xEsheet

56. Some more sheet…

57. A Few slides back No Batteryq0
q=C0Vo
When the dielectric is inserted, no charge
is added so the charge must be the same. + - V0 V qk

58. From this last equation

59. Another look + - Vo

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

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

62. SUMMARY OF RESULTS

63. APPLICATION OF GAUSS’ LAW

64. New Gauss for Dielectrics