Electrical Energy and Capacitance

1. Electrical Energy and Capacitance Chapter

2. Assembly of a system of charges • We know that V=kq/r. • To bring a charge in from infinity requires work to be done: Work = ΔU = qoV

3. Assembly of a system of charges • W2=q2V1=kq2q1/r1,2 • W3=q3V1 +q3V2 = kq3q1/r1,3+ kq3q2/r2,3 • And so U= W2+W3 q1

4. 24-1 The first two terms in 24-1 can be written: where V1 is the potential at the location of the charge q1 due to q2 and q3. Similarly the second and third terms represent the charge q3 times the potential due to charges q1 and q2 at the location of q3 and the first and third terms equal the charge q2 times the potential due to charges q1 and q3 at the location of q2. We can thus rewrite 24-1 as:

5. Energy of a system of point charges

6. Q Consider a spherical conductor of total charge Q and radius r: When only q is accumulated on conductor: r Work to bring an additional charge dq from infinity to the conductor. Q Total potential energy as charge builds up from 0 to its final value Q 0 Same result as for a collection of point charges! Good for any shape!

7. If we have a set of n conductors with the ith conductor carrying the charge Qi, and having the potential Vi, the total electrostatic potential energy is:

8. a) Calculate the total work to assemble the system. 1. To bring the first charge at point A: WA = 0 2. To bring the second charge at point B, WB= qVA, where VA is the potential at the location B due to the charge at point A a distance a away. 3. WC = qVC, where VC is the potential at point C due to q @ A a distance √2a away and q @ B a distance a away: 4. Similarly WD, the work needed to bring the forth charge at D is :

9. 5. The total work = sum of individual contributions: Wtotal = WA+WB+WC+WD= b) Calculate the work using 24-2: (the potential at the location of each charge is VD) Using

10. Capacitance • A capacitor is a device whose purpose is to store electrical energy which can then be released in a controlled manner during a short period of time. • A capacitor consists of 2 spatially separated conductors which can be charged to +Q and -Q respectively. • The capacitance is defined as the ratio of the charge on one conductor of the capacitor to the potential difference between the conductors. • The capacitance belongs only to the capacitor, independent of the charge and voltage.

11. A + + + + d - - - - - Example: Parallel Plate Capacitor • Calculate the capacitance. We assume+s, -scharge densities on each plate with potential difference V: • Need Q: • Need V: from def’n: • Use Gauss’ Law to find E

12. - + s s E=0 E=0 - + • Field outside the sheets is zero - + + - • Gaussian surface encloses zero net charge + - A + - + - + - • Field inside sheets is not zero: - + - + • Gaussian surface encloses non-zero net charge - A + - + - E Recall:Two Infinite Sheets(into screen)

13. A + + + + d - - - - - Þ Example:Parallel Plate Capacitor • Calculate the capacitance: • Assume +Q, -Q on plates with potential difference V. • As hoped for, the capacitance of this capacitor depends only on its geometry (A,d). • Note that C ~ length; this will always be the case!

14. d Current sensor Battery Moveable plate Fixed plate Practical Application: Microphone (“condenser”) Sound waves incident  pressure oscillations  oscillating plate separation d  oscillating capacitance ( )  oscillating charge on plate  oscillating current in wire ( )  oscillating electrical signal See this in action at http://micro.magnet.fsu.edu/electromag/java/microphone/ !

15. r a b L Example: Cylindrical Capacitor • Calculate the capacitance: • Assume +Q, -Q on surface of cylinders with potential difference V.

16. Er Er Q + + + + + + + • Apply Gauss' Law: L Þ Recall:Cylindrical Symmetry • Gaussian surface is cylinder of radius r and length L • Cylinder has charge Q

17. -Q r a +Q b L If we assume that inner cylinder has +Q, then the potentialV is positive if we take the zero of potential to be defined at r = b: Þ Example:Cylindrical Capacitor • Calculate the capacitance: • Assume +Q, -Q on surface of cylinders with potential difference V.

18. Capacitance: spherical conductor

19. How big is a Farad? • Find the radius of a spherical conductor with a capacitance of 1 Farad. • That’s 1400 times the radius of Earth!

20. Capacitance and Charge? • Q: A sphere of capacitance C1 has a charge of 20μC. If the charge is increased to 60 μC, what is the new capacitance? • A: C1; Capacitance is independent of charge because if the charge is tripled, potential is also tripled. C depends only on the radius of the sphere!

21. Parallel Plate Capacitors (again!) • Each plate contributes • So the total field is

22. Try this! • A parallel plate capacitor has square plates 10 cm on each side that are separated by 1 mm. • calculate C • If this capacitor is charged to 12 V, how much charge is transferred from one plate to the other?

23. Solution Q = CV = 8.85 x 10-11 F x 12 V = 106.2 x 10-11C = 1.062 nC

24. ConcepTest 16.8Capacitors 1) C1 2) C2 3) both have the same charge 4) it depends on other factors Capacitor C1 is connected across a battery of 5 V. An identical capacitor C2 is connected across a battery of 10 V. Which one has the most charge?

25. ConcepTest 16.8Capacitors 1) C1 2) C2 3) both have the same charge 4) it depends on other factors Capacitor C1 is connected across a battery of 5 V. An identical capacitor C2 is connected across a battery of 10 V. Which one has the most charge? Since Q = C V and the two capacitors are identical, the one that is connected to the greater voltage has the most charge, which is C2 in this case.

26. –Q +Q ConcepTest 16.9aVarying Capacitance I 1) increase the area of the plates 2) decrease separation between the plates 3) decrease the area of the plates 4) either (1) or (2) 5) either (2) or (3) What must be done to a capacitor in order to increase the amount of charge it can hold (for a constant voltage)?

27. –Q +Q ConcepTest 16.9aVarying Capacitance I 1) increase the area of the plates 2) decrease separation between the plates 3) decrease the area of the plates 4) either (1) or (2) 5) either (2) or (3) What must be done to a capacitor in order to increase the amount of charge it can hold (for a constant voltage)? Since Q = C V, in order to increase the charge that a capacitor can hold at constant voltage, one has to increase its capacitance. Since the capacitance is given by , that can be done by either increasing A or decreasing d.

28. –Q +Q 1) the voltage decreases 2) the voltage increases 3) the charge decreases 4) the charge increases 5) both voltage and charge change ConcepTest 16.9bVarying Capacitance II A parallel-plate capacitor initially has a voltage of 400 V and stays connected to the battery. If the plate spacing is now doubled, what happens?

29. Parallel Plate Lab • linear relationship between C and A. • inverse relationship between C and d.

30. Cylindrical Capacitors (again!) • -Q is placed on the outer cylinder while +Q is placed on the inner cylinder. • Since E depends on r, you must integrate to find C.

31. Charge is transferred from one plate to another. Work is done. Each charge experiences a change in its energy (U). What’s going on in there?

32. Moving charge • When a small positive charge, dq, is moved from the – plate to the + plate, its potential energy is increased by dU=V dq.

33. Total increase in Potential Energy

34. Potential Energy stored in a Capacitor • Using C= Q/V, we can express U in many ways:

35. Energy density • Using the equation for electrostatic potential energy for a capacitor

36. and Dielectrics A dielectric is an insulating material in which the individual molecules polarize in proportion to the the strength of an external electric field. This reduces the electric field inside the dielectric by a factor k, called the dielectric constant. For fixed charge Q on plates Capacitance is increased by .

37. Dielectric Strength • Dielectrics are insulators: charges are not free to move (beyond molecular distances) • Above a critical electric field strength, however, the electrostatic forces polarizing the molecules are so strong that electrons are torn free and charge flows. • This is called Dielectric Breakdown, and can disturb the mechanical structure of the material

38. Dielectric Properties of common materials

39. Charge, Voltage and Shocks • If a spark jumps across a 1mm gap when you reach for a doorknob, what was the potential difference between you and the knob and how much charge was on you? • Assume your finger and the knob form a parallel plate capacitor with area 1 cm2 and breakdown electric field = 3MV/m. C = Q/V V = Ed Q = CV=CEd = (A 0/d)(E d) = A 0 E Q = (0.01m)2 (8.8510-12 F/m)(3 x 106 V/m) Q = 2.7 10-15 (C/V)(Vm2/m 2) = 2.7 femto-Coulomb

40. Example2 • What plate area is required if an air-filled, parallel plate capacitor • with a plate separation of 2.6 mm is to have a capacitance of 12 pF? • (b) What is the maximum voltage that can be applied to this capacitor • without causing dielectric breakdown (Emax = 3MV/m)? • C = A 0/d, • A = Cd/ 0 = (12·10 -12 F)(2.6 ·10 -3 m) / (8.8510 -12 F/m) • A = 3.5 ·10-3 m2 • (b) Vmax = Emax d • Vmax = (3·10+6 V/m) (2.6 ·10 -3 m) = 7.8 ·10+3 V

41. -Q A r a + + + + +Q d b - - - - - L -Q a +Q b Summary • A Capacitor is an object with two spatially separated conducting surfaces. • The definition of the capacitance of such an object is: • The capacitance depends on the geometry : Parallel Plates Spherical Cylindrical

42. Combinations of Capacitors Parallel Capacitors • Upper plates of both are at a common potential; lower plates at common potential. • Same potential difference across both capacitors.

43. Combinations of Capacitors Series Capacitors • The magnitude of the charge on both capacitors must be the same. • ΔV across both capacitors is equal to the sum of the ΔV across each capacitor.

44. Parallel capacitors Ceq = C1 + C2 + C3… Qtotal = Q1 + Q2 + Q3… Series Capacitors Equivalent Capacitance