Capacitors January, 30 th , 2013

1. Chapter 18 Capacitors January, 30th, 2013

2. Electric Flux • Field lines penetrating an area A. • The flux, Φ is: ΦE= E A cosθ • The normal to the area A is at an angle θ to the field • If the area encloses a volume: • for E-field lines that go into the volume, the flux is negative • for E-field lines coming out of the volume the flux is positive

3. Gauss’s Law • Gauss’s Law: the electric flux through any closed surface is equal to the net charge Q inside the surface divided by εo • The constant εo is the permittivity of free space (εo=8.85 x 10-12 C2/Nm2) • The area used for Φ is an imaginary surface, a Gaussian surface, it does not have to coincide with the surface of a physical object

4. Electric Field: Flat Sheet of Charge • The large, flat sheet of charge has a charge per unit area =  • The flux through the sides of the surface is zero (cosq=0) • Through each end, E = EA • Since there are two ends, E = 2EA

5. Flat Sheet of Charge, cont. • The total charge is equal to the charge/area multiplied by the cross-sectional area of the cylinder • q =  A • Therefore, the electric field is • Note, the E-field is uniform everywhere on the sheet or other planes parallel to it, and does not depend on the distance from the sheet

6. Capacitor • A capacitor consists of two parallel metal plates that carry charges of +Q and –Q • The excess charge on the two plates will attract each other and draw all the excess charge to the inner sides of the two plates

7. Capacitor, cont. • Each plate has an area A and charge densities of + = +Q/A and - = -Q/A • Inside the capacitor: • The E-field is • The E-field is constant everywhere • The voltage drops V  as +- • You can store energy in a capacitor (problem: the charge drops rapidly)

8. Capacitors • A capacitor can be used to store charge and energy • This example is a parallel-plate capacitor • Connect the two metal plates to wires that can carry charge on or off the plates

9. Capacitors, cont. • Each plate produces a field E = Q / (2 εo A) • In the region between the plates, the fields from the two plates add, giving • This is the electric field between the plates of a parallel-plate capacitor • There is a potential difference across the plates • ΔV = Ed where d is the distance between the plates

10. Capacitance Defined • From the equations for electric field and potential, • Capacitance, C, is defined as • In terms of C, • A is the area of a single plate and d is the plates’ separation

11. demos Variation of voltage with plate separation Sample capacitors

12. demos Leyden jar and shorting wand Dissectible Leyden jar

13. Capacitance, Notes • Other configurations will have other specific equations • All will employ two plates of some sort • In all cases, the charge on the capacitor plates is proportional to the potential difference across the plates • the SI unit of capacitance is Coulomb/Volt, called the Farad • 1 F = 1 C/V • The Farad is named in honor of Michael Faraday (English, 1791-1867)

14. Storing Energy in a Capacitor • The use of capacitors depends on the capacitor’s ability to store energy and the relationship between charge and potential difference (voltage) • When there is a nonzero potential difference between the two plates, energy is stored in the device • The energy depends on the charge, the voltage, and the capacitance of the capacitor

15. Energy in a Capacitor, cont. • To move a charge ΔQ through a potential difference ΔV requires energy • The energy corresponds to the shaded area in the graph • The total energy stored is equal to the energy required to move all the packets of charge from one plate to the other

16. Energy in a Capacitor, Final • The total energy corresponds to the area under the ΔV – Q graph • Energy = Area = ½ QΔV = PEcap • Q is the final charge • ΔV is the final potential difference • From the definition of capacitance, the energy can be expressed in different forms • These expressions are valid for all types of capacitors • supercapacitor : http://www.batteryuniversity.com/partone-8.htm

17. Capacitors in Series • When dealing with multiple capacitors, the equivalent capacitance is useful • In series: • ΔVtotal = ΔVtop (1) +ΔVbottom (2)and remember that

18. Capacitors in Parallel • When dealing with multiple capacitors, the equivalent capacitance is useful • In parallel: • Qtotal = Q1 +Q2 and Cequiv = C1 +C2 remember that

19. Combinations of Three or More Capacitors • For capacitors in parallel: Cequiv = C1 +C2 + C3 + … • For capacitors in series: • These results apply to all types of capacitors • When a circuit contains capacitors in both series and parallel, the above rules apply to the appropriate combinations • A single equivalent capacitance can be found up to here 1

20. Problem 18.71 A defibrillator containing a 20 µF capacitor is used to shock the heart of a patient by attaching its plates to the patient’s chest. Just before discharging, the capacitor has a potential difference of 10,000 V across its plates. (a) what is the energy released into the patient? (b) If the energy is discharged over 20 ms, what is the power output of the defibrillator?

21. Dielectrics • Most real capacitors contain two metal “plates” separated by a thin insulating region • Many times these plates are rolled into cylinders • The region between the plates typically contains a material called a dielectric

22. Dielectrics, cont. • Adding a dielectric btw the plates of a capacitor increases the capacitance by a factor κ • and reduces the electric fieldinside the capacitor by a factor κ • The actual value of the dielectric constant κdepends on the material

23. Dielectrics, final • When the plates of a capacitor are charged, the electric field is established in the dielectric material • Most good dielectrics are highly ionic, thus charges separate inside the dielectric • As a result the charge, the E-field, the voltage all decreasewith time

24. Dielectric Summary • The results of adding a dielectric to a capacitor apply to any type of capacitor • Adding a dielectric increases the capacitance by a factor κ • Adding a dielectric reduces the electric fieldinside the capacitor by a factor κ • The actual value of the dielectric constantκdepends on the material

25. Dielectric Breakdown • As more and more charge is added to a capacitor, the electric field increases • For a capacitor containing a dielectric, the field can become so large that it rips the ions in the dielectric apart • This effect is called dielectric breakdown • The free ions are able to move through the material • They move rapidly toward the oppositely charged plate, and destroy the capacitor • The value of the field at which this occurs depends on the material • See table 18.1 for the values for various materials

26. Lightning • During a lightning strike, large amounts of electric charge move between a cloud (-) and the surface of the Earth (+), or between clouds • There is a dielectric breakdown of the air

27. Lightning • A man struck by lightning (Lichtenberg figure) http://205.243.100.155/frames/lichtenbergs.html

28. Lightning • A golf course struck by lightning (Lichtenberg figure) http://205.243.100.155/frames/lichtenbergs.html

29. Lightning, cont. • Most charge motion involves electrons • They are easier to move than protons • The electric field of a lightning strike is directed from the Earth to the cloud • After the dielectric breakdown, electrons travel from the cloud to the Earth

30. Lightning, final • In a thunderstorm, the water droplets and ice crystals gain a negative charge as they move in the cloud • They carry the negative charge to the bottom of the cloud • This leaves the top of the cloud positively charged • The negative charges in the bottom of the cloud repel electrons from the Earth’s surface • This causes the Earth’s surface to be positively charged and establishes an electric field similar to the field between the plates of a capacitor • Eventually the field increases enough to cause dielectric breakdown, which is the lightning bolt

31. t d Question Consider a capacitor made of two parallel metallic plates separated by a distance d. A new metal plate with thickness t is inserted between them without changing the charge on the original plates. The capacitance of the capacitor 1) increases 2) decreases 3) stays the same Effective “d” decreasesC = εoA/d increases up to here 2

32. t d Question Consider a capacitor made of two parallel metallic plates separated by a distance d. A dielectric slab with thickness t is inserted between them without changing the charge on the original plates. The energy stored in the capacitor 1) increases 2) decreases 3) stays the same Effective “εo”, or ε, increasesC=εA/d increasesEcap=Q2/2C decreases

33. Question - - - - + + + + -q +q d pull pull A parallel plate capacitor is given a charge q. The plates are then pulled a smalldistance further apart. What happens to the charge q on each plate of the capacitor? 1) Increases 2) Stays constant 3) Decreases Remember that charge is real/physical. There is no place for the charges to go.

34. Electric Potential Energy Revisited • One way to view electric potential energy is that the potential energy is stored in the electric field itself • Whenever an electric field is present in a region of space, potential energy is located in that region • The potential energy between the plates of a parallel plate capacitor can be determined in terms of the field between the plates:

35. Electric Potential Energy, final • The energy density of the electric field can be defined as the energy / volume: • These results give the energy density for any arrangement of charges • Potential energy is present whenever an electric field is present