1. Capacitance and Dielectrics AP Physics C

2. Commercial Capacitor Designs Section 16.10

3. Capacitors in Kodak Cameras Capacitors can be easily purchased at a local Radio Shack and are commonly found in disposable Kodak Cameras. When a voltage is applied to an empty capacitor, current flows through the capacitor and each side of the capacitor becomes charged. The two sides have equal and opposite charges. When the capacitor is fully charged, the current stops flowing. The collected charge is then ready to be discharged and when you press the flash it discharges very quickly released it in the form of light. Cylindrical Capacitor

4. Capacitance In the picture below, the capacitor is symbolized by a set of parallel lines. Once it's charged, the capacitor has the same voltage as the battery (1.5 volts on the battery means 1.5 volts on the capacitor) The difference between a capacitor and a battery is that a capacitor can dump its entire charge in a tiny fraction of a second, where a battery would take minutes to completely discharge itself. That's why the electronic flash on a camera uses a capacitor -- the battery charges up the flash's capacitor over several seconds, and then the capacitor dumps the full charge into the flash tube almost instantly

5. Measuring Capacitance Let’s go back to thinking about plates! The unit for capacitance is the FARAD, F.

6. Electric Potential for Conducting Sheets + + + + + + + E =0 Using Gauss’ Law we derived and equation to define the electric field as we move radially away from the charged sheet or plate. Electric Potential?

7. Capacitance This was derived from integrating the Gauss’ Law expression for a conducting plate. These variables represent a constant of proportionality between voltage and charge. That’s what C is ! What this is saying is that YOU CAN change the capacitance even though it represents a constant. That CHANGE, however, can only happen by physically changing the GEOMETRY of the capacitor itself.

8. Capacitor Geometry The capacitance of a capacitor depends on HOW you make it.

9. Capacitor Problems What is the AREA of a 1F capacitor that has a plate separation of 1 mm? 1.13x10 8 m 2 10629 m Is this a practical capacitor to build? NO! – How can you build this then? The answer lies in REDUCING the AREA. But you must have a CAPACITANCE of 1 F. How can you keep the capacitance at 1 F and reduce the Area at the same time? Add a DIELECTRIC!!!

10. Dielectric Remember, the dielectric is an insulating material placed between the conductors to help store the charge. In the previous example we assumed there was NO dielectric and thus a vacuum between the plates. All insulating materials have a dielectric constant associated with it. Here now you can reduce the AREA and use a LARGE dielectric to establish the capacitance at 1 F.

12. Dielectrics A nonconducting material is a dielectric. When a dielectric is placed in an electric field, the molecules of the dielectric become polarized, oftrn by the rotation of dipoles. This produces a surface charge on the dielectric. ©2008 by W.H. Freeman and Company

13. Dielectrics in Capacitors When a dielectric is placed in a capacitor, it weakens the electric field. The electric field produced at the surface of the dielectric opposes the electric field of the capacitor. Think of it as some of the capacitors E-field lines ending on the surface of the conductor. ©2008 by W.H. Freeman and Company

14. Dielectric Constant The dielectric constant relates the strength of the electric field without a dielectric to the strength with a dielectric. ©2008 by W.H. Freeman and Company

15. Dielectrics in Capacitors If the electric field is less, the potential across the capacitor is less. Since C = Q/V, when V is reduced, the capacitance increases. Dielectrics increase capacitance. ©2008 by W.H. Freeman and Company

16. Dielectrics in Parallel Plate Capacitors ©2008 by W.H. Freeman and Company

17. Capacitance of an Isolated Sphere Q V http://hyperphysics.phy-astr.gsu.edu/%E2%80%8Chbase/electric/capsph.html For a capacitor consisting of 2 concentric spheres go to

18. Coaxial Capacitors Consider two long coaxial conducting cylinders, with a charge of +Q on the inside cylinder and a charge of –Q on the outside cylinder. Find the capacitance of this capacitor. ©2008 by W.H. Freeman and Company

19. Coaxial Capacitors  Step 1: Find an expression for the electric field in the region between the cylinders.  Step 2: Use that expression to find the potential between the cylinders.  Step 3: Use the expression C=Q/V to find the capacitance. ©2008 by W.H. Freeman and Company

20. Coaxial Capacitors Step 1: Use Gauss’ Law to find the field in between the conductors.  For a long cylinder, the field must point radially outward, by symmetry.  The charge in the Gaussian container is given by ©2008 by W.H. Freeman and Company

21. Coaxial Capacitors The field is The potential is Note that we integrate from the negative charge to the positive charge. ©2008 by W.H. Freeman and Company

22. Coaxial Capacitor The capacitance is C=Q/V ©2008 by W.H. Freeman and Company

23. Capacitor Practice Problem A spherical capacitor consists of two concentric spherical shells of radii R 1 and R 2, where R 2 > R 1. Show that the capacitance is given by Show that for thin capacitors, i.e. R 2 – R 1 <<R, the formula is the same as the parallel plate capacitor with A the area of one sphere.

24. Energy Storage

25. Potential Energy of a System of Charges Consider a system of two equal charges, q 1 and q 2. Putting the first charge in place requires no energy. Putting the second charge requires q 2 V, where V is the potential of the charges. q 2 is ½ the total charge Q, so the energy can also be written

26. Potential Energy of a System of Charges The potential energy of a system of charges q i is given by

27. Energy Stored in a Capacitor Consider a capacitor that is charged one small charge at a time.  The energy stored in the capacitor when a small charge dq is moved from – to + is V dq.  V = q/C ©2008 by W.H. Freeman and Company

28. Energy Stored in a Capacitor Consider a capacitor that is charged one small charge at a time.  The first small charges face little potential, so they store little energy.  The charges moved at the end face the full potential of the charged capacitor. ©2008 by W.H. Freeman and Company

29. Energy Stored in a Capacitor ©2008 by W.H. Freeman and Company

30. Energy Stored in a Capacitor ©2008 by W.H. Freeman and Company

31. Energy Stored in a Capacitor ©2008 by W.H. Freeman and Company

32. Energy Stored in the Electric Field Consider a parallel plate capacitor. The electric field E is given by Thus the charge is ©2008 by W.H. Freeman and Company

33. Energy Stored in the Electric Field Since the electric field is constant, the potential is simply The energy stored in the capacitor is ©2008 by W.H. Freeman and Company

34. Energy Stored in the Electric Field The energy is stored in the electric field. Since the volume that the electric field occupies is A d, the energy density is given by ©2008 by W.H. Freeman and Company

35. Energy Density The energy density (energy per unit volume) in an electric field is given by This result is applicable to electric fields in general. If a dielectric is present, ©2008 by W.H. Freeman and Company

36. Energy of a Capacitor Practice 1 A dielectric is inserted between the plates of a parallel-plate capacitor. The capacitor was connected to a battery the entire time the dielectric was inserted. What happens to the capacitance of the capacitor, the charge on the plates, and the energy stored in the capacitor? Was the work done in inserting the dielectric positive or negative? ©2008 by W.H. Freeman and Company

37. Energy of a Capacitor Practice 2 A parallel plate capacitor of area A and separation d is charged to a potential difference V and then disconnected from the voltage source. The plates are plates are then pulled apart to a distance 3d. How much work was require to pull them apart? ©2008 by W.H. Freeman and Company

38. Using MORE than 1 capacitor Let’s say you decide that 1 capacitor will not be enough to build what you need to build. You may need to use more than 1. There are 2 basic ways to assemble them together Series – One after another Parallel – between a set of junctions and parallel to each other.

39. Capacitors in Series Capacitors in series each charge each other by INDUCTION. So they each have the SAME charge. The electric potential on the other hand is divided up amongst them. In other words, the sum of the individual voltages will equal the total voltage of the battery or power source.

40. Capacitors in Parallel In a parallel configuration, the voltage is the same because ALL THREE capacitors touch BOTH ends of the battery. As a result, they split up the charge amongst them.

41. Circuit Reduction Example Section 16.8

42. Finding V and Q First find C eq. Then work backwards, remembering that  In parallel, voltages are constant; charges add up.  In series, voltages add up, charges are constant. ©2008 by W.H. Freeman and Company

43. Capacitor Circuit Example 1 Find the charge on each capacitor and the potential difference across each. Ans.- 48 μC, 8.0 V and 4.0 V

44. Capacitor Circuit Example 2 This circuit is connected to a 6 V battery. Find the charge on and potential across each capacitor.

45. Example, Capacitors in Parallel and in Series:

54. Example, One Capacitor Charging up Another Capacitor: