1. 1/30/07184 Lecture 131 PHY 184 Spring 2007 Lecture 13 Title: Capacitors

2. 1/30/07184 Lecture 132NotesNotes  Homework Set 3 is done!  Homework Set 4 is open and Set 5 opens Thursday morning  Midterm 1 will take place in class Thursday, February 8.  One 8.5 x 11 inch equation sheet (front and back) is allowed.  The exam will cover Chapters 16 - 19 Homework Sets 1 - 4

3. 1/30/07184 Lecture 133ReviewReview  The capacitance of a spherical capacitor is r 1 is the radius of the inner sphere r 2 is the radius of the outer sphere  The capacitance of an isolated spherical conductor is R is the radius of the sphere

4. 1/30/07184 Lecture 134 Review (2)  The equivalent capacitance for n capacitors in parallel is  The equivalent capacitance for n capacitors in series is

5. 1/30/07184 Lecture 135 Example - System of Capacitors  Let’s analyze a system of five capacitors  If each capacitor has a capacitance of 5 nF, what is the capacitance of this system of capacitors?

6. 1/30/07184 Lecture 136 System of Capacitors (2)  We can see that C 1 and C 2 are in parallel and that C 3 is also in parallel with C 1 and C 2  We can define C 123 = C 1 + C 2 + C 3 = 15 nF  … and make a new drawing

7. 1/30/07184 Lecture 137 System of Capacitors (3)  We can see that C 4 and C 123 are in series  We can define  … and make a new drawing = 3.75 nF

8. 1/30/07184 Lecture 138 System of Capacitors (4)  We can see that C 5 and C 1234 are in parallel  We can define  And make a new drawing = 8.75 nF

9. 1/30/07184 Lecture 139 System of Capacitors (5)  So the equivalent capacitance of our system of capacitors  More than one half of the total capacitance of this arrangement is provided by C 5 alone.  This result makes it clear that one has to be careful how one arranges capacitors in circuits.

10. 1/30/07184 Lecture 1310 Clicker Question  Find the equivalent capacitance C eq A) B) C)

11. 1/30/07184 Lecture 1311 Clicker Question  Find the equivalent capacitance C eq C) First Step: C 1 and C 2 are in series Second Step: C 12 and C 3 are in parallel

12. 1/30/07184 Lecture 1312 A capacitor stores energy. Field Theory: The energy belongs to the electric field.

13. 1/30/07184 Lecture 1313  A battery must do work to charge a capacitor.  We can think of this work as changing the electric potential energy of the capacitor.  The differential work dW done by a battery with voltage V to put a differential charge dq on a capacitor with capacitance C is  The total work required to bring the capacitor to its full charge q is  This work is stored as electric potential energy Energy Stored in Capacitors

14. 1/30/07184 Lecture 1314  We define the energy density, u, as the electric potential energy per unit volume  Taking the ideal case of a parallel plate capacitor that has no fringe field, the volume between the plates is the area of each plate times the distance between the plates, Ad  Inserting our formula for the capacitance of a parallel plate capacitor we find Energy Density in Capacitors

15. 1/30/07184 Lecture 1315  Recognizing that V/d is the magnitude of the electric field, E, we obtain an expression for the electric potential energy density for parallel plate capacitor  This result, which we derived for the parallel plate capacitor, is in fact completely general.  This equation holds for all electric fields produced in any way The formula gives the quantity of electric field energy per unit volume. Energy Density in Capacitors (2)

16. 1/30/07184 Lecture 1316  An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. a) How much potential energy is stored in the electric field of the charged conductor? Key Idea: An isolated sphere has a capacitance of C=4  0 R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to Example - isolated conducting sphere … and substituting C=4  0 R gives

17. 1/30/07184 Lecture 1317  An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. b) What is the field energy density at the surface of the sphere? Key Idea: The energy density u depends on the magnitude of the electric field E according to so we must first find the E field at the surface of the sphere. Recall: Example - isolated conducting sphere (Why?)

18. 1/30/07184 Lecture 1318 Example: Thundercloud  Suppose a thundercloud with horizontal dimensions of 2.0 km by 3.0 km hovers over a flat area, at an altitude of 500 m and carries a charge of 160 C.  Question 1: What is the potential difference between the cloud and the ground?  Question 2: Knowing that lightning strikes require electric field strengths of approximately 2.5 MV/m, are these conditions sufficient for a lightning strike?  Question 3: What is the total electrical energy contained in this cloud?

19. 1/30/07184 Lecture 1319 Example: Thundercloud (2)  Question 1  We can approximate the cloud-ground system as a parallel plate capacitor whose capacitance is  The charge carried by the cloud is 160 C, which means that the “plate surface” facing the earth has a charge of 80 C  720 million volts ++++++++++++ …++++++++++++ …

20. 1/30/07184 Lecture 1320 Example: Thundercloud (3)  Question 2  We know the potential difference between the cloud and ground so we can calculate the electric field  E is lower than 2.5 MV/m, so no lightning cloud to ground May have lightning to radio tower or tree….  Question 3  The total energy stored in a parallel place capacitor is Enough energy to run a 1500 W hair dryer for more than 5000 hours

21. 1/30/07184 Lecture 1321 Clicker Question  A 1.0  F capacitor and a 3.0  F capacitor are connected in parallel across a 500 V potential difference V. What is the total energy stored in the capacitors? A) U=0.5 J B) U=0.27 J C) U=1.5 J D) U=0.02 J Hint: Use

22. 1/30/07184 Lecture 1322 Clicker Question  A 1.0  F capacitor and a 3.0  F capacitor are connected in parallel across a 500 V potential difference V. What is the total energy stored in the capacitors? A) U=0.5 J U = 0.5 J

23. 1/30/07184 Lecture 1323 Capacitors with Dielectrics  We have only discussed capacitors with air or vacuum between the plates.  However, most real-life capacitors have an insulating material, called a dielectric, between the two plates.  The dielectric serves several purposes: Provides a convenient way to maintain mechanical separation between the plates. Provides electrical insulation between the plates. Allows the capacitor to hold a higher voltage Increases the capacitance of the capacitor Takes advantage of the molecular structure of the dielectric material

24. 1/30/07184 Lecture 1324 Capacitors with Dielectrics (2)  Placing a dielectric between the plates of a capacitor increases the capacitance of the capacitor by a numerical factor called the dielectric constant,   We can express the capacitance of a capacitor with a dielectric with dielectric constant  between the plates as  … where C air is the capacitance of the capacitor without the dielectric  Placing the dielectric between the plates of the capacitor has the effect of lowering the electric field between the plates and allowing more charge to be stored in the capacitor.

25. 1/30/07184 Lecture 1325 Parallel Plate Capacitor with Dielectric  Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as  The constant  0 is the electric permittivity of free space  The constant  is the electric permittivity of the dielectric material