Understanding Capacitors: Energy Storage & Density

1. PHY 184 Spring 2007 Lecture 13 Title: Capacitors 184 Lecture 13

2. Notes • 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 - 4 184 Lecture 13

3. Review • The capacitance of a spherical capacitor is • r1 is the radius of the inner sphere • r2 is the radius of the outer sphere • The capacitance of an isolated spherical conductor is • R is the radius of the sphere 184 Lecture 13

4. Review (2) • The equivalent capacitance for n capacitors in parallel is • The equivalent capacitance for n capacitors in series is 184 Lecture 13

5. 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? 184 Lecture 13

6. System of Capacitors (2) • We can see that C1 and C2 are in parallel and that C3 is also in parallel with C1 and C2 • We can define C123 = C1 + C2 + C3 = 15 nF • … and make a new drawing 184 Lecture 13

7. System of Capacitors (3) = 3.75 nF • We can see that C4 and C123 are in series • We can define • … and make a new drawing 184 Lecture 13

8. System of Capacitors (4) = 8.75 nF • We can see that C5 and C1234 are in parallel • We can define • And make a new drawing 184 Lecture 13

9. 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 C5 alone. • This result makes it clear that one has to be careful how one arranges capacitors in circuits. 184 Lecture 13

10. Clicker Question • Find the equivalent capacitance Ceq A) B) C) 184 Lecture 13

11. Clicker Question • Find the equivalent capacitance Ceq C) First Step: C1 and C2 are in series Second Step: C12 and C3 are in parallel 184 Lecture 13

12. A capacitor stores energy. Field Theory: The energy belongs to the electric field. 184 Lecture 13

13. Energy Stored in Capacitors • 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 184 Lecture 13

14. Energy Density in Capacitors • 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 184 Lecture 13

15. Energy Density in Capacitors (2) • 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. 184 Lecture 13

16. Example - isolated conducting sphere • 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e0R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to … and substituting C=4pe0R gives 184 Lecture 13

17. Example - isolated conducting sphere (Why?) • 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: 184 Lecture 13

18. 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 betweenthe cloud and the ground? • Question 2: • Knowing that lightning strikes requireelectric field strengths of approximately2.5 MV/m, are these conditions sufficientfor a lightning strike? • Question 3: • What is the total electrical energy contained in this cloud? 184 Lecture 13

19. 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 …++++++++++++ … ++++++++++++ 184 Lecture 13

20. 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 184 Lecture 13

21. 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 184 Lecture 13

22. 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 184 Lecture 13

23. 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 184 Lecture 13

24. 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 Cair 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. 184 Lecture 13

25. 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 184 Lecture 13