Energy Stored in Capacitors

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1 Energy Stored in Capacitors U = 1 2 qv q = CV U = 1 2 CV 2 q 2 or U = 1 2 C 37

2 Energy Density in Capacitors (1) We define the, 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 38

3 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. 39

4 Example: Isolated Conducting Sphere (1) An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nc. Question 1: How much potential energy is stored in the electric field of the charged conductor? Answer: Key Idea: An isolated sphere has a capacitance of C=4πε 0 R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to and substituting C=4πε 0 R gives 40

5 Example: Isolated Conducting Sphere (2) An isolated conducting sphere whose radius R is 6.85 cm has a charge of q = 1.25 nc. Question 2: What is the field energy density at the surface of the sphere? Answer: Key Idea: The energy density u depends on the magnitude of the electric field E according to q so we must first find the E field at the surface of the sphere. Recall: 41

6 What is the total energy in E-field? U tot = R R udv = 1 4π 2 0E 2 r 2 dr = 2 1 q 2 2π 0 4π 0 r 4 r2 dr = R q π 0 R = 1 2 qv 6

7 What is the total energy in E-field? U tot = R R udv = 1 4π 2 0E 2 r 2 dr = 2 1 q 2 2π 0 4π 0 r 4 r2 dr = R q 2 4π 0 R = 2 qv Yes! 6

8 Example: Thundercloud (1) 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? 42

9 Example: Thundercloud (2) Question 1: What is the potential difference between the cloud and the ground? Answer: We can approximate the cloud-ground system as a parallel plate capacitor whose capacitance is The charge carried by the cloud is 160 C 720 million volts V = 1 q = C September 21,

10 Example: Thundercloud (3) Question 2: Knowing that lightning strikes require electric field strengths of approximately 2.5 MV/m, are these conditions sufficient for a lightning strike? Answer: 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. September 21,

11 Example: Thundercloud (4) Question 3: What is the total electrical energy contained in this cloud? Answer: The total energy stored in a parallel place capacitor is September 21,

12 Physics of a spark +q -q d V E V/d 11

13 Physics of a spark V d1 E V/d 1 E 0 +q -q d V E V/d 11

14 Physics of a spark V d1 E V/d 1 E 0 +q -q d V λ e E E V/d E k Eλ 1eV 11

15 Electric curcuit 12

16 Circuit diagram Lines represent conductors The battery or power supply is represented by The capacitor is represented by the symbol Battery provides (a DC) potential difference V 13

17 Charging/Discharging a Capacitor (2) Illustrate the charging processing using a circuit diagram. This circuit has a switch (pos c) When the switch is in position c, the circuit is open (not connected). (pos a) When the switch is in position a, the battery is connected across the capacitor. Fully charged, q = CV. (pos b) When the switch is in position b, the two plates of the capacitor are connected. Electrons will move around the circuit--a current will flow--and the capacitor will discharge. c c 8

18 + - 15

19 V

20 V

21 I V

22 I V

23 I V

24 I V

25 V V 15

26 V V 15

27 V V 15

28 V + - I + - V 15

29 V + - I + - V 15

30 V + - I + - V 15

31 Capacitors in Circuits A circuit is a set of electrical devices connected with conducting wires Capacitors can be wired together in circuits in parallel or series Capacitors in circuits connected by wires such that the positively charged plates are connected together and the negatively charged plates are connected together, are connected in parallel Capacitors wired together such that the positively charged plate of one capacitor is connected to the negatively charged plate of the next capacitor are connected in series

32 Capacitors in Parallel (1) Consider an electrical circuit with three capacitors wired in parallel Each of three capacitors has one plate connected to the positive terminal of a battery with voltage V and one plate connected to the negative terminal. The potential difference V across each capacitor is the same... key point for capacitors in parallel We can write the charge on each capacitor as 26

33 Capacitors in Parallel (2) We can consider the three capacitors as one equivalent capacitor C eq that holds a total charge q given by We can now define C eq by A general result for n capacitors in parallel is If we can identify capacitors in a circuit that are wired in parallel, we can replace them with an equivalent capacitance 27

34 Capacitors in Series (1) Consider a circuit with three capacitors wired in series The positively charged plate of C 1 is connected to the positive terminal of the battery The negatively charge plate of C 1 is connected to the positively charged plate of C 2 The negatively charged plate of C 2 is connected to the positively charge plate of C 3 The negatively charge plate of C 3 is connected to the negative terminal of the battery The battery produces an equal charge q on each capacitor because the battery induces a positive charge on the positive place of C 1, which induces a negative charge on the opposite plate of C 1, which induces a positive charge on C 2, etc... key point for capacitors in series 28

35 Capacitors in Series (2) Knowing that the charge is the same on all three capacitors we can write We can express an equivalent capacitance C eq as We can generalize to n capacitors in series If we can identify capacitors in a circuit that are wired in series, we can replace them with an equivalent capacitance 29

36 Review The equivalent capacitance for n capacitors in parallel is = The equivalent capacitance for n capacitors in series is = 31

37 iclicker Three capacitors, each with capacitance C, are connected as shown in the figure. What is the equivalent capacitance for this arrangement of capacitors? a) C/3 b) 3C c) C/9 d) 9C e) none of the above

38 iclicker Three capacitors, each with capacitance C, are connected as shown in the figure. What is the equivalent capacitance for this arrangement of capacitors? a) C/3 b) 3C c) C/9 d) 9C e) none of the above

39 Example: System of Capacitors (1) Question: If each capacitor has a capacitance of 5 nf, what is the capacitance of this system of capacitors? Answer: Find the equivalent capacitance Analyze each piece of the circuit individually, replacing pairs in series or in parallel by one capacitor with equivalent capacitance 32

40 Example: 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 find C 123 = C 1 + C 2 + C 3 and make a new drawing 33

41 Example: System of Capacitors (3) We can see that C 4 and C 123 are in series We find for the equivalent capacitance: and make a new drawing 34

42 Example: System of Capacitors (4) We can see that C 5 and C 1234 are in parallel We find for the equivalent capacitance and make a new drawing 35

43 Example: System of Capacitors (5) So the equivalent capacitance of our system of capacitors 36

44 Capacitors with Dielectrics (1) So far, we have 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 (plates attract!) 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 46

45 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, κ We can express the capacitance of a capacitor with a dielectric with dielectric constant κ 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. 47

46 Parallel Plate Capacitor with Dielectric Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as The constant ε 0 is the electric permittivity of free space The constant ε is the electric permittivity of the dielectric material 48

47 Microscopic Perspective on Dielectrics (1) Let s consider what happens at the atomic and molecular level when a dielectric is placed in an electric field There are two types of dielectric materials Polar dielectric Non-polar dielectric Polar dielectric material is composed of molecules that have a permanent electric dipole moment due to their molecular structure e.g., water molecules Normally the directions of the electric dipoles are randomly distributed: 53

48 Microscopic Perspective on Dielectrics (2) When an electric field is applied to these polar molecules, they tend to align with the electric field Non-polar dielectric material is composed of atoms or molecules that have no electric dipole moment 54

49 Microscopic Perspective on Dielectrics (3) These atoms or molecules can be induced to have a dipole moment under the influence of an external electric field This induction is caused by the opposite direction of the electric force on the negative and positive charges of the atom or molecule, which displaces the center of the relative charge distributions and produces an induced electric dipole moment

50 Microscopic Perspective on Dielectrics (4) In both the case of the polar and non-polar dielectric materials, the resulting aligned electric dipole moments tend to partially cancel the original electric field E0 The electric field inside the capacitor then is the original field minus the induced field = E κ 56

51 Microscopic Perspective on Dielectrics (4) In both the case of the polar and non-polar dielectric materials, the resulting aligned electric dipole moments tend to partially cancel the original electric field E0 Ed The electric field inside the capacitor then is the original field minus the induced field = E κ 56

52 Dielectric Strength The dielectric strength of a material measures the ability of that material to withstand voltage differences If the voltage across a dielectric exceeds the breakdown potential, the dielectric will break down and begin to conduct charge between the plates Real-life dielectrics enable a capacitor to provide a given capacitance and withstand the required voltage without breaking down Capacitors are usually specified in terms of their capacitance and rated (i.e., maximum) voltage 51

53 Dielectric Constant The dielectric constant of vacuum is defined to be 1 The dielectric constant of air is close to 1 and we will use the dielectric constant of air as 1 in our problems The dielectric constants of common materials are 52

54 Capacitor with Dielectric (1) Question 1: Consider a parallel plate capacitor with capacitance C = 2.00 µf connected to a battery with voltage V = 12.0 V as shown. What is the charge stored in the capacitor? Question 2: Now insert a dielectric with dielectric constant κ = 2.5 between the plates of the capacitor. What is the charge on the capacitor? The additional charge is provided by the battery. 57

55 Capacitor with Dielectric (2) We isolate the charged capacitor with a dielectric by disconnecting it from the battery. We remove the dielectric, keeping the capacitor isolated. Question 3: What happens to the charge and voltage on the capacitor? The charge on the isolated capacitor cannot change because there is nowhere for the charge to flow. Q remains constant. The voltage on the capacitor will be V increases The voltage went up because removing the dielectric increased the electric field and the resulting potential difference between the plates. 58

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iclicker A device has a charge q=10 nc and a potential V=100V, what s its capacitance? A: 0.1 nf B: 1nF C: 10nF D: F E: 1F

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