1. 2 Today’s agendum: Electric potential energy. You must be able to use electric potential energy in work-energy calculations. Electric potential. You.

2 Today’s agendum: Electric potential energy. You must be able to use electric potential energy in work-energy calculations. Electric potential. You must be able to calculate the electric potential for a point charge, and use the electric potential in work-energy calculations. Electric potential and electric potential energy of a system of charges. You must be able to calculate both electric potential and electric potential energy for a system of charged particles (point charges today, charge distributions next lecture). The electron volt. You must be able to use the electron volt as an alternative unit of energy.

3 This lecture introduces electric potential energy and something called “electric potential.” Electric potential energy is “just like” gravitational potential energy. Except that all matter exerts an attractive gravitational force, but charged particles exert either attractive or repulsive electrical forces—so we need to be careful with our signs. Electric potential is the electric potential energy per unit of charge. If you understand the symbols in the starting equations, and avoid sign and direction mistakes, homework and exams are not difficult.

4 Electric Potential Electric Potential Energy b a FEFE r dr ds q 2 (-) q 1 (+) Work done by Coulomb force when q 1 moves from a to b: rara rbrb q 1 (+) You don’t need to worry about the details of the math. They are provided for anybody who wants to study them later.

5 I did the calculation for a + charge moving away from a – charge; you could do a similar calculation for ++, -+, and ++. The important point is that the work depends only on the initial and final positions of q 1. In other words, the work done by the electric force is independent of path taken. The electric force is a conservative force. b a FEFE r dr ds q 2 (-) q 1 (+) rara rbrb Disclaimer: this is a “demonstration” rather than a rigorous proof.

6 The next two slides are intended to draw a parallel between the electric and gravitational forces. Instead of saying “ugh, this is confusing new stuff,” you are supposed to say, “oh, this is easy, because I already learned the concepts in Physics 103.”

7 If released, it gains kinetic energy and loses potential energy, but mechanical energy is conserved: E f =E i. The change in potential energy is U f - U i = -(W c ) i  f. A bit of review: y graphic “borrowed” from http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.htmlhttp://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html What force does W c ? Force due to gravity. x U i = mgy i U f = 0 yiyi Consider an object of mass m in a gravitational field. It has potential energy U(y) = mgy and “feels” a gravitational force F G = GmM/r 2, attractive.

8 + + + + + + + - - - - - - - - - - - - - - - - - - - + E A charged particle in an electric field has electric potential energy. It “feels” a force (as given by Coulomb’s law). It gains kinetic energy and loses potential energy if released. The Coulomb force does positive work, and mechanical energy is conserved. F

9 The next two slides define electrical potential energy.

10 ? on F E means the direction depends on the signs of the charges. Now that we realize the electric force is conservative, we can define a potential energy associated with it. The change in potential energy when a charge q 0 moves from point a to point b in the electric field of another charge q is b a FE?FE? r dr ds q q0q0 rara r ab The minus sign in this equation comes from the definition of change in potential energy. The sign from the dot product is “automatically” correct if you include the signs of q and q 0. The subscript “E” is to remind you this is electric potential energy. After this slide, I will drop the subscript “E.”

11 The next two slides use this definition of electrical potential energy to derive an equation for the electrical potential energy of two charged particles. is equivalent to your starting equation “i” and “f” refer to the two points for which we are calculating the potential energy difference. You could also use “a” and “b” like your text does, or “0” and “1” or anything else convenient. I use “i” and “f” because I always remember that  (anything) = (anything) f – (anything) i. (from the previous slide)

12 b a FE?FE? r dr ds q q0q0 rara r ab By convention, we choose electric potential energy to be zero at infinite separation of the charges. 0 0 If there are any math majors in the room, please close your eyes for a few seconds. We should be talking about limits. starting with an equation from two slides back… this diagram shows q 0 after it has moved from a to b

13 This provides us with the electric potential energy for a system of two point charges q and q 0, separated by a distance r: Example: calculate the electric potential energy for two protons separated by a typical proton-proton intranuclear distance of 2x10 -15 m. What is the meaning of the + sign in the result? You can call the charges q and q 0, or q 1 and q 2, or whatever you want. If you have more than two charged particles, simply add the potential energies for each unique pair of particles.

14 Really Important fact to keep straight. The change in potential energy is the negative of the work done by the conservative force which is associated with the potential energy (the electric force). If an external force* moves an object “against” the conservative force, then Always ask yourself which work you are calculating. *for example, if you push two negatively charged balloons together

16 Electric Potential Previously, we defined the electric field by the force it exerts on a test charge q 0 : Similarly, it is useful to define the potential of a charge in terms of the potential energy of a test charge q 0 : The electric potential V is independent of the test charge q 0.

17 From we see that the electric potential of a point charge q is The electric potential difference between points a and b is

18 Things to remember about electric potential:  Electric potential and electric potential energy are related, but are not the same. Electric potential difference is the work per unit of charge that must be done to move a charge from one point to another without changing its kinetic energy.  The terms “electric potential” and “potential” are used interchangeably.  The units of potential are joules/coulomb:

19 Things to remember about electric potential:  Only differences in electric potential and electric potential energy are meaningful. It is always necessary to define where U and V are zero. Here we defined V to be zero at an infinite distance from the sources of the electric field. Sometimes it is convenient to define V to be zero at the earth (ground). It should be clear from the context where V is defined to be zero, and I do not foresee you experiencing any confusion about where V is zero.

20 Two more starting equations: and so (potential is equal to potential energy per unit of charge) Potential energy and electric potential are defined relative to some reference point, so it is “better” to use

22 Two conceptual examples. Example: a proton is released in a region in space where there is an electric potential. Describe the subsequent motion of the proton. Example: an electron is released in a region in space where there is an electric potential. Describe the subsequent motion of the electron. The proton will move towards the region of lower potential. As it moves, its potential energy will decrease, and its kinetic energy and speed will increase. The electron will move towards the region of higher potential. As it moves, its potential energy will decrease, and its kinetic energy and speed will increase.

24 Electric Potential Energy of a System of Charges To find the electric potential energy for a system of two charges, we bring a second charge in from an infinite distance away: beforeafter q1q1 q1q1 q2q2 r

25 To find the electric potential energy for a system of three charges, we bring a third charge in from an infinite distance away: before q1q1 after q1q1 q2q2 r 12 q2q2 q3q3 r 13 r 23

26 Electric Potential and Potential Energy of a Charge Distribution Collection of charges: Charge distribution: P is the point at which V is to be calculated, and r i is the distance of the i th charge from P. P r dq Potential at point P.

27 Example: a 1  C point charge is located at the origin and a -4  C point charge 4 meters along the +x axis. Calculate the electric potential at a point P, 3 meters along the +y axis. q2q2 q1q1 3 m P 4 m x y

28 Example: how much work is required to bring a +3  C point charge from infinity to point P? An external force moves q 3 slowly from an infinite distance to the point P. q2q2 q1q1 3 m P 4 m x y q3q3 0 0 The work done by the external force was negative, so the work done by the electric field was positive. The electric field “pulled” q 3 in (keep in mind q 2 is 4 times as big as q 1 ). Positive work would have to be done by an external force to remove q 3 from P.

29 Example: find the total potential energy of the system of three charges. q2q2 q1q1 3 m P 4 m x y q3q3

31 The Electron Volt An electron volt (eV) is the energy acquired by a particle of charge e when it moves through a potential difference of 1 volt. This is a very small amount of energy on a macroscopic scale, but electrons in atoms typically have a few eV (10’s to 1000’s) of energy.

32 Today’s agendum: Electric potential of a charge distribution. You must be able to calculate the electric potential for a charge distribution. Equipotentials. You must be able to sketch and interpret equipotential plots. Potential gradient. You must be able to calculate the electric field if you are given the electric potential. Potentials and fields near conductors. You must be able to use what you have learned about electric fields, Gauss’ “Law”, and electric potential to understand and apply several useful facts about conductors in electrostatic equilibrium.

33 Electric Potential of a Charge Distribution Example: potential and electric field between two parallel conducting plates. Assume V 0 <V 1 (so we can determine the direction of the electric field). Also assume the plates are large compared to their separation, so the electric field is constant and perpendicular to the plates. V0V0 V1V1 E Also, let the plates be separated by a distance d. d

34 V0V0 V1V1 E d d x y z I’ll discuss in lecture why the absolute value signs are needed.

35 Important note: the derivation of did not require rectangular plates, or any plates at all. It works as long as E is uniform. In general, E should be replaced by the component of along the displacement vector.

36 Example: A rod of length L located along the x-axis has a total charge Q uniformly distributed along the rod. Find the electric potential at a point P along the y-axis a distance d from the origin. y x P d L dq x dx r =Q/L dq= dx

37 y x P d L dq x dx r A good set of math tables will have the integral:

38 Example: Find the electric potential due to a uniformly charged ring of radius R and total charge Q at a point P on the axis of the ring. P R dq r x x Every dq of charge on the ring is the same distance from the point P.

39 P R dq r x x Could you use this expression for V to calculate E? Would you get the same result as we obtained previously?

40 P a dq r x dE  

41 Example: A disc of radius R has a uniform charge per unit area  and total charge Q. Calculate V at a point P along the central axis of the disc at a distance x from its center. P r dq x x R The disc is made of concentric rings. The area of a ring at a radius r is 2  rdr, and the charge on each ring is  (2  rdr). We can use the equation for the potential due to a ring, replace R by r, and integrate from r=0 to r=R.

42 P r dq x x R

43 P r dq x x R Could you use this expression for V to calculate E?

44 See your text for other examples of potentials calculated from charge distributions, as well as an alternate discussion of the electric field between charged parallel plates.

45 PRACTICAL APPLICATION For some reason you think practical applications are important. Well, I found one!

47 Equipotentials Equipotentials are contour maps of the electric potential. http://www.omnimap.com/catalog/digital/topo.htm

48 The electric field must be perpendicular to equipotential lines. Why? Otherwise work would be required to move a charge along an equipotential surface, and it would not be equipotential. In the static case (charges not moving) the surface of a conductor is an equipotential surface. Why? Otherwise charge would flow and it wouldn’t be a static case. Equipotential lines are another visualization tool. They illustrate where the potential is constant. Equipotential lines are actually projections on a 2-dimensional page of a 3- dimensional equipotential surface. (“Just like” the contour map.)

49 Here are some electric field and equipotential lines I generated using an electromagnetic field program. Equipotential lines are shown in red.

51 Potential Gradient (Determining Electric Field from Potential) The electric field vector points from higher to lower potentials. More specifically, E points along shortest distance from a higher equipotential surface to a lower equipotential surface. You can use E to calculate V: You can use the differential version of this equation to calculate E from a known V:

52 For spherically symmetric charge distribution: In one dimension: In three dimensions:

53 Calculate -dV/d(whatever) including all signs. If the result is +, E vector points along the +(whatever) direction. If the result is -, E vector points along the –(whatever) direction.

54 Example: In a region of space, the electric potential is V(x,y,z) = Axy 2 + Bx 2 + Cx, where A = 50 V/m 3, B = 100 V/m 2, and C = -400 V/m are constants. Find the electric field at the origin.

56 When there is a net flow of charge inside a conductor, the physics is generally complex. Potentials and Fields Near Conductors When there is no net flow of charge, or no flow at all (the electrostatic case), then a number of conclusions can be reached using Gauss’ “Law” and the concepts of electric fields and potentials…

57 The electric field inside a conductor is zero. Summary of key points (electrostatic case): Any net charge on the conductor lies on the outer surface. The potential on the surface of a conductor, and everywhere inside, is the same. The electric field just outside a conductor must be perpendicular to the surface. Equipotential surfaces just outside the conductor must be parallel to the conductor’s surface.

58 Another key point: the charge density on a conductor surface will vary if the surface is irregular, and surface charge collects at “sharp points.” Therefore the electric field is large (and can be huge) near “sharp points.” Another Practical Application To best shock somebody, don’t touch them with your hand; touch them with your fingertip. Better yet, hold a small piece of bare wire in your hand and gently touch them with that.

59 Today’s agendum: Capacitance. You must be able to apply the equation C=Q/V. Capacitors: parallel plate, cylindrical, spherical. You must be able to calculate the capacitance of capacitors having these geometries, and you must be able to use the equation C=Q/V to calculate parameters of capacitors. Circuits containing capacitors in series and parallel. You must be understand the differences between, and be able to calculate the “equivalent capacitance” of, capacitors connected in series and parallel.

60 Capacitors and Dielectrics Capacitance A capacitor is basically two parallel conducting plates with air or insulating material in between. V0V0 V1V1 E L A capacitor doesn’t have to look like metal plates. Capacitor for use in high-performance audio systems.

61 When a capacitor is connected to an external potential, charges flow onto the plates and create a potential difference between the plates. Capacitor plates build up charge. The battery in this circuit has some voltage V. We haven’t discussed what that means yet. The symbol representing a capacitor in an electric circuit looks like parallel plates. Here’s the symbol for a battery, or an external potential. + - - - V

62 In a battery, chemical reactions release energy. The energy is expended by transporting charged particles from one terminal of the battery to the other. A “charge separation” is maintained.

63 If the external potential is disconnected, charges remain on the plates, so capacitors are good for storing charge (and energy). Capacitors are also very good at releasing their stored charge all at once. The capacitors in your tube-type TV are so good at storing energy that touching the two terminals at the same time can be fatal, even though the TV may not have been used for months. High-voltage TV capacitors are supposed to have “bleeder resistors” that drain the charge away after the circuit is turned off. I wouldn’t bet my life on it. Graphic from http://www.feebleminds-gifs.com/.http://www.feebleminds-gifs.com/. + + - - V conducting wires

64 assortment of capacitors

65 The magnitude of charge acquired by each plate of a capacitor is Q=CV where C is the capacitance of the capacitor. The unit of C is the farad but most capacitors have values of C ranging from picofarads to microfarads (pF to  F). micro  10 -6, nano  10 -9, pico  10 -12 (Know for exam!) C is always positive. +Q + -Q - V C Here’s this V again. It is the potential difference provided by the “external potential”. For example, the voltage of a battery.

67 Parallel Plate Capacitance V0V0 V1V1 E d We previously calculated the electric field between two parallel charged plates: This is valid when the separation is small compared with the plate dimensions. We also showed that E and  V are related: +Q-Q A This lets us calculate C for a parallel plate capacitor.

68 Reminders: Q is the magnitude of the charge on either plate. V is actually the magnitude of the potential difference between the plates. V is really |  V|. Your book calls it V ab. C is always positive.

69 V0V0 V1V1 E d +Q-Q A Parallel plate capacitance depends “only” on geometry. This expression is approximate, and must be modified if the plates are small, or separated by a medium other than a vacuum. Greek letter Kappa. For today’s lecture use Kappa=1.

70 Isolated Sphere Capacitance An isolated sphere can be thought of as concentric spheres with the outer sphere at an infinite distance and zero potential. We already know the potential outside a conducting sphere: The potential at the surface of a charged sphere of radius R is so the capacitance at the surface of an isolated sphere is

71 Capacitance of Concentric Spheres Let’s calculate the capacitance of a concentric spherical capacitor of charge Q. In between the spheres You need to do this derivation if you have a problem on spherical capacitors! +Q -Q b a

72 Let a  R and b  to get the capacitance of an isolated sphere. +Q -Q b a alternative calculation of capacitance of isolated sphere

73 We can also calculate the capacitance of a cylindrical capacitor (made of coaxial cylinders). L Coaxial Cylinder Capacitance The next slide shows a cross-section view of the cylinders.

74 Q -Q b r a E d Gaussian surface Lowercase c is capacitance per unit length: This derivation is sometimes needed for homework problems!

75 Example: calculate the capacitance of a capacitor whose plates are 20 cm x 3 cm and are separated by a 1.0 mm air gap. d = 0.001 area = 0.2 x 0.03 If you keep everything in SI (mks) units, the result is “automatically” in SI units.

76 Example: what is the charge on each plate if the capacitor is connected to a 12 volt* battery? 0 V +12 V  V= 12V *Remember, it’s the potential difference that matters. If you keep everything in SI (mks) units, the result is “automatically” in SI units.

77 Example: what is the electric field between the plates? 0 V +12 V  V= 12V d = 0.001 E If you keep everything in SI (mks) units, the result is “automatically” in SI units.

79 Capacitors in Circuits Recall: this is the symbol representing a capacitor in an electric circuit. And this is the symbol for a battery… + - …or this… …or this.

80 Capacitors connected in parallel: C1C1 C2C2 C3C3 + - V The potential difference (voltage drop) from a to b must equal V. a b V ab = V = voltage drop across each individual capacitor. V ab Circuits Containing Capacitors in Parallel Note how I have introduced the idea that when circuit components are connected in parallel, then the voltage drops across the components are all the same. You may use this fact in homework solutions.

81 C1C1 C2C2 C3C3 + - V a Q = C V  Q 1 = C 1 V & Q 2 = C 2 V & Q 3 = C 3 V Now imagine replacing the parallel combination of capacitors by a single equivalent capacitor. By “equivalent,” we mean “stores the same total charge if the voltage is the same.” C eq + - V a Q 1 + Q 2 + Q 3 = C eq V = Q Q3Q3 Q2Q2 Q1Q1 + - Q Important!

82 Q 1 = C 1 VQ 2 = C 2 VQ 3 = C 3 V Q 1 + Q 2 + Q 3 = C eq V Summarizing the equations on the last slide: Using Q 1 = C 1 V, etc., gives C 1 V + C 2 V + C 3 V = C eq V C 1 + C 2 + C 3 = C eq (after dividing both sides by V) Generalizing: C eq =  C i (capacitors in parallel) C1C1 C2C2 C3C3 + - V a b

83 Capacitors connected in series: C1C1 C2C2 + - V C3C3 An amount of charge +Q flows from the battery to the left plate of C 1. (Of course, the charge doesn’t all flow at once). +Q-Q An amount of charge -Q flows from the battery to the right plate of C 3. Note that +Q and –Q must be the same in magnitude but of opposite sign. Circuits Containing Capacitors in Series

84 C1C1 C2C2 + - V C3C3 +Q A -Q B The charges +Q and –Q attract equal and opposite charges to the other plates of their respective capacitors: -Q +Q These equal and opposite charges came from the originally neutral circuit regions A and B. Because region A must be neutral, there must be a charge +Q on the left plate of C 2. Because region B must be neutral, there must be a charge --Q on the right plate of C 2. +Q -Q

85 C1C1 C2C2 + - V C3C3 +Q A -Q B +Q -Q Q = C 1 V 1 Q = C 2 V 2 Q = C 3 V 3 The charges on C 1, C 2, and C 3 are the same, and are But we don’t know V 1, V 2, and V 3 yet. a b We do know that V ab = V and also V ab = V 1 + V 2 + V 3. V3V3 V2V2 V1V1 V ab Note how I have introduced the idea that when circuit components are connected in series, then the voltage drop across all the components is the sum of the voltage drops across the individual components. This is actually a consequence of the conservation of energy.

86 C eq + - V +Q-Q V Let’s replace the three capacitors by a single equivalent capacitor. By “equivalent” we mean V is the same as the total voltage drop across the three capacitors, and the amount of charge Q that flowed out of the battery is the same as when there were three capacitors. Q = C eq V

87 Collecting equations: Q = C 1 V 1 Q = C 2 V 2 Q = C 3 V 3 V ab = V = V 1 + V 2 + V 3. Q = C eq V Substituting for V 1, V 2, and V 3 : Substituting for V: Dividing both sides by Q: Important!

88 Generalizing: (capacitors in series)

89 C3C3 C2C2 C1C1 I don’t see a series combination of capacitors, but I do see a parallel combination. C 23 = C 2 + C 3 = C + C = 2C Example: determine the capacitance of a single capacitor that will have the same effect as the combination shown. Use C 1 = C 2 = C 3 = C.

90 C 1 = C C 23 = 2C Now I see a series combination.

91 Example: for the capacitor circuit shown, C 1 = 3  F, C 2 = 6  F, C 3 = 2  F, and C 4 =4  F. (a) Find the equivalent capacitance. (b) if  V=12 V, find the potential difference across C 4. I’ll work this at the blackboard. C3C3 C2C2 C1C1 C4C4 VV

92 Overview Electric charge and electric force Coulomb’s Law Electric field calculating electric field motion of a charged particle in an electric field Gauss’ “Law” electric flux calculating electric field using Gaussian surfaces properties of conductors

93 Overview Electric potential and electric potential energy calculating potentials and potential energy calculating fields from potentials equipotentials potentials and fields near conductors Capacitors capacitance of parallel plates, concentric cylinders, (concentric spheres not for this exam) equivalent capacitance of capacitor network Don’t forget the concepts from Physics 103 that were frequently used!

94 F1F1 F2F2 +Q -Q +Q P a x y   Three charges +Q, +Q, and –Q, are located at the corners of an equilateral triangle with sides of length a. What is the force on the charge located at point P (see diagram)?

95 F1F1 F2F2 +Q -Q +Q P a x y  

96 F1F1 F2F2 +Q -Q +Q P a x y   You can repeat the above calculation, replacing F by E. Or… What is the electric field at P due to the two charges at the base of the triangle?

97 An insulating spherical shell has an inner radius a and outer radius b. The shell has a total charge Q and a uniform charge density. Find the magnitude of the electric field for r<a.

98 An insulating spherical shell has an inner radius a and outer radius b. The shell has a total charge Q and a uniform charge density. Find the magnitude of the electric field for a<r<b.

99 An electron has a speed v. Calculate the magnitude and direction of an electric field that will stop this electron in a distance D. Do you understand that these E’s have different meanings? Use magnitudes if you have determined direction of E by other means. Know where this comes from!

100 Q Q P a Two equal positive charges Q are located at the base of an equilateral triangle with sides of length a. What is the potential at point P (see diagram)?

101 Q Q P a Q Three equal positive charges Q are located at the corners of an equilateral triangle with sides of length a. What is the potential energy of the charge located at point P (see diagram)?

102 For the capacitor system shown, C 1 =6.0  F, C 2 =2.0  F, and C 3 =10.0  F. (a) Find the equivalent capacitance. V0V0 C1C1 C2C2 C3C3

103 For the capacitor system shown, C 1 =6.0  F, C 2 =2.0  F, and C 3 =10.0  F. The charge on capacitor C 3 is found to be 30.0  C. Find V 0. V0V0 C1C1 C2C2 C3C3

104 Today’s agendum: Energy Storage in Capacitors. You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered. Dielectrics. You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

105 Energy Storage in Capacitors Let’s calculate how much work it takes to charge a capacitor. The work required for an external force to move a charge dq through a potential difference  V is dW = dq  V. VV +- +q-q + dq From Q=C  V (   V = q/C): q is the amount of charge on the capacitor during the time the charge dq is being moved. We start with zero charge on the capacitor, and end up with Q, so

106 The work required to charge the capacitor is the amount of energy you get back when you discharge the capacitor (because the electric force is conservative). Thus, the work required to charge the capacitor is equal to the potential energy stored in the capacitor. Because C, Q, and V are related through Q=CV, there are three equivalent ways to write the potential energy.

107 All three equations are valid; use the one most convenient for the problem at hand. It is no accident that we use the symbol U for the energy stored in a capacitor. It is just another “version” of electrical potential energy. You can use it in your energy conservation equations just like any other form of potential energy!

108 Example: a camera flash unit stores energy in a 150  F capacitor at 200 V. How much electric energy can be stored? If you keep everything in SI (mks) units, the result is “automatically” in SI units.

109 Example: compare the amount of energy stored in a capacitor with the amount of energy stored in a battery. A 12 V car battery rated at 100 ampere-hours stores 3.6x10 5 C of charge and can deliver at least 4.3x10 6 joules of energy (we’ll learn how to calculate that later in the course). If a battery stores so much more energy, why use capacitors? 10 6 joules of energy are stored at high voltage in capacitor banks, and released during a period of a few milliseconds. The enormous current produces incredibly high magnetic fields.high magnetic fields A 100  F capacitor that stores 3.6x10 5 C at 12 V stores an amount of energy U=CV 2 /2=7.2x10 -3 joules. If you want your capacitor to store lots of energy, store it at a high voltage.

110 Energy Stored in Electric Fields VV +- +Q-Q E d area A Energy is stored in the capacitor: The “volume of the capacitor” is Volume=Ad

111 Energy stored per unit volume (u): VV +- +Q-Q E d area A The energy is “stored” in the electric field! We’ve gone from the concrete (electric charges experience forces)… …to the abstract (electric charges create electric fields)… …to an application of the abstraction (electric field contains energy).

112 VV +- +Q-Q E f area A This is not a new “kind” of energy. It’s the electric potential energy resulting from the coulomb force between charged particles. Or you can think of it as the electric energy due to the field created by the charges. Same thing. “The energy in electromagnetic phenomena is the same as mechanical energy. The only question is, ‘Where does it reside?’ In the old theories, it resides in electrified bodies. In our theory, it resides in the electromagnetic field, in the space surrounding the electrified bodies.”—James Maxwell

113 Today’s agendum: Energy Storage in Capacitors. You must be able to calculate the energy stored in a capacitor, and apply the energy storage equations to situations where capacitor configurations are altered. Dielectrics. You must understand why dielectrics are used, and be able include dielectric constants in capacitor calculations.

114 If an insulating sheet (“dielectric”) is placed between the plates of a capacitor, the capacitance increases by a factor , which depends on the material in the sheet.  is the dielectric constant of the material. dielectric In general, C =  0 A / d.  is 1 for a vacuum, and  1 for air. (You can also define  =  0 and write C =  A / d). Dielectrics

115 The dielectric is the thin insulating sheet in between the plates of a capacitor. dielectric Any reasons to use a dielectric in a capacitor?  Lets you apply higher voltages (so more charge).  Lets you place the plates closer together (make d smaller).  Increases the value of C because  >1.  Makes your life as a physics student more complicated.

116 Example: a parallel plate capacitor has an area of 10 cm 2 and plate separation 5 mm. 300 V is applied between its plates. If neoprene is inserted between its plates, how much charge does the capacitor hold.  V=300 V d=5 mm A=10 cm 2  =6.7

117 Example: how much charge would the capacitor on the previous slide hold if the dielectric were air? The calculation is the same, except replace 6.7 by 1. Or just divide the charge on the previous page by 6.7 to get.  V=300 V d=5 mm A=10 cm 2  =1

118 A capacitor connected as shown acquires a charge Q. V While the capacitor is still connected to the battery, a dielectric material is inserted. Will Q increase, decrease, or stay the same? Why? V V=0 Conceptual Example

119 Example: find the energy stored in the capacitor in slide 13.  V=300 V d=5 mm A=10 cm 2  =6.7

120 Example: the battery is now disconnected. What are the charge, capacitance, and energy stored in the capacitor?  V=300 V d=5 mm A=10 cm 2  =6.7 The charge and capacitance are unchanged, so the voltage drop and energy stored are unchanged.

121 Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor?  V=300 V d=5 mm A=10 cm 2  =6.7  V=? d=5 mm

122  V=? d=5 mm A=10 cm 2 Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor? The charge remains unchanged, because there is nowhere for it to go.

123  V=? d=5 mm A=10 cm 2 Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor? Knowing C and Q we can calculate the new potential difference.

124  V=2020 V d=5 mm A=10 cm 2 Example: the dielectric is removed without changing the plate separation. What are the capacitance, charge, potential difference, and energy stored in the capacitor?

125 Huh?? The energy stored increases by a factor of  ?? Sure. It took work to remove the dielectric. The stored energy increased by the amount of work done.

1. 2 Today’s agendum: Electric potential energy. You must be able to use electric potential energy in work-energy calculations. Electric potential. You.

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1. 2 Today’s agendum: Electric potential energy. You must be able to use electric potential energy in work-energy calculations. Electric potential. You.

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Presentation on theme: "1. 2 Today’s agendum: Electric potential energy. You must be able to use electric potential energy in work-energy calculations. Electric potential. You."— Presentation transcript:

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About project

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