1. Chapter 20 Electric Potential and Capacitance

2. Electric Potential Energy When a point charge, q o, is placed in an electric field, it experiences a force The force is conservative To find the work associated with the force, look at ds, an infinitesimal displacement vector that is oriented tangent to a path through space

3. Electric Potential Energy, cont The work done by the electric field is As this work is done by the field, the potential energy of the charge-field system is changed by  U For a finite displacement of the charge from A to B,

4. Electric Potential Energy, final Because is conservative, the line integral does not depend on the path taken by the charge This is the change in potential energy of the system

5. Electric Potential The potential energy per unit charge, U/q o, is the electric potential The potential is independent of the value of q o The potential has a value at every point in an electric field The electric potential is

6. The potential is a scalar quantity Since energy is a scalar The potential is a property of only the field Not the charge-field system As a charged particle moves in an electric field, it will experience a change in potential Electric Potential, cont

7. Electric Potential, final We often take the value of the potential to be zero at some convenient point in the field Sometimes called ground Electric potential is a scalar characteristic of an electric field, independent of any charges that may be placed in the field

8. Potential and Potential Energy The potential is characteristic of the field only It is independent of the charge placed in the field The difference in potential is proportional to the difference in potential energy Potential energy is characteristic of the charge-field system Due to an interaction between the field and a charged particle placed in the field

9. Work and Electric Potential The electric potential at an arbitrary point due to source charges equals the work required by an external agent to bring a test charge from infinity to that point divided by the charge on the test particle Assumes a charge moves in an electric field without any change in its kinetic energy The work performed on the charge is W =  V = q  V

10. Units 1 V = 1 J/C V is a Volt It takes one Joule of work to move a 1 Coulomb charge through a potential difference of 1 Volt In addition, 1 N/C = 1 V/m This indicates we can interpret the electric field as a measure of the rate of change with position of the electric potential

11. Electron-Volts Another unit of energy that is commonly used in atomic and nuclear physics is the electron-volt One electron-volt is defined as the energy a charge-field system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt 1 eV = 1.60 x 10 -19 J

12. Potential Difference in a Uniform Field The equations for electric potential can be simplified if the electric field is uniform: The negative sign indicates that the electric potential at B is lower than at point A

13. Energy and the Direction of Electric Field When the electric field is directed downward, point B is at a lower potential than point A When a positive test charge moves from A to B, the charge- field system loses potential energy

14. More About Directions A system consisting of a positive charge and an electric field loses electric potential energy when the charge moves in the direction of the field An electric field does work on a positive charge when the charge moves in the direction of the electric field The charged particle gains kinetic energy equal to the potential energy lost by the charge-field system Another example of Conservation of Energy

15. Directions, cont If q o is negative, then  U is positive A system consisting of a negative charge and an electric field gains potential energy when the charge moves in the direction of the field In order for a negative charge to move in the direction of the field, an external agent must do positive work on the charge

16. Equipotentials Point B is at a lower potential than point A Points B and C are at the same potential The name equipotential surface is given to any surface consisting of a continuous distribution of points having the same electric potential

17. Charged Particle in a Uniform Field, Example A positive charge is released from rest and moves in the direction of the electric field The change in potential is negative The change in potential energy is negative The force and acceleration are in the direction of the field

18. Potential and Point Charges A positive point charge produces a field directed radially outward The potential difference between points A and B will be

19. Potential and Point Charges, cont The electric potential is independent of the path between points A and B It is customary to choose a reference potential of V = 0 at r A =  Then the potential at some point r is

20. Electric Potential with Multiple Charges The electric potential due to several point charges is the sum of the potentials due to each individual charge This is another example of the superposition principle The sum is the algebraic sum V = 0 at r = 

21. Potential Energy of Multiple Charges Consider two charged particles The potential energy of the system is

22. More About U of Multiple Charges If the two charges are the same sign, U is positive and work must be done to bring the charges together If the two charges have opposite signs, U is negative and work is done to keep the charges apart

23. U with Multiple Charges, final If there are more than two charges, then find U for each pair of charges and add them Add them algebraically The total electric potential energy of a system of point charges is equal to the work required to bring the charges, one at a time, from an infinite separate to their final positions

24. Potential of a Charge Now remove one charge The potential due to charge q 2 is The equipotential surfaces for an isolated point charge are a family of spheres concentric with the charge

25. Finding E From V Assume, to start, that E has only an x component Similar statements would apply to the y and z components Equipotential surfaces must always be perpendicular to the electric field lines passing through them

26. E and V for an Infinite Sheet of Charge The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines

27. E and V for a Point Charge The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines

28. E and V for a Dipole The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines

29. Electric Field from Potential, General In general, the electric potential is a function of all three dimensions Given V (x, y, z) you can find E x, E y and E z as partial derivatives

30. Electric Potential for a Continuous Charge Distribution Consider a small charge element dq Treat it as a point charge The potential at some point due to this charge element is

31. V for a Continuous Charge Distribution, cont To find the total potential, you need to integrate to include the contributions from all the elements This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions

32. Problem Solving Strategies – Electric Potentials Conceptualize Think about the charges or the charge distribution Image the type of potential they would create This establishes a mental representation Use any symmetry in the arrangement of the charges to help you visualize the potential

33. Problem Solving Strategies – Electric Potentials Categorize Individual charges or a distribution? Analyze Scalar, so no components Superposition principle is algebraic sum Signs are important Changes in potential are what is important The point where V = 0 is arbitrary But usually at a point infinitely far from the charges

34. Problem Solving Strategies – Electric Potentials, cont Analyze, cont For a group of individual charges, use the superposition principle For a continuous charge distribution, integrate over the entire distribution If is known, the line integral of can be evaluated

35. Problem Solving Strategies – Electric Potentials, final Finalize Check to see if your result is consistent with the mental representation Be sure the result reflects any symmetry you noted Image varying parameters to see if the mathematical result changes in a reasonable way

36. V for a Uniformly Charged Ring P is located on the perpendicular central axis of the uniformly charged ring The ring has a radius a and a total charge Q

37. V for a Uniformly Charged Sphere A solid sphere of radius R and total charge Q For r  R, For r  R,

38. V for a Uniformly Charged Sphere, Graph The curve for V D is for the potential inside the curve It is parabolic It joins smoothly with the curve for V B The curve for V B is for the potential outside the sphere It is a hyperbola

39. V Due to a Charged Conductor Consider two points on the surface of the charged conductor as shown is always perpendicular to to the displacement Therefore, = 0 Therefore, the potential difference between A and B is also zero

40. V Due to a Charged Conductor, cont V is constant everywhere on the surface of a charged conductor in equilibrium  V = 0 between any two points on the surface The surface of any charged conductor in electrostatic equilibrium is an equipotential surface Because the electric field is zero inside the conductor, we conclude that the electric potential is constant everywhere inside the conductor and equal to the value at the surface

41. E Compared to V The electric potential is a function of r The electric field is a function of r 2 The effect of a charge on the space surrounding it The charge sets up a vector electric field which is related to the force The charge sets up a scalar potential which is related to the energy

42. Irregularly Shaped Objects The charge density is high where the radius of curvature is small And low where the radius of curvature is large The electric field is large near the convex points having small radii of curvature and reaches very high values at sharp points

43. Cavity in a Conductor Assume an irregularly shaped cavity is inside a conductor Assume no charges are inside the cavity The electric field inside the conductor is must be zero

44. Cavity in a Conductor, cont The electric field inside does not depend on the charge distribution on the outside surface of the conductor For all paths between A and B, A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity

45. Capacitors Capacitors are devices that store electric charge The capacitor is the first example of a circuit element A circuit generally consists of a number of electrical components (called circuit elements) connected together by conducting wires forming one or more closed loops

46. Definition of Capacitance The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors The SI unit of capacitance is a farad (F)

47. Makeup of a Capacitor A capacitor consists of two conductors When the conductors are charged, they carry charges of equal magnitude and opposite directions A potential difference exists between the conductors due to the charge The capacitor stores charge

48. More About Capacitance Capacitance will always be a positive quantity The capacitance of a given capacitor is constant The capacitance is a measure of the capacitor’s ability to store charge The Farad is a large unit, typically you will see microfarads (  F) and picofarads (pF) The capacitance of a device depends on the geometric arrangement of the conductors

49. Capacitance – Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity Note, this is independent of the charge and the potential difference

50. Parallel Plate Capacitor Each plate is connected to a terminal of the battery If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires

51. Capacitance – Parallel Plates The charge density on the plates is  = Q/A A is the area of each plate, which are equal Q is the charge on each plate, equal with opposite signs The electric field is uniform between the plates and zero elsewhere

52. Capacitance – Parallel Plates, cont. The capacitance is proportional to the area of its plates and inversely proportional to the plate separation

53. Parallel Plate Assumptions The assumption that the electric field is uniform is valid in the central region, but not at the ends of the plates If the separation between the plates is small compared with the length of the plates, the effect of the non-uniform field can be ignored

54. Energy in a Capacitor – Overview Consider the circuit to be a system Before the switch is closed, the energy is stored as chemical energy in the battery When the switch is closed, the energy is transformed from chemical to electric potential energy

55. Energy in a Capacitor – Overview, cont The electric potential energy is related to the separation of the positive and negative charges on the plates A capacitor can be described as a device that stores energy as well as charge

56. Capacitance of a Cylindrical Capacitor From Gauss’ Law, the field between the cylinders is E = 2 k e / r  V = -2 k e ln (b/a) The capacitance becomes

57. Circuit Symbols A circuit diagram is a simplified representation of an actual circuit Circuit symbols are used to represent the various elements Lines are used to represent wires The battery’s positive terminal is indicated by the longer line

58. Capacitors in Parallel When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged

59. Capacitors in Parallel, 2 The flow of charges ceases when the voltage across the capacitors equals that of the battery The capacitors reach their maximum charge when the flow of charge ceases The total charge is equal to the sum of the charges on the capacitors Q = Q 1 + Q 2 The potential difference across the capacitors is the same And each is equal to the voltage of the battery

60. Capacitors in Parallel, 3 The capacitors can be replaced with one capacitor with a capacitance of C eq The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors

61. Capacitors in Parallel, final C eq = C 1 + C 2 + … The equivalent capacitance of a parallel combination of capacitors is the algebraic sum of the individual capacitances and is larger than any of the individual capacitances

62. Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery

63. Capacitors in Series, 2 As this negative charge accumulates on the right plate of C 2, an equivalent amount of negative charge is removed from the left plate of C 2, leaving it with an excess positive charge All of the right plates gain charges of –Q and all the left plates have charges of +Q

64. Capacitors in Series, 3 An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage

65. Capacitors in Series, final The equivalent capacitance of a series combination is always less than any individual capacitor in the combination

66. Summary and Hints Be careful with the choice of units In SI, capacitance is in F, distance is in m and the potential differences in V Electric fields can be in V/m or N/c When two or more capacitors are connected in parallel, the potential differences across them are the same The charge on each capacitor is proportional to its capacitance The capacitors add directly to give the equivalent capacitance

67. Summary and Hints, cont When two or more capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor

68. Equivalent Capacitance, Example The 1.0  F and 3.0  F are in parallel as are the 6.0  F and 2.0  F These parallel combinations are in series with the capacitors next to them The series combinations are in parallel and the final equivalent capacitance can be found

69. Energy Stored in a Capacitor Assume the capacitor is being charged and, at some point, has a charge q on it The work needed to transfer a charge from one plate to the other is The total work required is

70. Energy, cont The work done in charging the capacitor appears as electric potential energy U This applies to a capacitor of any geometry The energy stored increases as the charge increases and as the potential difference increases In practice, there is a maximum voltage before discharge occurs between the plates

71. Energy, final The energy can be considered to be stored in the electric field For a parallel plate capacitor, the energy can be expressed in terms of the field as U = 1 / 2 (  o Ad)E 2 It can also be expressed in terms of the energy density (energy per unit volume) u E = 1 / 2  o E 2

72. Capacitors with Dielectrics A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance Dielectrics include rubber, plastic, or waxed paper With a dielectric, C =  C o The capacitance is multiplied by the factor κ when the dielectric completely fills the region between the plates For a parallel plate capacitor, this becomes C = =  o (A/d)

73. Dielectrics, cont In theory, d could be made very small to create a very large capacitance In practice, there is a limit to d d is limited by the electric discharge that could occur though the dielectric medium separating the plates For a given d, the maximum voltage that can be applied to a capacitor without causing a discharge depends on the dielectric strength of the material

74. Dielectrics, final Dielectrics provide the following advantages Increase in capacitance Increase the maximum operating voltage Possible mechanical support between the plates This allows the plates to be close together without touching This decreases d and increases C

75. Dielectrics – An Atomic View The molecules that make up the dielectric are modeled as dipoles The molecules are randomly oriented in the absence of an electric field

76. Dielectrics – An Atomic View, cont An external electric field is applied This produces a torque on the molecules The molecules partially align with the electric field

77. Dielectrics – An Atomic View, final An external field can polarize the dielectric whether the molecules are polar or nonpolar The charged edges of the dielectric act as a second pair of plates producing an induced electric field in the direction opposite the original electric field

78. Table of Some Dielectric Values

79. Types of Capacitors – Tubular Metallic foil may be interlaced with thin sheets of paper or Mylar The layers are rolled into a cylinder to form a small package for the capacitor

80. Types of Capacitors – Oil Filled Common for high voltage capacitors A number of interwoven metallic plates are immersed in silicon oil

81. Types of Capacitors – Variable Variable capacitors consist of two interwoven sets of metallic plates One plate is fixed and the other is moveable The capacitor generally vary between 10 and 500 pF

82. Types of Capacitors – Electrolytic Is used to store large amounts of charge at relatively low voltages The electrolyte is a solution that conducts electricity by virtue of motion of ions contained in the solution

83. The Atmosphere as a Capacitor A negative charge occurs on the Earth’s surface Positive charges are distributed through the atmosphere This separation of charge can be modeled as a capacitor

84. Atmosphere, cont. The charge distribution on the surface is assumed to be spherically symmetric The effective distance between the plates is 5 km Based on modeling the charge distribution in the atmosphere The capacitance of this system is approximately 0.9 F