Electric Potential

1. Electric Potential

2. Electric Potential Energy and Potential Difference • Relation between Electric Potential and Electric Field • Electric Potential Due to Point Charges • Potential Due to Any Charge Distribution • Equipotential Surfaces • Electric Dipole Potential

3. E Determined from V • ElectrostaticPotential Energy; the Electron Volt • Cathode Ray Tube: TV and Computer Monitors, Oscilloscope

4. Electrostatic Potential Energy and Potential Difference The electrostatic force is conservative – potential energy can be defined. Change in electric potential energy is negative of work done by electric force:

5. Electrostatic Potential Energy and Potential Difference Electric potential is defined as potential energy per unit charge: Unit of electric potential: the volt (V): 1 V = 1 J/C.

6. Electrostatic Potential Energy and Potential Difference Only changes in potential can be measured, allowing free assignment of V = 0:

7. Electrostatic Potential Energy and Potential Difference A negative charge. Suppose a negative charge, such as an electron, is placed near the negative plate at point b, as shown here. If the electron is free to move, will its electric potential energy increase or decrease? How will the electric potential change?

8. Electrostatic Potential Energy and Potential Difference Analogy between gravitational and electrical potential energy:

9. Electric Potential Energy II 1)proton 2)electron 3)both feel the same acceleration 4) neither – there is no acceleration 5) they feel the same magnitude acceleration but opposite direction A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. Which has the larger acceleration? Electron electron - + Proton proton

10. Electric Potential Energy II 1)proton 2)electron 3)both feel the same acceleration 4) neither – there is no acceleration 5) they feel the same magnitude acceleration but opposite direction A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. Which has the larger acceleration? Electron Since F = maand the electron is much less massive than the proton, the electron experiences the larger acceleration. electron - + Proton proton

11. Electric Potential Energy III 1)proton 2)electron 3)both acquire the same KE 4) neither – there is no change of KE 5) they both acquire the same KE but with opposite signs A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. When it strikes the opposite plate, which one has more KE? Electron electron - + Proton proton

12. Electric Potential Energy III 1)proton 2)electron 3)both acquire the same KE 4) neither – there is no change of KE 5) they both acquire the same KE but with opposite signs A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. When it strikes the opposite plate, which one has more KE? Since PE = qV and the proton and electron have the same charge in magnitude, they both have the same electric potential energy initially. Because energy is conserved, they both must have the same kinetic energy after they reach the opposite plate. Electron electron - + Proton proton

13. Electrostatic Potential Energy and Potential Difference Electrical sources such as batteries and generators supply a constant potential difference. Here are some typical potential differences, both natural and manufactured:

14. Electrostatic Potential Energy and Potential Difference Electron in CRT. Suppose an electron in a cathode ray tube is accelerated from rest through a potential difference Vb – Va = Vba = +5000 V. (a) What is the change in electric potential energy of the electron? (b) What is the speed of the electron (m = 9.1 × 10-31 kg) as a result of this acceleration?

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16. Relation between Electric Potential and Electric Field The general relationship between a conservative force and potential energy: Substituting the potential difference and the electric field:

17. Relation between Electric Potential and Electric Field The simplest case is a uniform field:

18. Relation between Electric Potential and Electric Field Electric field obtained from voltage. Two parallel plates are charged to produce a potential difference of 50 V. If the separation between the plates is 0.050 m, calculate the magnitude of the electric field in the space between the plates.

19. Solution:

20. Relation between Electric Potential and Electric Field Charged conducting sphere. Determine the potential at a distance r from the center of a uniformly charged conducting sphere of radius r0 for (a) r > r0, (b) r = r0, (c) r < r0. The total charge on the sphere is Q.

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22. Solution:

23. Relation between Electric Potential and Electric Field The previous example gives the electric potential as a function of distance from the surface of a charged conducting sphere, which is plotted here, and compared with the electric field:

24. Relation between Electric Potential and Electric Field Breakdown voltage. In many kinds of equipment, very high voltages are used. A problem with high voltage is that the air can become ionized due to the high electric fields: free electrons in the air (produced by cosmic rays, for example) can be accelerated by such high fields to speeds sufficient to ionize O2 and N2 molecules by collision, knocking out one or more of their electrons. The air then becomes conducting and the high voltage cannot be maintained as charge flows. The breakdown of air occurs for electric fields of about 3.0 × 106 V/m. (a) Show that the breakdown voltage for a spherical conductor in air is proportional to the radius of the sphere, and (b) estimate the breakdown voltage in air for a sphere of diameter 1.0 cm.

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26. Point Charges To find the electric potential due to a point charge, we integrate the field along a field line:

27. Point Charges Setting the potential to zero at r =∞ gives the general form of the potential due to a point charge:

28. Point Charges Work required to bring two positive charges close together. What minimum work must be done by an external force to bring a charge q = 3.00 μC from a great distance away (take r = ∞) to a point 0.500 m from a charge Q = 20.0 µC?

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30. Electric Potential I 1)V > 0 2)V = 0 3)V < 0 What is the electric potential at point A? A B

31. A B Electric Potential I 1)V > 0 2)V = 0 3)V < 0 What is the electric potential at point A? Since Q2(which is positive) is closer to point A than Q1(which is negative) and since the total potential is equal to V1 + V2, the total potential is positive.

32. A B Electric Potential II 1)V > 0 2)V = 0 3)V < 0 What is the electric potential at point B?

33. A B Electric Potential II 1)V > 0 2)V = 0 3)V < 0 What is the electric potential at point B? Since Q2and Q1 are equidistant from point B, and since they have equal and opposite charges, the total potential is zero. Follow-up: What is the potential at the origin of the x y axes?

34. Point Charges Potential above two charges. Calculate the electric potential (a) at point A in the figure due to the two charges shown, and (b) at point B.

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36. Charge Distribution The potential due to an arbitrary charge distribution can be expressed as a sum or integral (if the distribution is continuous): or

37. Charge Distribution Potential due to a ring of charge. A thin circular ring of radius R has a uniformly distributed charge Q. Determine the electric potential at a point P on the axis of the ring a distance x from its center.

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39. Charge Distribution Potential due to a charged disk. A thin flat disk, of radius R0, has a uniformly distributed charge Q. Determine the potential at a point P on the axis of the disk, a distance x from its center.

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42. Equipotential Surfaces An equipotential is a line or surface over which the potential is constant. Electric field lines are perpendicular to equipotentials. The surface of a conductor is an equipotential.

43. Equipotential Surfaces Point charge equipotential surfaces. For a single point charge with Q = 4.0 × 10-9 C, sketch the equipotential surfaces (or lines in a plane containing the charge) corresponding to V1 = 10 V, V2 = 20 V, and V3 = 30 V.

44. Equipotential Surfaces Equipotential surfaces are always perpendicular to field lines; they are always closed surfaces (unlike field lines, which begin and end on charges).

45. Equipotential Surfaces A gravitational analogy to equipotential surfaces is the topographical map – the lines connect points of equal gravitational potential (altitude).

46. 1 2 3 +Q –Q 4 Equipotential Surfaces I 5) all of them At which point does V = 0?

47. 1 2 3 +Q –Q 4 Equipotential Surfaces I 5) all of them At which point does V = 0? All of the points are equidistant from both charges. Since the charges are equal and opposite, their contributions to the potential cancel outeverywhere along the mid-plane between the charges. Follow-up: What is the direction of the electric field at all 4 points?

48. +2mC +2mC +2mC +1mC -2mC +1mC x x x -1mC -2mC +1mC -2mC -1mC -1mC 1) 2) 3) Equipotential Surfaces II Which of these configurations gives V = 0 at all points on the x axis? 4)all of the above 5)none of the above

49. +2mC +2mC +2mC +1mC -2mC +1mC x x x -1mC -2mC +1mC -2mC -1mC -1mC 1) 2) 3) Equipotential Surfaces II Which of these configurations gives V = 0 at all points on the x axis? 4)all of the above 5)none of the above Only in case (1), where opposite charges lie directly across the x axis from each other, do the potentials from the two charges above the x axis cancel the ones below the x axis.

50. +2mC +2mC +2mC +1mC -2mC +1mC x x x -1mC -2mC +1mC -2mC -1mC -1mC 1) 2) 3) Equipotential Surfaces III Which of these configurations gives V = 0 at all points on the y axis? 4)all of the above 5)none of the above