1. Oct. 4, 20071 From last time(s)… Work, energy, and (electric) potential Electric potential and charge Electric potential and electric field. Electric charges, forces, and fields Motion of charged particles in fields. Today… No honors lecture this week

2. Oct. 4, 20072 Forces, work, and energy Particle of mass m at rest Apply force to particle - what happens? Particle accelerates Stop pushing - what happens? Particle moves at constant speed Particle has kinetic energy

3. Oct. 4, 20073 Work and energy Work-energy theorem: Change in kinetic energy of isolated particle = work done Total work

4. Oct. 4, 20074 Electric forces, work, and energy Consider bringing two positive charges together They repel each other Pushing them together requires work Stop after some distance How much work was done? + +

5. Oct. 4, 20075 Calculating the work E.g. Keep Q 2 fixed, push Q 1 at constant velocity Net force on Q 1 ? Force from hand on Q 1 ? + + Q1Q1 Q2Q2 Zero Total work done by hand Force in direction of motion R + x initial x final

6. Oct. 4, 20076 Conservation of Energy Work done by hand Where did this energy go? Energy is stored in the electric field as electric potential energy for pos charges

7. Oct. 4, 20077 Electric potential energy of two charges Define electric potential energy U so that Work done on system Change in kinetic energy Change in electric potential energy Works for a two-charge system if Define: potential energy at infinite separation = 0 Then Units of Joules for two charges

8. Oct. 4, 20078 Quick Quiz Two balls of equal mass and equal charge are held fixed a distance R apart, then suddenly released. They fly away from each other, each ending up moving at some constant speed. If the initial distance between them is reduced by a factor of four, their final speeds are A.Two times bigger B.Four times bigger C.Two times smaller D.Four times smaller E.None of the above

9. Oct. 4, 20079 More About U of 2 Charges Like charges  U > 0 and work must be done to bring the charges together since they repel (W>0) Unlike charges  U < 0 and work is done to keep the charges apart since the attract one the other (W<0)

10. Oct. 4, 200710 Electric Potential Energy of single charge Work done to move single charge near charge distribution. Other charges provide the force, q is charge of interest. + + + + q q1q1 q2q2 q3q3 Superposition of individual interactions Generalize to continuous charge distribution.

11. Oct. 4, 200711 Electric potential Electric potential V usually created by some charge distribution. V used to determine electric potential energy U of some other charge q V has units of Joules / Coulomb = Volts Electric potential  U energy proportional to charge q Electric potential

12. Oct. 4, 200712 Electric potential of point charge Consider one charge as ‘creating’ electric potential, the other charge as ‘experiencing’ it Q q

13. Oct. 4, 200713 Electric Potential of point charge Potential from a point charge Every point in space has a numerical value for the electric potential y x Distance from ‘source’ charge +Q +Q+Q

14. Oct. 4, 200714 Potential energy, forces, work U=q o V Point B has greater potential energy than point A Means that work must be done to move the test charge q o from A to B. This is exactly the work to overcome the Coulomb repulsive force. Electric potential energy= q o V A B q o > 0 Work done = q o V B -q o V A = Differential form:

15. Oct. 4, 200715 Quick Quiz Two points in space A and B have electric potential V A =20 volts and V B =100 volts. How much work does it take to move a +100µC charge from A to B? A.+2 mJ B.-20 mJ C.+8 mJ D.+100 mJ E.-100 mJ

16. Oct. 4, 200716 V(r) from multiple charges Work done to move single charge near charge distribution. Other charges provide the force, q is charge of interest. q q1q1 q2q2 q3q3 Superposition of individual electric potentials

17. Oct. 4, 200717 Quick Quiz 1 At what point is the electric potential zero for this electric dipole? +Q+Q-Q-Q x=+ a x=- a A B A.A B.B C.Both A and B D.Neither of them

18. Oct. 4, 200718 Superposition: the dipole electric potential +Q+Q-Q-Q x=+ a x=- a Superposition of potential from +Q potential from -Q Superposition of potential from +Q potential from -Q += V in plane

19. Oct. 4, 200719 Electric Potential and Field for a Continuous Charge Distribution If symmetries do not allow an immediate application of the Gauss’ law to determine E often it is better to start from V! Consider a small charge element dq The potential at some point due to this charge element is To find the total potential, need to integrate over all the elements This value for V uses the reference of V = 0 when P is infinitely far away from the charge distribution

20. Oct. 4, 200720 Quick Quiz Two points in space have electric potential V A =200V & V B =150V. A particle of mass 0.01kg and charge 10 -4 C starts at point A with zero speed. A short time later it is at point B. How fast is it moving? A.0.5 m/s B.5 m/s C.10 m/s D.1 m/s E.0.1 m/s

21. Oct. 4, 200721 E-field and electric potential If E-field known, don’t need to know about charges creating it. E-field gives force From force, find work to move charge q + + + + q Electric potential Non-constant potential Non-zero E-field

22. Oct. 4, 200722 Potential of spherical conductor Zero electric field in metal -> metal has constant potential Charge resides on surface, so this is like the spherical charge shell. Found E = k e Q / R 2 in the radial direction. What is the electric potential of the conductor? Integral along some path, from point on surface to inf. difficult path easy path Easy because is same direction as E,

23. Oct. 4, 200723 Electric potential of sphere Conducting spheres connected by conducting wire. Same potential everywhere. So conducting sphere of radius R carrying charge Q is at a potential R1R1 R2R2 Q1Q1 Q2Q2 But  not same everywhere

24. Oct. 4, 200724 Connected spheres Since both must be at the same potential, Surface charge densities? Charge proportional to radius Surface charge density proportional to 1/R Electric field? Since Local E-field proportional to 1/R (1/radius of curvature)

25. Oct. 4, 200725 Varying E-fields on conductor Expect larger electric fields near the small end. Can predict electric field proportional to local radius of curvature. Large electric fields at sharp points, just like square Fields can be so strong that air is ionized and ions accelerated.

26. Oct. 4, 200726 Quick Quiz Four electrons are added to a long wire. Which of the following will be the charge distribution? A) B) C) D)

27. Oct. 4, 200727 Conductors: other geometries Rectangular conductor (40 electrons) Edges are four lines Charge concentrates at corners Equipotential lines closest together at corners So potential changes faster near corners. So electric field is larger at corners.

28. Oct. 4, 200728 E-field and potential energy

29. Oct. 4, 200729 Zero What is electric potential energy of isolated charge?

30. Oct. 4, 200730 The Electric Field is the Electric Field It is independent of the test charge, just like the electric potential It is a vector, with a magnitude and direction, When potential arises from other charges, = Coulomb force per unit charge on a test charge due to interaction with the other charges. We’ll see later that E-fields in electromagnetic waves exist w/o charges!

31. Oct. 4, 200731 Electric field and potential Electric field strength/direction shows how the potential changes in different directions For example, Potential decreases in direction of local E field at rate Potential increases in direction opposite to local E-field at rate potential constant in direction perpendicular to local E-field Said before that

32. Oct. 4, 200732 Potential from electric field Electric field can be used to find changes in potential Potential changes largest in direction of E-field. Smallest (zero) perpendicular to E-field V=VoV=Vo

33. Oct. 4, 200733 Quick Quiz 3 Suppose the electric potential is constant everywhere. What is the electric field? A)Positive B)Negative C)Zero

34. Oct. 4, 200734 Electric Potential - Uniform Field Constant E-field corresponds to linearly increasing electric potential The particle gains kinetic energy equal to the potential energy lost by the charge-field system E cnst A B x +

35. Oct. 4, 200735 Electric field from potential Said before that Spell out the vectors: This works for Usually written

36. Oct. 4, 200736 Equipotential lines Lines of constant potential In 3D, surfaces of constant potential

37. Oct. 4, 200737 Electric Field and equipotential lines for + and - point charges The E lines are directed away from the source charge A positive test charge would be repelled away from the positive source charge The E lines are directed toward the source charge A positive test charge would be attracted toward the negative source charge Blue dashed lines are equipotential

38. Oct. 4, 200738 Quick Quiz 1  C  C  C  m 1.W = +19.8 mJ 2.W = 0 mJ 3.W = -19.8 mJ Question: How much work would it take YOU to assemble 3 negative charges? Likes repel, so YOU will still do positive work!

39. Oct. 4, 200739 Work done to assemble 3 charges W 1 = 0 1C 3C 2C  m W 2 = k q 1 q 2 /r W 3 = k q 1 q 3 /r + k q 2 q 3 /r (9  10 9 )(1  10 -6 )(3  10 -6 )/5 + (9  10 9 )(2  10 -6 )(3  10 -6 )/5 =16.2 mJ W = +19.8 mJ W E = -19.8 mJ U E = +19.8 mJ =(9  10 9 )(1  10 -6 )(2  10 -6 )/5 =3.6 mJ q3q3 q2q2 q1q1 Similarly if they are all positive:

40. Oct. 4, 200740 Quick Quiz 2 QQ QQ  Q  m 1.positive 2.zero 3.negative The total work required for YOU to assemble the set of charges as shown below is:

41. Oct. 4, 200741 Why  U/q o ? Why is this a good thing?  V=  U/q o is independent of the test charge q o Only depends on the other charges.  V arises directly from these other charges, as described last time. Last week’s example: electric dipole potential -Q-Q+Q+Q x=+ a x=- a Superposition of potential from +Q potential from -Q Superposition of potential from +Q potential from -Q

42. Oct. 4, 200742 Dipole electric fields Since most things are neutral, charge separation leads naturally to dipoles. Can superpose electric fields from charges just as with potential But E-field is a vector, -add vector components +Q+Q-Q-Q x=+ a x=- a

43. Oct. 4, 200743 Quick Quiz 2 In this electric dipole, what is the direction of the electric field at point A? A) Up B) Left C) Right D) Zero +Q+Q-Q-Q x=+ a x=- a A

44. Oct. 4, 200744 Dipole electric fields +Q+Q -Q-Q Note properties of E-field lines

45. Oct. 4, 200745 Conservative forces Conservative Forces: the work done by the force is independent on the path and depends only on the starting and ending locations. It is possible to define the potential energy U  W conservative  U = U initial - U final = = -(K final - K initial ) = -  K FgFg

46. Oct. 4, 200746 Potential Energy of 2 charges Consider 2 positive charged particles. The electric force between them is The work that an external agent should do to bring q 2 at a distance r f from q 1 starting from a very far away distance is equal and opposite to the work done by the electric force. Charges repel  W>0! r 12 F

47. Oct. 4, 200747 Potential Energy of 2 charges Since the 2 charges repel, the force on q 2 due to q 1 F 12 is opposite to the direction of motion The external agent F = -F 12 must do positive work! W > 0 and the work of the electric force W E < 0 r 12 F dr

48. Oct. 4, 200748 Potential Energy of 2 charges Since W E = -  U = U initial - U final = = -W  W =  U We set U initial = U(  ) = 0 since at infinite distance the force becomes null The potential energy of the system is

49. Oct. 4, 200749 More than two charges?

50. Oct. 4, 200750 U with Multiple Charges If there are more than two charges, then find U for each pair of charges and add them For three charges: