From last time(s)…

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

2. 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. Work and energy • Work-energy theorem: • Change in kinetic energy of isolated particle = work done • Total work

4. + + 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. + + + Force in direction of motion Calculating the work Zero • E.g. Keep Q2 fixed, push Q1 at constant velocity • Net force on Q1 ? • Force from hand on Q1 ? Q1 R Q2 xfinal xinitial • Total work done by hand

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

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

8. 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 Two times bigger Four times bigger Two times smaller Four times smaller None of the above

9. 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. 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 + q1 + q2 + + q3 Superposition of individual interactionsGeneralize to continuous charge distribution.

11. 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. Electric potential of point charge • Consider one charge as ‘creating’ electric potential, the other charge as ‘experiencing’ it q Q

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

14. B Electric potential energy=qoV A qo > 0 Potential energy, forces, work • U=qoV • Point B has greater potential energy than point A • Means that work must be done to move the test charge qo from A to B. • This is exactly the work to overcome the Coulomb repulsive force. Work done = qoVB-qoVA = Differential form:

15. Quick Quiz Two points in space A and B have electric potential VA=20 volts and VB=100 volts. How much work does it take to move a +100µC charge from A to B? +2 mJ -20 mJ +8 mJ +100 mJ -100 mJ

16. V(r) from multiple charges • Work done to move single charge near charge distribution. • Other charges provide the force, q is charge of interest. q1 q2 q q3 Superposition of individual electric potentials

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

18. x=-a x=+a +Q -Q Superposition: the dipole electric potential • Superposition of • potential from +Q • potential from -Q + = V in plane

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

21. Non-zero E-field Non-constant potential 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

22. difficult path easy path Easy because is same direction as E, 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 = keQ / R2 in the radial direction. • What is the electric potential of the conductor? Integral along some path, from point on surface to inf.

23. Electric potential of sphere So conducting sphere of radius R carrying charge Q is at a potential Conducting spheres connected by conducting wire. Same potential everywhere. Q1 Q2 R1 R2 But not same everywhere

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

25. 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. Quick Quiz • Four electrons are added to a long wire. Which of the following will be the charge distribution?

27. 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. E-field and potential energy

29. What is electric potential energy of isolated charge? Zero

30. 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. Electric field and potential Said before that • 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

32. 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=Vo

33. Quick Quiz 3 • Suppose the electric potential is constant everywhere. What is the electric field? Positive Negative Zero

34. B A x Electric Potential - Uniform Field + E cnst • 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

35. Electric field from potential • Said before that • Spell out the vectors: • This works for Usually written

36. Equipotential lines • Lines of constant potential • In 3D, surfaces of constant potential

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

38. Quick Quiz 1 Question: How much work would it take YOU to assemble 3 negative charges? • W = +19.8 mJ • W = 0 mJ • W = -19.8 mJ -3mC 5 m 5 m Likes repel, so YOU will still do positive work! -1mC -2mC 5 m

39. Work done to assemble 3 charges Similarly if they are all positive: • W1 = 0 • W2 = k q1 q2 /r =3.6 mJ =(9109)(110-6)(210-6)/5 • W3 = k q1 q3/r + k q2 q3/r • (9109)(110-6)(310-6)/5 + (9109)(210-6)(310-6)/5=16.2 mJ • W = +19.8 mJ • WE = -19.8 mJ • UE = +19.8 mJ q3 3C 5 m 5 m 2C 1C q2 5 m q1

40. +Q 5 m 5 m - Q +Q 5 m Quick Quiz 2 The total work required for YOU to assemble the set of charges as shown below is: • positive • zero • negative

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

42. x=-a x=+a +Q -Q 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

43. x=-a x=+a +Q -Q 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 A

44. +Q -Q Dipole electric fields Note properties of E-field lines

45. Conservative forces Fg • 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 • Wconservative = -D U = Uinitial - Ufinal = = -(Kfinal - Kinitial) = -DK

46. 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 q2 at a distance rf from q1 starting froma very far away distance is equal and opposite to the work done by the electric force. Charges repel W>0! F r12

47. Potential Energy of 2 charges • Since the 2 charges repel, the force on q2 due to q1 F12 is opposite to the direction of motion • The external agent F = -F12 must do positive work! W > 0 and the work of the electric force WE < 0 F dr r12

48. Potential Energy of 2 charges • Since WE = -DU = Uinitial - Ufinal = = -W  W = DU • We set Uinitial = U() = 0 since at infinite distance the force becomes null • The potential energy of the system is

49. More than two charges?

50. 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: