§8-5 Electric potential

1. q0 §8-5 Electric potential Electrostatic field does work for moving charge --E-field possesses energy 1.Work done by electrostatic force  Let test charge q0 moves a b along arbitrary path in the E-field set up by point charge q . The work done by electrostatic force=?

2. : displacement 

3. q0 The work depends on only the initial position and final position of qo, and has nothing to do with the path.

4. When qo moves in the E-field set up by charges’ system q1, q2, qn , ---has nothing to do with path

5.  When qo moves in the E-field set up by charged body, -- the workhas nothing to do with path Conclusion: Electrostatic force is conservative force. 2. Circular theorem of electrostatic field When q0 moves along a closed path L， E-force does work:

6. 、 Circular theorem of electrostatic field Electrostatic field is conservative field. The work done by electrostatic force = the decrease of the electric potential energy 3. Electric potential energy q0 moves in E-field a b， --The E-potential energy when q0 at point a and b.

7. Z Z The work done by E-force for q0=- increment of E-potential energy. Notes (1) EPa is relative quantity.If we want to decide the magnitude of EPwhen q0 at a point , we must choose zero reference point of E-potential energy. Z--zero E-potential energy point

8. (3) EPdepends on E-field and ，it belongs the system. The choice of zero E-potential energy point ：  Choose zero point at  when the charge distribution is finite.  Choose zero point at the finite distance point when the charge distribution is infinite. (2) EPis scalar. It can be positive, negative or zero.

9. Z 4. Electric potential --Describe the character of E-field. Definition E-potential difference: Work:

10. The distance from q to a Take U=0，then the E-potential of a point a : 5.Calculation of E-potential (1) The E-potential of a point chargeq

11. U r + U r Discussion Ifq>0，U>0 for any point in the space. when r，U  U()  0 Ifq<0，U<0 for any point in the space. when r，U  U()  0

12.  a ri r1 qi q1 （2）The E-potential of a system The system of point charges：q1,q2,,qn --superposition principle of E-potential

13. charge element a dq （3）The E-potential of charged body Divide q many ofdq For any dq： For entire charged body ：

14. Integrating for charged body Caution ！！ This method can be used for the finite distribution charged body.

15. 6. Examples of calculating E-potential [Example 1] Four point charges q1＝q2＝q3＝q4＝q is put on the vertexes of a square with edge of a respectively. Calculate (1)The E-potential at point 0. (2)If test charge q0 is moved from  to 0, how much work does the E-force do?

16. （1） （2）

17. Use the definition of E-potential as the distribution of is known. – definition method Z Integrating for path Calculate the E-potential set up by a charged body Two methods Definition method

18. Integrating for charged body  Use the superposition principle of E-potential--superposition method. superposition method

19. [Example 2] Calculate the E-potential on the axis of a uniform charged ring. q、 R are known. Solution Method  Use the E-field distribution of the ring that was calculated before. The direction: along x axis

21. q dq Method -- superposition method

22. If the charged body is a half circle, ？ 0 R (2)  Discussion (1)

23. q + + + R r R  ） （ + + + + + ） （  [Example 3] Calculate the E-potential distribution of a charged spherical surface Solution  Definition method E-field distribution: Zero potential point： 

24.  ） （ P r q + + + R + + + + + (1) For any point P outside the sphere  (2) For any point inside the sphere surface

25.  分段积分 + + + + + R + + q + + + + +  P The sphere is equipotential.

26.  ） （ ） （  Conclusion

27. The distribution curve ofE-field： E 1 8 r2 O r R  1 8 r The distribution curve ofE-potential r R O

28. Integrating for charged body superposition method： It’s very complex！！

29. Conclusion  When the E-field distribution is symmetry and it can be calculated by using Gauss’s Law conveniently, it is simpler to calculate potential by using definition method.  When the E-field distribution is not symmetry and it can’t be calculated by using Gauss’s Law conveniently, it is simpler to calculate potential by using superposition method.

30. [Example 4] Calculate the E-potential distribution of an infinite line with uniform charge(the linear density isλ). Solution Use  definition method

31. Finite distance to the charged line How to choose thezero potential point? Choose any point b as zero potential point  When rp<rb， U >0. When rp>rb， U <0.

32. §8-6 Equipotential surface and Potential gradient 1. Equipotential surface --the potential has the same value at all points on the surface.

33. Real line-- -line Dash line-- equipotential surface Positive point charge Electric dipole

34. A parallel plate capacitor + + + + + + + + +

35. a q0 b The properties of equipotential surfaces  No net work is done by the E-field as a charge moves between any two points on the same equipotential surface. Prove： Assume q0 moves along equipotential surface a  b， then：

36.  -lines are always normal to equipotential surfaces Assume on an equipotential surface, the field at point P is q0 moves along equipotential surface, q0 P Prove： then：

37.  -line points on the direction of the increase of the potential. a b Assume there are two equipotential surface U a , Ub Prove： then

38. E2 r2 r1 E1 Uc Ua Ub  the density of equipotential surfaces shows the magnitude of E-field. Prove： Assume there is a family of equipotential surfaces Ua、 U b、 Uc、 then：

39. -line If the distribution of U is known, How can we calculate ? 2. Potential gradient Equipotential surface

40. C  Express that the ’s direction is the direction of U decrease.  b a • Special example： uniform field unit positive chargeab, the work done by E-force: q=1

41. ’s component at the direction of ’s component at any direction = the negative magnitude of the rate of U change with distance on that direction. q=1 moves ac , the work done by E-force ：

42. The normal direction of equipotential surface, point on U increasing (2) Any E-field: definition： potential gradient grad U =U grad U U

43. In Cartesian coordinate system

44. conclusion a.the magnitude of at any point = themaximumof the rate of U change with distance on that point. the direction of is perpendicular to the equipotential surface at that point and along the the direction of U decreasing. b. In the space that U is constant, U= 0，  E = 0 c.E is not sure = 0 in the space of U= 0. U is not sure = 0 in the space of E= 0.

45. P( ) [Example 1] Calculate the E-field of an electric dipole.

46. 

48. [example 2] calculate the E-field on the axis of the round charged plate using its potential gradient .

49. The potential of any ring on the axis: Solution

50. i.e., the potential on the axis: