1. Dr. Hugh Blanton ENTC 3331

2. Energy & Potential

3. Dr. Blanton - ENTC 3331 - Energy & Potential 3 The work done, or energy expended, in moving any object a vector differential distance d l under the influence of a force is: Work, or energy, is measured in joules (J).

4. Dr. Blanton - ENTC 3331 - Energy & Potential 4 The differential electric potential energy dW per unit charge is called the differential electric potential (or differential voltage) dV. Convention: potential at infinity  0  ground. minus sign is convention

5. Dr. Blanton - ENTC 3331 - Energy & Potential 5 Electrical potential of a point charge & E-field of a point charge, q, located at the origin?

6. Dr. Blanton - ENTC 3331 - Energy & Potential 6 Compare the electric field and potential of a point charge. Expressions are rather similar.

7. Dr. Blanton - ENTC 3331 - Energy & Potential 7 Refer to the gradient in the spherical coordinates no dependence on  or 

8. Dr. Blanton - ENTC 3331 - Energy & Potential 8 It follows that for a point charge:

9. Dr. Blanton - ENTC 3331 - Energy & Potential 9 Generalizations: All complicated charge distributions are always superposition of fields due to many point charges. All these fields superimpose linearly.

10. Dr. Blanton - ENTC 3331 - Energy & Potential 10 Multiple charges Charge distributions

11. Dr. Blanton - ENTC 3331 - Energy & Potential 11 And importantly, for all distributions. Note potential energy {do not confuse with potential energy} Coulomb force

12. Dr. Blanton - ENTC 3331 - Energy & Potential 12 From mechanics For both Coulomb and mechanical forces, the force is equal to the gradient of the potential energy. Gravitational force potential energy

13. Dr. Blanton - ENTC 3331 - Energy & Potential 13 This isn’t so surprising, since Coulomb’s law And the gravitational force are so similar.

14. Dr. Blanton - ENTC 3331 - Energy & Potential 14 Forces for which holds are conservative. That is, the energy associated with the conservative force is conserved.

15. Dr. Blanton - ENTC 3331 - Energy & Potential 15 Determine the electric potential, V, at the origin for the figure. For multiple charges: For all four charges: + ++ + r is the origin

16. Dr. Blanton - ENTC 3331 - Energy & Potential 16 Important observation while

17. Dr. Blanton - ENTC 3331 - Energy & Potential 17 Find the E-field above a circular disk of radius a with  S = constant. What are the potential, V, and E-field at point P(0,0,z) right above the circular disk? y z x

18. Dr. Blanton - ENTC 3331 - Energy & Potential 18 y z x

19. Dr. Blanton - ENTC 3331 - Energy & Potential 19 The question is, what is the field

20. Dr. Blanton - ENTC 3331 - Energy & Potential 20 Since

21. Dr. Blanton - ENTC 3331 - Energy & Potential 21 The field points along the z-axis: This result is identical to that obtained earlier. By first calculating the potential V and then using, considerable geometrical considerations can be avoided.

22. Dr. Blanton - ENTC 3331 - Energy & Potential 22 Summary Energy conservation requires that Because of energy conservation the Coulomb force is conservative: Consequently,

23. Dr. Blanton - ENTC 3331 - Energy & Potential 23 Stoke’s Theorem General mathematical theorem of Vector Analysis. any surface any vector field closed boundary of that surface a surface closed path C along boundary

24. Dr. Blanton - ENTC 3331 - Energy & Potential 24 Since Stoke’s Theorem is of general validity, it can certainly be applied to the electrostatic fields: and

25. Dr. Blanton - ENTC 3331 - Energy & Potential 25 any closed path through consequence of: energy conservation Coulomb’s force is conservative & fields are conservative

26. Dr. Blanton - ENTC 3331 - Energy & Potential 26 Since C and S are arbitrary, this is only possible if:

27. Dr. Blanton - ENTC 3331 - Energy & Potential 27 Recall that the curl of a gradient field is always zero. That is: Thus:

28. Dr. Blanton - ENTC 3331 - Energy & Potential 28 The curl of the electrostatic field is zero. The circulation of any electrostatic field is zero An electrostatic field is non-rotational An electrostatic field is conservative. All of the preceding statements are equivalent.

29. Dr. Blanton - ENTC 3331 - Energy & Potential 29 The complete set of postulates for electrostatics are: and These are Maxwell’s equations of Electrostatics The postulates are accepted to be true for any E or D field as long as  is independent of time.

30. Dr. Blanton - ENTC 3331 - Energy & Potential 30 Cornerstones of Electrostatics Principle of Linear Superposition charges are the sources of the field Energy Conservation generalization for multiple charges SI units Coulomb’s Law conservative central force the field is non- rotational  Postulates of Electrostatics  theoretical experimental empirical facts field concept