2. ELECTRIC POTENTIAL Summer, 2008

3. Chapter 24 Electric Potential In this chapter we will define the electric potential ( symbol V ) associated with the electric force and accomplish the following tasks: Calculate V if we know the corresponding electric field Calculate the electric field if we know the corresponding potential V Determine the potential V generated by a point charge Determine the potential V generated by a continuous charge distribution Determine the electric potential energy U of a system of charges Define the notion of an equipotential surface Explore the geometric relationship between equipotential surfaces and electric field lines Explore the potential of a charged isolated conductor (24 - 1)

4. Picture a Region of space Where there is an Electric Field Imagine there is a particle of charge q at some location. Imagine that the particle must be moved to another spot within the field. Work must be done in order to accomplish this.

5. So, when we move a charge in an Electric Field.. Move the charge at constant velocity so it is in mechanical equilibrium all the time. Ignore the acceleration at the beginning because you have to do the same amount of negative work to stop it when you get there.

6. When an object is moved from one point to another in an Electric Field,  It takes energy (work) to move it.  This work can be done by an external force (you). FIELD negative  You can also think of this as the FIELD doing the negative of this amount of work on the particle.

7. The net work done by a conservative (field) force on a particle moving around a closed path is ZERO!

8. IMPORTANT The work necessary to move a charge from an initial point to a final point is INDEPENDENT OF THE PATH CHOSEN!

9. Electric Potential Energy When an electrostatic force acts between two or more charged particles, we can assign an ELECTRIC POTENTIAL ENERGY U to the system. The change in potential energy of a charge is the amount of work that is done by an external force in moving the charge from its initial position to its new position. It is the negative of the work done by the FIELD in moving the particle from the initial to the final position.

10. Definition – Potential Energy PE or U is the work done by an external agent in moving a charge from a REFERENCE POSITION to a different position. A Reference ZERO is placed at the most convenient position  Like the ground level in many gravitational potential energy problems.

11. Example: E Zero Level q F d Work by External Agent W external = F  d = qEd= U Work done by the Field is: W field = -qEd = -W external

12. AN IMPORTANT DEFINITION Just as the ELECTRIC FIELD was defined as the FORCE per UNIT CHARGE: We define ELECTRICAL POTENTIAL as the POTENTIAL ENERGY PER UNIT CHARGE: VECTOR SCALAR

13. UNITS OF POTENTIAL

14. Furthermore… If we move a particle through a potential difference of  V, the work from an external “person” necessary to do this is q  V

15. Consider Two Plates OOPS …

17. Look at the path issue

18. An Equipotential Surface is defined as a surface on which the potential is constant. It takes NO work to move a charged particle between two points at the same potential. The locus of all possible points that require NO WORK to move the charge to is actually a surface.

19. (24 - 10)

20. S A B  V q (24 - 11) S V

21. Field Lines are Perpendicular to the Equipotential Lines

22. Equipotential

23. Keep in Mind Force and Displacement are VECTORS! Potential is a SCALAR.

24. UNITS 1 VOLT = 1 Joule/Coulomb For the electric field, the units of N/C can be converted to: 1 (N/C) = 1 (N/C) x 1(V/(J/C)) x 1J/(1 NM) Or 1 N/C = 1 V/m So an acceptable unit for the electric field is now Volts/meter. N/C is still correct as well.

25. In Atomic Physics It is sometimes useful to define an energy in eV or electron volts. One eV is the additional energy that an proton charge would get if it were accelerated through a potential difference of one volt. 1 eV = e x 1V = (1.6 x 10 -19 C) x 1(J/C) = 1.6 x 10 -19 Joules.

26. Coulomb Stuff: A NEW REFERENCE: INFINITY Consider a unit charge (+) being brought from infinity to a distance r from a Charge q: q r To move a unit test charge from infinity to the point at a distance r from the charge q, the external force must do an amount of work that we now can calculate. x

27. Set the REFERENCE LEVEL OF POTENTIAL at INFINITY so (1/r A )=0.

28. For point charges

29. r1r1 r2r2 r3r3 P q1q1 q3q3 q2q2 (24 - 5)

30. For a DISTRIBUTION of charge:

31. dq O A (24 - 8)

32. A B V V+dV (24 - 13) Calculating the electric field E from the potential V

33. A B V V+dV (24 - 14)

34. x y O q1q1 q2q2 q3q3 r 12 r 23 r 13 (24 - 15)

35. (24 - 16) O x y  q1q1 Step 1 x y O r 12 q1q1 q2q2  Step 2 Step 3 x y O r 12 r 13 r 23 q1q1 q2q2 q3q3 

36. path A B conductor (24 - 17)

37. (24 - 18)

38. (24 - 19)

39. In the figure, point P is at the center of the rectangle. With V = 0 at infinity, what is the net electric potential in terms of q/d at P due to the six charged particles?

40. Continuing s 1 23 4 56