1. Physics for Scientists and Engineers Chapter 23: Electric Potential Copyright © 2004 by W. H. Freeman & Company Paul A. Tipler Gene Mosca Fifth Edition Lecture 4

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5. Learning Objectives To provide a definition of Electric Potential V To show how E is calculated from V

6. dU = -mg.dl dW = mg.dl m m h W=mgh

7. dldl a b q Charged Particle in an E-field From Classical Mechanics: work done = force  distance moved in direction of force dW = F.dl F

8. dldl a b q dW = qE.dl Charged Particle in an E-field From Classical Mechanics: work done = force  distance moved in direction of force dW = F.dl

9. This work done by an E-field represents a decrease in the electric potential energy ( dU = -dW ) dW = qE.dl dU = -qE.dl

10. If the E-field varies as particle moves from a point a to b we need to integrate line integral

11. a b dldl a b dldl Positive charges move from high field (high U) to low field (low U).

12. ab dldl For a negative charge, q is -ve, U b -U a is positive. The E-field tends to move the charge from b to a. Negative charges move from low field (high U) to high field (low U). When a charge is moved in an E-field, its potential energy is a function of position. This leads to the definition of the electric potential.

13. Definition of Electric Potential: Potential Energy of the system per unit charge It is a property of a point in an E-field. It is a scalar. The unit of potential, Joule Coulomb -1, is called a Volt (V).

14. Alessandro Volta (1745-1827) The first electric battery

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17. Electric Potential Difference (23-1) The electric potential difference between two points is: Most commonly used for electric fields [Volt] = [N/C] [N/C]= [V/m] [m]  V b and V a are unique for points b and a

18. If the charge returns to its original position, by any route, NO WORK IS DONE The Uniqueness of Electric Potential E-field is conservative (electrostatics)

19. Electric Potential at a distance r from a point charge q q r = V(r) – V(  ) = V(r) – 0

20. Coulomb Potential q can be positive or negative, so is the potential.

21. The U of a charge q 0 at a distance r from q is:

22. Potential due to a collection of point charges (23-2) r i is the distance from the i th charge, q i, to the point at which V is being evaluated

23. E-field lines and Equipotential Surfaces (23-5) An E-field line traces the path that a +ve test charge would follow under the action of electrostatic forces. If released the positive charge will accelerate in the direction of the electric field

24. E-field lines and Equipotential Surfaces (23-5) Note that lines of force are always perpendicular to the equipotentials Surfaces over which V is constant are called equipotentials

25. Why do we use potentials?

26. Calculation of E from the Electric Potential: the Gradient of Potential

27. In general The negative of the gradient of the electric potential

28. In Plane Polar coordinates

29. The Coulomb potential: This is easy because V is a function of r only.

30. Summary We can characterise an electric field through the electric potential Electric potentials can be easily superposed (numbers not vectors)

31. Summary

32. Application: Field ion microscope - used to image atoms Physicist’s approximation of a needle is: a Radius b Long conducting wire Q Charge q Works by having a high electric field around a point of a needle. How is this high electric field achieved?

33. Electric potential of larger sphere: Electric potential of smaller sphere:

34. But potentials are equal as connected: Compare electric fields at the surface of each sphere Smaller radius of curvature, the higher the E-field

36. Practical Importance: High Voltage Power Lines Losses are higher than normal in damp weather. Why? Charged water droplets on wire become elongated to a point because of repulsion. The resulting high E-field leads to ionisation and heating of the air (energy loss) Results in TV and radio interference

37. Summary

38. Next Lecture Further properties of Electrostatic Fields Gauss’s Law

39. Classwork A positive (small) test charge q 0 in the neighbourhood of other charges experiences an electric force F which varies in magnitude and direction at each point. The electric field strength at any point is defined as

40. Potential – arbitrary zero of potential taken at infinite separation. Suppose an external agent does work W in bringing a small test charge q 0 from infinity to a particular point in an electric field. The electric potential at that point is defined by the equation:

42. Electric Potential: Numerically equal to the work done in bringing a unit positive charge to the body from the arbitrary point chosen as the zero of potential (often chosen to be infinite separation). Suppose an external agent does work W in bringing a small test charge q from infinity to a particular point in an electric field. The electric potential V at that point is numerically equal to:

44. Work done by the gravitational field g is equal to the decrease in P.E. Tipler Fig. 23-1 Work done by the E is equal to the decrease in P.E.

46. Figure 23-3

48. 23-19

49. Similarly