Chapter 22: Electric Potential

1. Chapter 22: Electric Potential • Review of work, potential and kinetic energy • Consider a force acts on a particle moving from point a to point b. • The work done by the force WAB is given by: Electric Potential Energy • If the force is conservative, namely when the work done by the force • depends only on the initial and final position of the particle but not on • the path taken along the particle’s path, the work done by the force F • can always be expressed in terms of a potential energy U.

2. Review of work, potential and kinetic energy • In case of a conservative force, the work done by the force can be • expressed in terms of a potential energy U: • The change in kinetic energy DK of a particle during any displacement is • equal tothe total work done on the particle: Electric Potential Energy • If the force is conservative, then

3. Electric potential energy in a uniform field • Consider a pair of charged parallel metal plates that generate a uniform • downward electric field E and a test charge q0 >0 A + + + + + + + + + + d conservative force Electric Potential Energy - - - - - - - - - - the force is in the same direction as the net displacement of the test charge B • In general a force is a vector: m g Fg Note that this force is similar to the force due to gravity:

4. Electric potential energy in a uniform field (cont’d) • In analogy to the gravitational force, a potential can be defined as: • When the test charge moves from height ya to height yb , the work done • on the charge by the field is given by: Electric Potential Energy • U increases (decreases) if the test charge moves in the direction • opposite to (the same direction as) the electric force DUAB<0 DUAB <0 DUAB>0 DUAB >0 A B A B + + - - + B + A - B - A

5. Electric potential energy of two point charges • The force on the test charge at a distance r b rb • The work done on the test charge Electric Potential Energy q0 r a ra + q

6. Electric potential energy of two point charges (cont’d) • In more general situation tangent to the path B r Electric Potential Energy A Natural and consistent definition of the electric potential

7. Electric potential energy of two point charges (cont’d) • Definition of the electric potential energy • Reference point of the electric potential energy Potential energy is always defined relative to a reference point where U=0. When r goes to infinity, U goes to zero. Therefore r= is the reference point. This means U represents the work to move the test charge from an initial distance r to infinity. Electric Potential Energy If q and q0 have the same sign, this work is POSITIVE ; otherwise it is NEGATIVE. U U 0 qq0>0 qq0<0 0

8. Electric potential energy with several point charges • A test charge placed in electric field by several particles Electric Potential Energy • Electric potential energy to assemble particles in a configuration

9. Example : A system of point charges q1=-e q2=+e q3=+e - + + x=a x=2a x=0 Work done to take q3 from x=2a to x=infinity Electric Potential Work done to take q1,q2 and q3 to infinity

10. Two interpretations of electric potential energy • Work done by the electric field on a charged particle moving in the field Work done by the electric force when the particle moves from A to B • Work needed by an external force to move a charged particle • slowly from the initial to the final position against the electric force Electric Potential Energy Work done by the external force when the particle moves from B to A

11. Electric potential or potential • Electric potential V is potential energy per unit charge 1 V = 1 volt = 1 J/C = 1 joule/coulomb Electric Potential potential of A with respect to B work done by the electric force when a unit charge moves from A to B work needed to move a unit charge slowly from b to a against the electric force

12. Electric potential or potential (cont’d) • Electric potential due to a single point charge • Electric potential due to a collection of point charges Electric Potential • Electric potential due to a continuous distribution of charge

13. From E to V • Sometimes it is easier to calculate the potential from the known • electric field Electric Potential The unit of electric field can be expressed as: 1 V/m = 1 volt/meter = 1 N/C = 1 newton / coulomb

14. Example : Electric Potential Replace R with r

15. Example: q1 q2 m, q0 + - = 0 A B Electric Potential

16. Unit: electron volt (useful in atomic & nuclear physics) • Consider a particle with charge q moves from a point where the potential • is VA to a point where it is VB , the change in the potential energy U is: • If the charge q equals the magnitude e of the electron charge • 1.602 x 10-19 C and the potential difference VAB= 1 V, the change • in energy is: Electric Potential meV, keV, MeV, GeV, TeV,…

17. Example: A charged conducting sphere Using Gauss’s law we calculated the electric field. Now we use this result to calculate the potential and we take V=0 at infinity. + + + + + R + + + E the same as the potential due to a point charge Calculating Electric Potential r 0 V inside of the conductor E is zero. So the potential stays constant and is the same as at the surface r 0

18. Equipotential surface • An equipotential surface is a 3-d surface on which the electric potential V • is the same at every point • No point can be at two different potentials, so equipotential surfaces for • different potentials can never touch or intersect • Because potential energy does not change as a test charge moves over an • equipotential surface, the electric field can do no work Equipotential Surface • E is perpendicular to the surface at every point • Field lines and equipotential surfaces are always mutually perpendicular

19. Examples of equipotential surface Equipotential Surface

20. Equipotentials and conductors • E = 0 everywhere inside a conductor - At any point just inside the conductor the component of E tangent to the surface is zero - The tangential component of E is also zero just outside the surface If it were not, a charge could move around a rectangular path partly inside and partly outside and return to its starting point with a net amount of work done on it. vacuum Equipotential Surface conductor • When all charges are at rest, the electric field just outside a conductor must • be perpendicular to the surface at every point • When all charges are at rest, the surface of a conductor is always an • equipotential surface

21. Equipotentials and conductors (cont’d) • Consider a conductor with a cavity without any charge inside the cavity • - The conducting cavity surface is an equipotential surface A • Take point P in the cavity at a different potential and it is on a • different equipotential surface B • The field goes from surface B to A or A to B • Draw a Gaussian surface which surrounds the surface B inside • cavity Guassian surface Equipotential Surface equipotential surface through P B A P • The net flux that goes through this Gaussian surface is not zero • because the electric field is perpendicular to the surface • Gauss’s law says this flux is zero as there is no charge inside • Then the surfaces A and B are at the same potential conductor surface of cavity • In an electrostatic situation, if a conductor contains a cavity and if no • charge is present inside the cavity, there can be no net charge anywhere • on the surface of the cavity

22. Electrostatic shielding Equipotential Surface

23. Potential gradient • Potential difference and electric field • Potential difference and electric field Potential Gradient

24. Potential gradient (cont’d) • E from V PotentialGradient • Gradient of a function f If E is radial with respect to a point or an axis

25. Potential gradient (cont’d) PotentialGradient

26. Exercise 1 Exercises

27. Exercise 1 (cont’d) Exercises

28. Exercise 1 (cont’d) Exercises

29. Exercise 2 Exercises

30. Exercise 3 Exercises

31. Exercise 4 Exercises

32. Exercise 4 (cont’d) Exercises

33. Exercise 4 (cont’d) Exercises

34. Exercise 4 (cont’d) Exercises

35. Exercise 5: An infinite line charge + a conducting cylinder Q -Q Outer metal braid r r Exercises Signal wire line charge density l