Electric Potential

1. Electric Potential AP Physics: Electricity & Magnetism

2. Potential Energy What is potential energy? The ability to do work The energy possessed by an object due to its position in a force field.

3. What is Work? Work is an energy transfer A constant force applied over a distance in the same direction.

4. Work and Potential Energy The work done by a conservative force equals the decrease in the potential energy of the particle.

5. Work done by an electric field This is called a line or path integral which is independent of the path taken from point A to B.

6. The change in potential energy per unit charge: This value is called Electric Potential and is independent of the value of q The potential at any point in the field:

7. Electric Potential: Be careful, electric potential is not potential energy. Units? J/C The work per unit charge that an external agent must perform to move a test charge from A to B without a change in kinetic energy

8. Potential at a point P The electric field at an arbitrary point equals the work required per unit charge to bring a positive test charge from infinity to that point

9. A note on Units:

10. The electron Volt The energy that an electron or proton gains or loses by moving through a potential difference of 1V

11. Potential Differences in a uniform electric field: What is the potential difference between two points if the displacement is parallel to the field lines?

12. Potential at points B & C E Cross-section of equipotential surface B C A

13. Equipotential Surface The name given to any surface consisting of a continuous distribution of points having the same electric potential. No work is done in moving a test charge between any two points on an equipotential surface.

14. Equipotential Surface • Red lines are electric field lines • Blue lines are equipotential surfaces • Equipotential surfaces must be perpendicular to field lines.

15. Equipotential Surface

16. Equipotential Surface

17. Gain or loss? A negative charge gains electric potential energy when it moves in the direction of the electric field. Explain this. A proton loses potential energy when moving in the direction of the electric field, but picks up an equal amount of kinetic energy. Work done by a field is positive when energy is given to an object from the field (object is lowered in a gravitational field or moved in the direction of E filed lines). This occurs when Uf <Uiso, W=-ΔU

18. The gap between electrodes in a spark plug is 0.06cm, To produce an electric spark in a gasoline air mixture, an E field of 3x106V/m must be achieved. When starting the car, what minimum potential difference must be supplied by the ignition circuit?

19. Calculate the speed of a proton that is accelerated from rest through a potential difference of 120V. An electron?

20. An ion accelerated through a potential difference of 115V experiences an increase in kinetic energy of 7.37x10-17 J.Calculate the charge of the ion

21. How much work is done in moving Avogadro's number of electrons from an initial point where the electric potential is 9V to a point where the potential is -5V?

22. Potential energy due to a point charge? What is the electric field due to a point charge (q)? Field is radially outward (or inward), and the Gaussian surface is a sphere chosen to match symmetry of field.

23. Potential energy due to a point charge? The following simplification can be made because the potential difference is not dependant upon the path taken through the field, only the endpoints.

24. Potential energy due to a point charge? If we want to know the potential at one particular location near a point charge we chose the first point to be infinity.

25. Superposition principle Potential at a point due to many point charges. This is a scalar value so it is much easier to calculate than E field.

26. Potential Energy due to many charges. The potential energy of a pair of charged particles separated by a distance r12. this is also the work done to move q2 from infinity to point P where q1 is located. V1 is the electric potential at point P due to charge q1. Notice that if the charges are like then positive work is done to bring the charges together (they have more energy together because they will repel). Energy delivered to the system requires positive work.

27. For multiple charges: Imagine q1 is at a fixed position. We want to bring q2 and q3 from infinity to a position near q1. Each term represents the the work required to bring each charge to a location near the other. It does not matter which order we bring them in from infinity. r12 q1 q2 r13 r23 q3

28. You know potential and you want to know Electric Field… How can we use this to solve for E field? If the E field only has one component…

29. You know potential and you want to know Electric Field… Remember that the potential is zero for displacements perpendicular to the field If your potential is a function of all three spatial coordinates (x, y, z)

30. To find the electric field, we must take the partial derivative of this potential… Partial derivative: Treat other variables as constant while differentiating with respect to one variable.

32. In vector notation this is often written as… If the potential is constant in some region, what is the electric field? If the electric field is zero in some region, what is the electric potential?

33. Question If the potential is constant in some region, what is the electric field? If the electric field is zero in some region, what is the electric potential? The difference in electric potential (voltage) measured when moving from point A to point B is equal to the work which would have to be done, per unit charge, against the electric field to move the charge from A to B.

34. Electric field due to continuous charge distributions. The electric potential at point P due to a continuous charge distribution can be calculated by dividing the charged body into segments of charge dq and summing the potential contributions over all segments.

35. Potential due to a uniformly charged ring: dq a P x All segments of charge are equidistant from point P

36. Potential due to a uniformly charged ring at the center: dq The field is zero here, therefore the potential must be constant: a P this potential is the work necessary to bring a test charge from infinity to the location in the center of the ring.

37. Potential of a charged conductor This oddly shaped conductor with an excess positive charge is in equilibrium A B Along this surface path E is always perpendicular to the displacement ds.

38. Potential of a charged conductor The surface of any charged conductor in equilibrium is an equipotential surface. Furthermore, since the electric field is zero inside the conductor, we conclude that the potential is constant everywhere inside the conductor and equal to its value at the surface. A B

39. The electric field is large near points having small convex radii of curvature and reaches very high values at sharp points.

40. A cavity within a conductor Because the potential on the surface of the cavity is an equipotential surface. B A A cavity in a conductor with no charge in it…The electric field inside the cavity must be zero, regardless of the charge distribution on the outside surface of the conductor.

41. Electric Shielding • Sensitive circuits and people can be protected if placed in a cavity inside of a conductor.

42. Corona Discharge: Air can become a conductor as a result of the ionization or air molecules in regions of high electric fields. At STP this happens around 3x106 V/m

43. Practice Quiz 1 +2Q .E .A .B .D .C -Q -Q 2a Three charges are arranged in an equilateral triangle, as shown. At which of these points (a,b,c bisect sides, d is equidistant from other points) is the electric potential smallest?

44. Practice Quiz 1 +2Q .E .A .B .D .C -Q -Q 2a C) A small positive test charge will move towards an area of low potential. Ask yourself “Where would an small positive charge end up if released near these charges?”

45. Practice Quiz 2 .P 10V 40V 70V The diagram shows a set of equipotential surfaces. At point P, what is the direction of the electric field? left B) right C) up the slide D) down the slide E) either left or right, which one cannot be determined

46. Practice Quiz 2 .P 10V 40V 70V A) A small positive charge placed at P would move to a location of low potential (left). The force that moves it is caused by an electric field which will be in the same direction or opposite it. Equipotential surfaces are always perpendicular to electric field lines.

47. Practice Quiz 3 a Q Q What is the electric potential at a point halfway between the two charges? A) kQ/a B) 2kQ/a C) zero D) 4kQ/a E) 8kQ/a D) Electric potential is a scalar

48. Practice Quiz 4 A solid conducting sphere carries a charge +Q. Which of the following is true of the electric field E and the electric potential V inside the sphere? A) E=0 and V=0 B) E=0 and V≠0 C) E≠0 and V=0 D) E≠0 and V≠0 E) It cannot be determined without knowing the radius of the sphere. B) The field in a conductor is always zero. E is the derivative of V, so to be zero, V must have been a constant.

49. Practice Quiz 5 A negatively charged rod is brought near a metal object on an insulating stand, as shown. When charges stop moving, the left side of the object has an excess of positive charge, and the right side of the object, where the radius of curvature is less, has an excess of negative charge. Which of the following best describes the electric potential on the metal object? It is greatest on the + side B) It is greatest on the – side C) It is greatest at the center D) It is the same everywhere on the object E) It cannot be determined from the       information given + + + + - - - - - - - - - -

50. Practice Quiz 5 + + + + - - - - - - - - - It is greatest on the + side B) It is greatest on the – side C) It is greatest at the center D) It is the same everywhere on the object E) It cannot be determined from the       information given - D) It is the same everywhere on the object. The surface of a conductor is an equipotential surface. The E field is 0 inside the conductor and V is constant.