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Ch. 23

Ch. 23. Electric Potential. Chapter Overview. Review Work and Potential Energy Define Potential Difference Compute the Potential Difference from the Electric Field Find the electric potential due to point charges Calculate the electric potential for a continuous charge distribution

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Ch. 23

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  1. Ch. 23 Electric Potential

  2. Chapter Overview • Review Work and Potential Energy • Define Potential Difference • Compute the Potential Difference from the Electric Field • Find the electric potential due to point charges • Calculate the electric potential for a continuous charge distribution • Compute the Electric Field from the Potential • Describe Properties of Equipotential Surfaces

  3. Review of Work • A force of 20 N is applied at an angle of 20° above the horizontal to a block sitting on a frictionless horizontal surface. a) Sketch the situation. b) If the force moves the block 2.5 m, how much work was done?

  4. W = Fd cosθ= 20 N x 2.5 m x cos 20°= 47 J

  5. Electrostatic Potential Energy • Work done by an electrostatic force is • dW = F∙dr • Electrostatic Force is a “conservative” force • dW = -dUelectrostatic

  6. A -2.0 µC charge is placed in a uniform electric field of 500 N/Ca) Sketch the situation. b) What is the work done by the electric field on the charge if it moves 5.5 m? c) What is the change in electrostatic potential energy of the charge?

  7. How would your answer for the ΔUelecchange if the charge moved was twice as big? (CT) • There would be no change • It would be twice as big • It would be ½ as big • It cannot be determined

  8. How would your answer for the ΔUelecchange if the charge moved was half as big? (CT) • There would be no change • It would be twice as big • It would be ½ as big • It cannot be determined

  9. What type of relationship is there between the charge and the ΔUelec? (GR)

  10. Definition of the Potential Difference • Since the ΔUelec is proportional to the charge, we define a quantity which is independent of the charge by dividing the potential energy by the charge.

  11. Definition of the Potential Difference • If a charge goes from point A to point B and has a change in electric potential energy of dUelec, then the potential difference between A and B is

  12. Definition of the Potential Difference • Remember that E = F/q for a test charge • so

  13. Definition of Potential Difference • If we add up the small changes in potential then we can obtain the finite potential difference between the points in space A and B B A

  14. Definition of the Potential Difference • SI Units? • This combination of units is called the volt, V • 1 V = 1 J/C • volt is named after Alessandro Volta who invented the battery

  15. Ex. Two points in space are in a region of uniform electric field of magnitude 550 N/C with the field pointing in the direction from one point to the other. a) Sketch the situation and depict the electric field. b) Find the potential difference between the two points.

  16. Connection between Electrostatic Potential Energy and Potential Difference • Knowing the potential difference between two point tells you the change in electrostatic potential energy between those two points • ΔUelec = q0ΔV = -Welec

  17. Ex. 2.0 J of work are done when a charge of 5.0 pC moves from point A to point B. a) Sketch the situation. b) Find the change in Uelec of the charge. c) Find the potential difference between points A and B.

  18. Ex. A battery maintains a constant potential difference between two pieces of metal of 6.0 V. A charge of -2.0 nC moves between the plates. a) Sketch the situation b) What is the change of electric potential energy for the charge.

  19. Ex. Two parallel plates of metal are placed in a vacuum chamber. A battery is used to place a potential difference of 400 V between the plates. An electron is released from rest at the plate with the lower potential. a) Sketch the situation. b) What will happen to the electron and why? c) What will be the speed of the electron once it has crossed the gap between the plates?

  20. Energy Conservation • The electrostatic force is a conservative force, so work done by it is stored as a potential energy • Review: The Work – Kinetic Energy Theorem Wnet = ΔK where K = ½ mv2

  21. Energy Conservation • If no non-conservative work is done then mechanical energy is conserved • E = K + U • If Wnc = 0 then ΔE = 0

  22. The Electric Potential of a Point Charge Find the Potential difference between the points A and B along the same radial line outwards from the positive point charge Q Q A B

  23. Suppose the points A and B were at the same radii but in different directions as shown. Would the answer to the potential difference change? • Yes • No • Depends on the exact position B Q A

  24. The Electric Potential of a Point Charge • The potential difference is given by the difference of • The answer depends only on the radius of each point and not the direction

  25. The Electric Potential of a Point Charge • We can conclude, if a point charge of charge Q is a distance r from a point P, then the electric potential at the point P is given by

  26. At what distance from the charge Q is the electric potential 0? (TPS) • r = 0 (at the charge) • r = ∞ (very far from the charge • The electric potential is never 0 • Cannot be determined

  27. At what distance from the charge Q is the electric potential 0? (TPS) • r = 0 (at the charge) • r = ∞ (very far from the charge • The electric potential is never 0 • Cannot be determined

  28. The Electric Potential of a Point Charge • We can conclude, if a point charge of charge Q is a distance r from a point P, then the electric potential at the point P is given by • The electric potential of a point charge has built in that the potential is 0 when r = ∞

  29. Ex. A point P is 5.0 cm away from a small charged object with charge 2.0 μC. a) Sketch the situation. b) Find the electric potential at the point P c) What work would be done to place a -2.0 μC at the point P from very far away. d) What does the work?

  30. Superposition • The electric potential due to several charges is simply the sum of the potentials of the individual charges • The electric potential is a scalar. No need to take into account direction, but do need to include signs in calculation

  31. Ex. a) Find the electric potential at the point P. b) How much work would it require to bring a 7.5 pC charge to point P from very far away? c) Does the field do work or is work done on the field? -5.0 pc 3.0 PC

  32. The Potential of a Continuous Distribution of Charge • Suppose we a continuous distribution of charge • We can divide the object into small pieces that we treat like point objects dqi ri ++++++++ ++++++ ++++++ P

  33. The Potential of a Continuous Distribution of Charge • To find the total potential at P, we add up all the little pieces

  34. Ex. Find the potential on the axis of a uniform positively charged ring of charge at a point P a distance x from the center of the ring.

  35. The Potential of a Continuous Distribution of Charge • Each small piece of charge dqi contributes a small piece to the potential dVi given by

  36. The Potential of a Continuous Distribution of Charge 0 • To find the total potential at the point P

  37. Determining the Potential from the Field • We can in principle find the potential for any distribution of charge using • However if we know the electric field, it can be easier to use the definition of the potential difference to find the potential

  38. Ex. A very long, insulating, solid cylinder of radius is uniformly charged with a positive charge. a) Sketch the cylinder and discuss what the symmetry tells you about the E-field. b) Use Gauss’s Law to find the Electric Field for r < R and R > r. c) Use the electric field to find the potential for r < R and r > R

  39. Determining E from V • If you know the potential in a region of space, you can determine the electric field • Since • We can find E from V by taking a derivative • In 1D we can write

  40. Ex. A potential along the x-axis is given by V(x) = kQ/|x|. Find the electric field as a function of x. If Q is positive, what is the direction of the electric field in the +x and –x directions?

  41. Ex. Two parallel metal plates are separated by 2.0 mm. The plates are connected to the opposite terminals of a 6.0 V battery. a) Sketch the situation. b) Find the average electric field in the region between the plates. c) Indicate the direction of the electric field in the region between the plates

  42. A region of space has a constant positive electric potential. What can you say about the electric field in that region? • It is constant and positive • It is constant and negative • It is 0 • Cannot be determined

  43. A region of space has an electric field of o. What can you say about the electric potential in that region? • It is constant and positive • It is constant and negative • It is 0 • Cannot be determined

  44. Equipotential Surfaces • A surface at a constant electric potential is an equipotential surface • Electric field lines are perpendicular to equipotential surfaces and point form higher ot lower potential • Two conductors in contact at electrostatic equilibrium will be equipotential • Electric Field is higher near more sharply curved points on an equipotential surface

  45. The figure shows a set of equi- potential surfaces measured by a student. Find the average electric field and indicate the direction of the electric field in each region.

  46. Equipotential Surfaces • An equipotential surface is one in which the potential difference between any two points on the surface is 0 • Is a conductor in electrostatic equilibrium an equipotential surface?

  47. Dielectric Beakdown

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