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MIT Class: Electric Potential

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  1. Workshop: Using Visualization in Teaching Introductory E&MAAPT National Summer Meeting, Edmonton, Alberta, Canada.Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy

  2. MIT Class: Electric Potential 2

  3. Potential Energyand Potential Start with Gravity

  4. Gravity: Force and Work Gravitational force on m due to M: Work done by gravity moving m from A to B: PATH INTEGRAL

  5. Work done by gravity moving m from A to B: Work Done by Earth’s Gravity

  6. PRS Question:Sign of Wg

  7. PRS: Sign of Wg Thinking about the sign and meaning of this… Moving from rA to rB: • Wg is positive – we do work • Wg is positive – gravity does work • Wg is negative – we do work • Wg is negative – gravity does work • I don’t know :19

  8. PRS Answer: Sign of Wg Answer: 3. Wg is negative – we do work Wg is the work that gravity does. This is the opposite of the work that we must do in order to move an object in a gravitational field. We are pushing against gravity  we do positive work

  9. Work done by gravity moving m from A to B: Work Near Earth’s Surface G roughly constant: Wg depends only on endpoints – not on path taken – Conservative Force

  10. Potential Energy (Joules) • U0: constant depending on reference point • Only potential difference DU has physical significance

  11. Define gravitational potential difference: Gravitational Potential(Joules/kilogram) That is, two particle interaction  single particle effect

  12. PRS Question:Masses in Potentials

  13. PRS: Masses in Potentials Consider 3 equal masses sitting in different gravitational potentials: A) Constant, zero potential B) Constant, non-zero potential C) Linear potential (V  x) but sitting at V = 0 Which statement is true? • None of the masses accelerate • Only B accelerates • Only C accelerates • All masses accelerate, B has largest acceleration • All masses accelerate, C has largest acceleration • I don’t know :19

  14. PRS Answer: Masses in Potentials Answer: 3. Only C (linear potential) accelerates When you think about potential, think “height.” For example, near the Earth: U = mgh so V = gh Constant potential (think constant height) does not cause acceleration! The value of the potential (height) is irrelevant. Only the slope matters

  15. Move to Electrostatics

  16. Gravity - Electrostatics Mass M Charge q (±) Both forces are conservative, so…

  17. Potential & Potential Energy Units: Joules/Coulomb = Volts Change in potential energy in moving the charged object (charge q) from A to B: Joules

  18. Potential & External Work Change in potential energy in moving the charged object (charge q) from A to B: Joules The external work is If the kinetic energy of the charged object does not change, then the external work equals the change in potential energy 18

  19. How Big is a Volt? Know These! • AA, C, D Batteries 1.5 V • Car Battery 12 V • US Outlet 120 V (AC) • Residential Power Line • Our Van de Graaf • Big Tesla Coil

  20. Potential: Summary Thus Far Charges CREATE Potential Landscapes

  21. Potential Landscape Positive Charge Negative Charge

  22. Potential: Summary Thus Far Charges CREATE Potential Landscapes Charges FEEL Potential Landscapes We work with DU (DV) because only changes matter

  23. 2 PRS Questions:Potential & Potential Energy

  24. 11 PRS: Positive Charge Place a positive charge in an electric field. It will accelerate from • higher to lower electric potential; lower to higher potential energy • higher to lower electric potential; higher to lower potential energy • lower to higher electric potential; lower to higher potential energy • lower to higher electric potential; higher to lower potential energy

  25. PRS Answer: Positive Charge Answer: 2. + acc. from higher to lower electric potential; higher to lower potential energy Objects always “move” (accelerate) to reduce their potential energy. Positive charges do this by accelerating towards a lower potential

  26. PRS: Negative Charge Place a negative charge in an electric field. It will accelerate from • higher to lower electric potential; lower to higher potential energy • higher to lower electric potential; higher to lower potential energy • lower to higher electric potential; lower to higher potential energy • lower to higher electric potential; higher to lower potential energy 18

  27. PRS Answer: Negative Charge Answer: 4. Neg. acc. from lower to higher electric potential higher to lower potential energy Objects always “move” (accelerate) to reduce their potential energy. Negative charges do this by accelerating towards a higher potential:

  28. Potential Landscape Positive Charge Negative Charge

  29. Creating Potentials:Calculating from E,Two Examples

  30. Potential in a Uniform Field Just like gravity, moving in field direction reduces potential

  31. Potential Created by Pt Charge Take V = 0 at r = ∞:

  32. PRS Question:Point Charge Potential

  33. -q +q P PRS: Two Point Charges The work done in moving a positive test charge from infinity to the point P midway between two charges of magnitude +q and –q: • is positive. • is negative. • is zero. • can not be determined – not enough info is given. • I don’t know :16

  34. -q +q P PRS Answer: Two Point Charges 3. Work from  to P is zero The potential at  is zero. The potential at P is zero because equal and opposite potentials are superimposed from the two point charges (remember: V is a scalar, not a vector)

  35. Potential Landscape Positive Charge Negative Charge

  36. Group Problem: Superposition Consider the 3 point charges at left. What total electric potential do they create at point P (assuming V = 0)

  37. Deriving E from V

  38. Deriving E from V A = (x,y,z), B=(x+Dx,y,z) Ex = Rate of change in V with y and z held constant

  39. Deriving E from V If we do all coordinates: Gradient (del) operator:

  40. PRS Questions:E from V

  41. PRS: E from V Consider the point charges you looked at earlier: You calculated V(P). From that can you derive E(P)? • Yes, its kQ/a2 (up) • Yes, its kQ/a2 (down) • Yes in theory, but I don’t know how to take a gradient • No, you can’t get E(P) from V(P) • I don’t know 15

  42. PRS Answer: E from V 4. No, you can’t get E(P) from V(P) The electric field is the gradient (spatial derivative) of the potential. Knowing the potential at a single point tells you nothing about its derivative. People commonly make the mistake of trying to do this. Don’t!

  43. PRS: E from V larger than that for x < 0 smaller than that for x < 0 equal to that for x < 0 I don’t know The graph above shows a potential V as a function of x. The magnitude of the electric field for x > 0 is :20 43

  44. PRS Answer: E from V The slope is smaller for x > 0 than x < 0 Translation: The hill is steeper on the left than on the right. Answer: 2. The magnitude of the electric field for x > 0 is smaller than that for x < 0 44

  45. PRS: E from V Ex > 0 is > 0 and Ex < 0 is > 0 Ex > 0 is > 0 and Ex < 0 is < 0 Ex > 0 is < 0 and Ex < 0 is < 0 Ex > 0 is < 0 and Ex < 0 is > 0 I don’t know The above shows potential V(x). Which is true? 20 45

  46. PRS Answer: E from V E is the negative slope of the potential, negative on the left, positive on the right Translation: “Downhill” is to the left on the left and to the right on the right. Answer: 2. Ex > 0 is > 0 and Ex < 0 is < 0 46

  47. Group Problem: E from V A potential V(x,y,z) is plotted above. It does not depend on x or y. What is the electric field everywhere? Are there charges anywhere? What sign?

  48. Demonstration:Making & Measuring Potential(Lab Preview)

  49. Configuration Energy

  50. Configuration Energy How much energy to put two charges as pictured? • First charge is free • Second charge sees first: