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Potential

Potential. Line Integrals Force -> Work Electric Field -> Potential Conservative Loops Adding Potentials Single charge Multiple charges Continuum Examples (ring, sheet, line) Gradient Potential - > Electric Field Examples (ring, sheet, point) Example – Electric Dipole Stored Energy

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Potential

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  1. Potential • Line Integrals • Force -> Work • Electric Field -> Potential • Conservative Loops • Adding Potentials • Single charge • Multiple charges • Continuum • Examples (ring, sheet, line) • Gradient • Potential - > Electric Field • Examples (ring, sheet, point) • Example – Electric Dipole • Stored Energy • Stored in Charge • Stored in Field

  2. Topographical Analogy • Topographical Analogy • Elevation -> Potential • Slope -> Electric Field • Observations • Slope (E) is 3-D gradient of elevation (V) or contour lines • Change in Elevation (VA-VB) is independent of path to get there

  3. Topographical Analogy(little more complicated) Taos, NM

  4. Work moving charge in electric field • Field points to right • Positive charge feels force to right • Applied force must be to left to move in opposite direction • Work of applied force to move to left (opposite direction) over path • Positive work to move “up” the field to left • Increase in Potential Energy

  5. The integral expression for work is completely general: Any shape path may be taken, with the component of force evaluated on each differential path segment. The integral expression involving the scalar product of the field with a differential path vector is called a line integral or a contour integral. Work moving Charge over Arbitrary Path Reverse initial and final points to go “up” the field TO LEFT (sign of Eˑ dL)

  6. We wish to find: where and using these: Line Integral in Rectangular Coordinates

  7. An electric field is given as: We wish to find the work done in moving a point charge of magnitude Q = 2 over the shorter arc of the circle given by The initial point is B(1, 0, 1) and the final point is A(0.8, 0.6 ,1): This is the basic setup, in which the path has not yet been specified. Work moving charge along Circular Path in z= 1 plane

  8. We now have W and we need to include the y dependence on x in the first integral, and the x dependence on y in the second integral: Note that the third integral vanishes because there is no motion along the z direction. Using the given equation for the circular path, , we rewrite the integrals: Work moving charge along Circular Path (cont)

  9. Work moving charge along Straight-Line Path • Moving between same endpoints B(1, 0, 1) and A(0.8, 0.6 ,1) along straight line path: • So the integration becomes • So the line integral is the same, independent of path taken!

  10. Differential Path Lengths in 3 Coordinate Systems

  11. Work moving charge near Line of Charge Moving Q in radial direction in cylindrical coordinates • Work to move from b to a • Simplifies to: • Which is positive for b>a • Note path length is directed outward, integration supplies (-) sign • Simplifies to

  12. where as expected! Work moving charge near Line of Charge Moving Q in azimuthal direction in cylindrical coordinates

  13. We now have the work done in moving charge Q from initial to final positions. This is the potential energy gained by the charge as a result of this position change. The potential difference is defined as the work done (or potential energy gained) per unit charge. We express this quantity in units of Joules/Coulomb, or volts: Finally: Definition of Potential Difference Just lose the “Q”

  14. + . . B A rB rA Q Potential Difference of Point Charge Field In this exercise, we evaluate the work done in moving a unit positive charge from point B to point A, within the field associated with point charge Q where and where in general: The path used in getting to point A from point B is immaterial, since only changes in radius affect the result. Path independence would also qualify this field as conservative, but we need to show this.

  15. To complete the problem: we use along with: + . . B A rB rA to obtain: Q Potential Difference of Point Charge Field (cont) For rb > ra VAB is positive

  16. Absolute Potential for Point Charge Field • The difference in potential between rb and ra is • The difference relative to infinity (rb = ∞) is • So absolute potential can be defined as • Where C1 is arbitrary additive constant (doesn’t change gradient!)

  17. Introduce a second point charge, and the two scalar potentials simply add: For n charges, the process continues: Absolute Potential For Two or More Point Charges Electric potential from point charge

  18. As we allow the number of elements to become infinite, we obtain the integral expression: Absolute Potential for Continuous Charge Distributions 1. No sin or cos terms - just scalars! 2. Approaches zero at infinity

  19. Line Charge: Surface Charge: Volume Charge: Compare to our earlier expression for electric field --- generally a more difficult integral to evaluate: Potential Functions Associated with Line, Surface, and Volume Charge Distributions charge-density/length * length charge-density/area * area charge-density/volume * volume

  20. We use: with Example - Potential for ring of charge The problem is to find the potential anywhere on the z axis arising from a circular ring of charge in the x-y plane, centered at the origin.

  21. So now becomes: Potential for ring of charge (continued) charge-density/length * length 1. No sin or cos terms - just scalars! 2. Approaches zero at infinity

  22. Potential for ring of charge No sin or cos terms, Just scalars! (compare with previous) http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potlin.html#c1

  23. Potential for disk of charge No sin or cos terms, Just scalars! http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potlin.html#c1

  24. Potential for line of charge No sin or cos terms, Just scalars! http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potlin.html#c1

  25. Electric Field as Potential Gradient • Potential as line integral of field • Differential form • If ΔL in direction of E • Electric field as gradient of Potential

  26. The maximum rate of increase in potential should occur in a direction exactly opposite the electric field: unit vector normal to an equipotential surface and in the direction of increasing potential Equipotential surfaces E points in the direction of maximum rate of decrease in potential -- in the direction of the negative gradient of V. aN E Relation Between Electric Field and Potential • Skier’s analogy • Fall Line points opposite direction from maximum elevation increase • Constant elevation contour perpendicular to fall line.

  27. Skier’s Analogy – Slope http://www.skitaos.org/Taos, NM

  28. Skier’s Analogy – Elevation Lines http://topomaps.usgs.gov Taos, NM

  29. Electric Field from Potential (wire loop) Now get Electric Field from Potential! (Slide 21) • If Potential is: • Then Electric Field is • Comparing E with problem 2.24 (ρ = a, using only az term)

  30. The differential voltage change can be written as the sum of changes of V in the three coordinate directions: We also know that: So that: We therefore identify: Electric Field from V in Rectangular Coordinates

  31. This is obtained by using the del operator, on V Electric Field as Negative Gradient of Potential We now have the relation between E and V A more compact relation therefore emerges, which is applicable to static electric fields: E is equal to the negative gradient of V The direction of the gradient is that of the maximum rate of increase in the scalar field, or normal to all equipotential surfaces.

  32. Gradient of V in Three Coordinate Systems

  33. Electric Field from Potential (charge disk) Now get Electric Field from Potential! (Slide 23) • If Potential is: • Then Electric Field is • As

  34. Electric Field from Potential (point charge) Now get Electric Field from Potential! (Slide 16) • If Potential is: • Then Electric Field is

  35. Other examples • Example 4.4 • D4.8 • D4.7

  36. Example D4.8 • B) Electric Field at (3, 60°,2) • C) Magnitude of E at (3, 60°,2) • D) dV/dN at (3, 60°,2) • E) Normal to equipotential surface an at (3, 60°,2)

  37. Example D4.8 (continued) • F) Charge density at (3, 60°,2)

  38. Example

  39. Electric Dipole • Dipole • Positive and negative charge separated by d. • Used in describing atoms and molecules. • Spherically symmetric at great distances. • Azimuthally symmetric everywhere. • Put +Q at +d/2, -Q at –d/2

  40. Electric Dipole Approach • 2 approaches • Find electric field, use line integral for potential • Involves adding vectors – yech! • Line integration • Find potential, use gradient for electric field • Involves adding scalars – much simpler! • Simple 3-D differentiation

  41. Electric Dipole Potential • Put +Q at + d/2 (R1) , -Q at – d/2 (R2) • Make approximations • Get Dipole Potential

  42. Having found the potential: Electric field is found by taking the negative gradient: or.. from which finally: Electric Dipole Field (from Gradient) Much easier doing potential first!

  43. Dipole Equipotential lines • Equipotential lines obtained by plotting r vs. θ • fixed V’s • fixed dipole strength • Equipotential equations • Estimate r as function of θ (from vertical)

  44. Dipole field lines • Electric field lines obtained by finding streamlines • Taking differentials in r and θ direction • Estimate r as function of θ (from vertical)

  45. Dipole Moment • Define dipole moment • Then • Becomes • V max top and bottom (red) • V min bisector (red) • E always perpendicular (black)

  46. Q1 Q2 R2,1 + + Charge Q2 is brought into position from infinity. Q1 has zero energy if isolated The work done in bringing Q2 into position is: This is the stored energy in the “system”, consisting of the two assembled charges. Potential Energy of Two Point Charges

  47. Q1 Q2 R2,1 + + + R3,2 Charge Q3 is brought into position from infinity, with Q1 and Q2 already situated. R3,1 Q3 The system energy is now the previous 2-charge energy plus the work done in bringing Q3 into position: where and Potential Energy of Three Point Charges

  48. Three Point Charges in Reverse Order • Start 1, then 2, then 3 • Start 3, then 2, then 1 • Adding

  49. Extending the previous result, we can write the energy expression for n charges: where the local potential (at the position of charge m) is: Note that this is the potential due to all charges except charge m, evaluated at the location of charge m. Potential Energy for n Point Charges

  50. If we have a continuous charge, characterized by a charge density function, we use implicitly the expression but the charge Q is replaced by the quantity dq = v dv, and the summation becomes an integral over the charge volume: where V is the position-dependent potential function within the charge volume. Potential Energy for Continuous Charge Distribution

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