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Electric Potential (III)

Electric Potential (III). - Fields Potential Conductors. Potential and Continuous Charge Distributions. We can use two completely different methods: Or, Find from Gauss’s Law, then…. Potential and Electric Field. Since .

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Electric Potential (III)

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  1. Electric Potential (III) • - Fields • Potential • Conductors

  2. Potential and Continuous Charge Distributions • We can use two completely different methods: • Or, Find from Gauss’s Law, then…

  3. Potential and Electric Field • Since therefore, we have (in Cartesian coords): Hence:

  4. Ex 1a:Given V=3x2y+y2+yz, find E.

  5. Ex 1b:Given V=(10/r2)sinθcos (spherical coords) • a) find E. • b) find the work done in moving a 10μC charge from A(1, 30o, 120o) to B(4, 90o, 60o).

  6. There are many coordinate systems that can be used: • Bipolarcylindrical, bispherical, cardiodal, cardiodcylindrical, Cartesian, casscylindrical, confocalellip, confocalparab, conical, cylindrical, ellcylindrical, ellipsoidal, hypercylindrical, invcasscylindrical, invellcylindrical, invoblspheroidal, invprospheroidal, logcoshcylindrical, logcylindrical, maxwellcylindrical, oblatespheroidal, paraboloidal, paracylindrical, prolatespheroidal, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal.

  7. Ex 2: Find the potential of a finite line charge at P, • AND the y-component of the electric field at P. P r d dq x L

  8. Solution

  9. Example: The Electric Potential of a Dipole y a a x P -q +q Find: a) Potential V at point P along the x-axis. b) What if x>>a ? c) Find E.

  10. Solution

  11. Example: Find the potential of a uniformly charged sphere of radius R, inside and out. R

  12. Uniformly Charged Sphere,radius R E r R V r R

  13. Example: Recall that the electric field inside a solid conducting sphere with charge Q on its surface is zero. Outside the sphere the field is the same as the field of a point charge Q (at the center of the sphere). The point charge is the same as the total charge on the sphere. Find the potential inside and outside the sphere. +Q R

  14. Solution (solid conducting) • Inside (r<R), E=0, integral of zero = constant, so V=const • Outside (r>R), E is that of a point charge, integral gives • V=kQ/r

  15. Solid Conducting Sphere,radius R E r R V r R

  16. Quiz A charge +Q is placed on a spherical conducting shell. What is the potential (relative to infinity) at the centre? +Q • keQ/R1 • keQ/R2 • keQ/ (R1 - R2) • zero R1 R2

  17. Calculating V from Sources: • Point source: (note: V0 as r ) or ii) Several point sources: (Scalar) iii) Continuous distribution: OR … I. Find from Gauss’s Law (if possible) II. Integrate, (a “line integral”)

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