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Chapter 25

Chapter 25. Electric Potential. Introduction. We’ve used conservation of Energy and the idea of potential energy associated with conservative forces (spring/gravity) in our study of mechanics. The electrostatic force is also conservative (note similarities to gravity)

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Chapter 25

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  1. Chapter 25 Electric Potential

  2. Introduction • We’ve used conservation of Energy and the idea of potential energy associated with conservative forces (spring/gravity) in our study of mechanics. • The electrostatic force is also conservative (note similarities to gravity) • We can use Electric Potential Energy to study phenomena and also to define a new scalar quantity, Electric Potential.

  3. 25.1 Potential Difference and Electric Potential • When a test charge qo is placed into an E-field, E, created by a charge distribution, we know the electric force on qo is found by (conservative) • If some external agent moves the charge within the electric field, the work done by the electric field on the charge is equal to the negative work done by the agent. • Gravitational Analog

  4. 25.1 • When talking about Electric and Magnetic fields often ds will be used to represent infinitesimal displacement vectors tangent to a path through space. • Adding up the work done along the path (either curved or straight) is called the “path integral” or “line integral”

  5. 25.1 • For an infinitely small displacement, ds, of a charge, qo, the work (W = Fd) done by the electric field is • As this amount of work is done by the field, the potential energy of the charge-field system is changed by a small amount

  6. 25.1 • If we look at a finite path from points A to B, the change in potential energy of the system is • This integration is performed along the path that qo follows as it moves from A to B. • Because qoE is conservative, this line integration does not depend on the path taken from A to B.

  7. 25.1 • If we divide qo, we can define a term that measures “Potential Energy per unit test charge” which now solely depends on the source charge distribution. • This quantity U/qo is called the electric potential (or simply “potential”) V.

  8. 25.1 • Since U is a scalar, V is also a scalar. • If we move a test charge between points A and B, the system experiences a change in potential energy. • The Potential Difference between A and B is found by

  9. 25.1 • Just as we saw with (grav) potential energy, only the differences are meaningful. • We will take the value of electric potential to be zero at some convenient point in the field. • Electric Potential is a scalar characteristic of an electric field, independentofanycharges that may be placed in the field.

  10. 25.1 • If an external agent moves a test charge from A to B, without changing the kinetic energy, the work done is simply equal to the change in potential energy. and therefore

  11. 25.1 • Units- Electric Potential is a measure of potential energy per unit charge, the SI unit is a Joule/Coulomb, defined as a volt (V). • Or, 1 J of work must be done to move a 1-C charge through a potential difference of 1 volt.

  12. 25.1 • Since potential difference also has units of E-field times distance, E-field can also be expressed in volts per meter. • E-field can now be interpreted as a measure of the rate of change of electric potential, with position.

  13. 25.1 • A common energy unit for Atomic and Nuclear physics is the electron volt (eV) • Defined as the energy gained/lost by a system when an electron/proton moves through a potential difference of 1 V. • Since 1 V = 1 J/C, and e = 1.60 x 10-19 J

  14. 25.1 Example The electron beam of a typical CRT television reaches a speed of 3.0 x 107 m/s. a. What is the kinetic energy (in eV) of a single electron? b. What potential difference is required to accelerate this electron from rest?

  15. 25.1 Quick Quizzes p 765

  16. 25.2 Potential Differences in a Uniform Electric Field • While the equations for Electric Potential Energy and Potential Difference hold in any field, they can be simplified if the field is uniform. • First consider a uniform E field in the negative y direction.

  17. 25.2 • We can calculate the potential difference between points A and B, separated by a distance |s| = d, where displacement vector s is parallel to the field lines.

  18. 25.2 • Since E is constant we can remove it giving • The negative indicates that the potential at B is lower than potential at A. • Electric field lines point in the direction of decreasing electric potential.

  19. 25.2 • Now if we move a test charge qo from A to B, we can calculate the change in potential energy of the charge-field system • A system of a positive charge and electric field loses potential energy when the charge moves in the direction of the field.

  20. 25.2 • We can imagine what would happen if we release a positive test charge from rest in a field. • The net force would be qoE • The charge would accelerate. • A gain of kinetic energy • Loss of Potential Energy

  21. 25.2 • If the test charge qo is negative, the opposite is true. The system gains potential energy if the charge moves in the direction of the field. • A negative charge would accelerate in a direction opposite to the field, gaining K, losing U.

  22. 25.2 • A more general case is if the charge moves a displacement vector s, that is not parallel to the field lines.

  23. 25.2 • Again with a uniform E field, it can be removed from the integral. • So the Potential Energy of the charge-field system is.

  24. 25.2 • Now, all points in a plane that is perpendicular to the uniform field have the same Electric Potential. • We can see this is true from the cosine component within the dot product. • The Potential Difference VB-VA is equal to VC-VA

  25. 25.2 • Equipotential Surface- any surface having a continuous distribution of points having the same electric potential • For a uniform E-Field, equipotential surfaces are a family of parallel planes that are all perpendicular to the field. • Quick Quizzes p 766 • Examples 25.1, 25.2

  26. 25.3 Electric Potential and Potential Energy Due to Point Charges • We can determine the electric potential around a single point charge. • Consider points A and B near a source charge q. • As it moves through ds, its radial distance changes by dr, where dscosθ = dr

  27. 25.3 • We can then determine the change in electric potential from points A to B.

  28. 25.3 • We see that this result is independent of the path from A to B, and therefore the Electric field of a fixed point charge is conservative. • Electric Potential at any distance from a charge is • V = 0 at ∞

  29. 25.3 • As a scalar quantity the electric potential around multiple charges is simply the sum of electric potentials. • Example Dipole The steep slope indicates A strong E field between the charges.

  30. 25.3 • Potential energy is • And therefore, between two charges

  31. 25.3 • And with several charges (Example of 3) • Using q’s +/- takes into account whether postive or negative work must be done to keep the charges in place.

  32. 25.3 • Like V, U = 0 at ∞ • Quick Quizzes p 770 • Example 25.3

  33. 25.4 Obtaining E-Field from Electric Potential • Equipotential Surfaces are must always be perpendicular to the electric field lines passing through them. • Uniform Field

  34. 25.4 • Point Charge

  35. 25.4 • Electric Dipole

  36. 25.5 Electric Potential from Continuous Charge Distributions • Adding up each little potential of each piece of charge. • So the total potential will equal

  37. 25.5 • Examples 25.5-25.8

  38. 25.6 Electric Potential due to Charged Conductors • Charged Conductors • Charge resides on the surface. • E-field just outside the surface is perpendicular and equal to σ/εo • E-field inside is zero.

  39. 25.6 • Consider two points on a charged conductor. • Since E is always perp. to the surface, any small displacement along the surface ds will be perp to E. (E.ds = 0) • Therefore ΔV = 0 • The surface of a charged conductor is equipotential

  40. 25.6 • Also, since the E-Field inside the conductor is zero, the rate of change of voltage dV is zero, so V must be constant. • The potential inside is equal to the potential at the surface.

  41. 25.6 • The surface density is uniform on a conducting sphere. • An irregular conductor will have greater charge density (and also E) at convex points with small radii of curvature. • Sharp points on the conductor will have the highest charge density.

  42. 25.6 • Example 25.9 p 780

  43. 25.6 • Corona Discharge- High Voltage conductors can cause ionization in the air molecules • Separated electrons are accelerated away from parent molecules, causing additional ionizations • Eventually the electrons/molecules recombine, giving off a dim glow (excited state -> ground state)

  44. 25.6 • The Corona Discharge effect tends to occur in at sharp points and edges of conductors. • Useful for identifying fraying wire strands, broken insulators etc. • Still difficult because the majority of the radiation is in the UV band, washed out by sunlight.

  45. 25.7-25.8 • Read p. 781-784 • Millikan Oil-Drop Experiement • Determined the value of e. • Other Applications of Electrostatics

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