From Last Time:

1. From Last Time: • Electric field of a single charge • Electric field from multiple charges • Superposition • Ex: calculate electric field of dipole Today: • Continuous charge distributions • Electric field of line, ring, plane • Conductors : electrostatic equilibrium • Electrostatic potential Physics 208 Lecture 9

2. Exam 1 results B BC • Average = 66% • Curve • Ave at B/BC boundary • Contributes to course grade as curved score. • Exams returned in Monday discussion AB C A D F Physics 208 Lecture 9

3. From Last Time: • Electric field of a single charge • Electric field from multiple charges • Superposition • Ex: calculate electric field of dipole Today: • Continuous charge distributions • Electric field of line, ring, plane • Conductors : electrostatic equilibrium • Electrostatic potential Physics 208 Lecture 9

4. Charge Densities • Volume charge density: when a charge is distributed evenly throughout a volume •  = Q / V dq =  dV • Surface charge density: when a charge is distributed evenly over a surface area •  = Q / A dq = dA • Linear charge density: when a charge is distributed along a line •  = Q / dq =  d Physics 208 Lecture 9

5. Infinite line of charge + + + + + + + + + + + + + + + + + + + An infinite line of charge has a uniform charge density  Coulombs / meter. What direction is the electric field at point x? B. C. x A. D. E. Physics 208 Lecture 9

6. E-field: infinite line of charge r x y • Charge density Coulombs/meter + + + + + + + + + + + + + + + + + + + + Physics 208 Lecture 9

7. Add all these up • Units? Physics 208 Lecture 9

8. Non-infinite (finite) line of charge + + + + + + + + + + + + + + + + + + + + What direction is the E-field above the end of the line charge? C D B E A Physics 208 Lecture 9

9. Ring of uniform positive charge A) B) C) D) E) z Which is the graph of on the z-axis? y x z Physics 208 Lecture 9

10. Quick quiz We have an infinite sheet of charge of uniform charge density . The electric field increases with distance from plane decreases with distance from plane is independent of distance from plane changes direction with distance from plane Physics 208 Lecture 9

11. E-field of plane of charge Surface charge density • Text does this as adding up rings of charge. Physics 208 Lecture 9

12. Electric fields and forces • Original definite of E-field was(Coulomb Force) / charge • E-field produced force on charged particle. Physics 208 Lecture 9

13. Force on charged particle • Electric field produces force qE on charged particle • Force produces an acceleration a = FE / m • Uniform E-field (direction & magnitude) produces constant acceleration • Positive charge • accelerates in direction of the field • Negative charge • accelerates in direction opposite the electric field Physics 208 Lecture 9

14. Motion of charged particle • If no other forces, positive charge accelerates in direction of E-field. • But many systems have drag forces (e.g. molecules in a liquid, etc) • Drag force is complex, but usually depends on velocity. • Particle reaches terminal velocity, determined by force balance Physics 208 Lecture 9

15. Application: Gel Electrophoresis Steady state: FE+FD=0 • Charged macromolecules in ‘gel’ with applied E-field • Electric force: FE = qEDrag force: FD ~ -cv • v = qE / c • Speed depends on charge and drag (molecule size) • Sometimes too complex to interpret • Protein electrophoresis:soak in detergent to give proteinsall the same charge density. • Result, small proteins move faster Physics 208 Lecture 9

16. Conductors: electrostatic equilibrium Because charges are mobile in conductor : • E-field inside conductor = 0 • Non-zero E-field causes charges to move • Stop at surface, generate canceling E-field • Local net charge resides at surface • Repulsive force between charges pushes them away from each other – stop at surface • E-field at locally perpendicular to surface • A parallel component would cause charge motion • Has magnitude Surface charge density

17. Physics 208 Lecture 9 Charge distribution on conductor • What charge is induced on sphere to make zero electric field? +

18. Electrogenic fish • Dipole + nearby conducting object Some fish generate charge separation - electric dipole. Dipole is induced in nearby (conducting) fish Small changes detected by fish.

19. - - + + - + - + - - + + Summary of conductors • everywhere inside a conductor • Charge in conductor is only on the surface • surface of conductor

20. + + Electric forces, work, and energy • Consider positive particle charge q, mass m at rest in uniform electric field E • Force on particle from field • Opposite force on particle from hand • Let particle go - it moves a distance d • How much work was done on particle? • How fast is particle moving? v>0 v=0

21. In our case, Work and kinetic energy • Work-energy theorem: • Change in kinetic energy of isolated particle = work done • Total work

22. + Electric forces, work, and energy • Same particle, but don’t let go • How much force does hand apply? • Move particle distance d, keep speed ~0 • How much work is done by hand on particle? • What is change in K.E. of particle? Conservation of energy? W stored in field as potential energy +

23. Work done on system Change in electric potential energy Change in kinetic energy Work, KE, and potential energy • If particle is not isolated, Works for constant electric field if • Only electric potential energy difference • Sometimes a reference point is chosen • E.g. • Then for uniform electric field

24. Electric potential V • Electric potential difference V is the electric potential energy / unit charge = U/q • For uniform electric field, This is only valid for a uniform electric field

25. Quick Quiz Two points in space A and B have electric potential VA=20 volts and VB=100 volts. How much work does it take to move a +100µC charge from A to B? +2 mJ -20 mJ +8 mJ +100 mJ -100 mJ

26. + Check for uniform E-field Push particle against E-field, or across E-field Which requires work? Constant electric potential in this direction + Increasing electric potential in this direction Decreasing electric potential in this direction

27. Potential from electric field • Potential changes largest in direction of E-field. • Smallest (zero) perpendicular to E-field V=Vo

28. Electric potential: general • Electric field usually created by some charge distribution. • V(r) is electric potential of that charge distribution • V has units of Joules / Coulomb = Volts Electric potential energy difference U proportional to charge q that work is done on Electric potential difference Depends only on charges that create E-fields

29. for point charge Electric potential of point charge • Electric field from point charge Q is • What is the electric potential difference? Define Then

30. Electric Field and equipotential lines for + and - point charges • The E lines are directed toward the source charge • A positive test charge would be attracted toward the negative source charge • The E lines are directed away from the source charge • A positive test charge would be repelled away from the positive source charge Blue dashed lines are equipotential