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Understanding Electric Fields, Gauss's Law, and Induced Dipoles in Conductors and Dielectrics

This comprehensive overview explores the concepts of electric fields, Gauss's Law, and the behavior of induced dipoles in various materials. It highlights the significance of uniform electric fields and the symmetrical properties of conductors and dielectrics. The relationship between electric fields and induced dipoles in atoms such as helium and sulfur is discussed, alongside the principles of magnetic fields and their continuous loops as per Ampere's Law. Key formulas and principles are presented to aid in understanding these fundamental concepts in electrostatics and magnetism.

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Understanding Electric Fields, Gauss's Law, and Induced Dipoles in Conductors and Dielectrics

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  1. +qfreeon inner surface + + + + + + + + + + + + -qbound Symmetry – fields must be uniform – field lines perpendicular to plates +qbound - - - - - - - - - - - - -qfreeon inner surface Interior points electric field must be zero

  2. + + + + + conductor - - - dielectric Gauss’s Law

  3. dielectric constant frequency

  4. + + + + + + + + + + + + - - - - - - - - - - - - V = 0

  5. iron core BFe HFe gap region Bgap Hgap i coil windings Bair Hair

  6. . . . . . . . . 3 4 X X X X X X Current i out of page Circulation loop: square of length L 1 2 Current i into page Cross-section through electromagnet

  7. thickness t width L q = - e area A electrons are the charge carriers in copper

  8. + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + -

  9. + + + + + + + + + dy + + + + + + + + + F - - - - - - - - - -q +q

  10. + + + + + + + + + V r - - - - - - - - - x L-x C = CA + CB

  11. Induced dipole moment – helium atom +2e +2e -e -e -e -e Zero electric field – helium atom symmetric  zero dipole moment A B effectively charge +2e at A and -2e at B dipole moment p = 2ed

  12. Induced dipole moment – sulfur atom +16e +16e -8e -8e -8e -8e Zero electric field – helium atom symmetric  zero dipole moment A B effectively charge +16e at A and -16e at B dipole moment p = 16ed

  13. Er E P r2  r + (d/2)cos r1  r – (d/2)cos r (d/2)cos  +q -q

  14. dA +f + + + + + + + + +  -b   +b  - - - - - - - - - -f

  15. -q +q

  16. S +f -b O r +b -f

  17. - - - S d  r + Pcos + +  The area of the shaded ring between  and  + d is equal to

  18. +Ze +Ze d a a d << a

  19. F F +Q  d F - Q

  20. + p E U 0 - p E π π/2 0 

  21. Po T

  22. 1 slope = 1/3 0 10 pE/kT

  23. Gaussian surface S conducting sphere q air a r non-conducting liquid Symmetry  field lines must be radial

  24. conducting sphere q Eairt air Eliquidt non-conducting liquid Symmetry  Eairt = Eliquidt Eair = Eliquid = E

  25. field lines of E field lines of D

  26. - + + - - + + shift in atoms due to ionic nature of bond induced dipoles due to shift in electron cloud rotation orientation of polar molecules

  27. 6 5 4 S N HFe Hair Circulation loop: square side L 2 1 3

  28. Cylindrical Gaussian surface Gauss’s Law for magnetism B-field lines – form continuous loops

  29. N pole Bound surface currents (right hand screw rule)  

  30. un-magnetized piece of iron Bar magnet bought near un-magnetized piece of iron N N N  Bar magnet will attract the iron that was initially un-magnetized north pole attracts south pole

  31. Cu ramp Fe ramp plastic ramp N N N

  32. Circulation loop for circulation integration used in applying Ampere’s Law N N Hair Hiron

  33. if if d

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