Explore examples of magnetostatic fields, magnetic dipoles, and hyperfine interaction in this detailed lecture covering various forms of Amperes law, vector potential, current density sources, electron orbital magnetic dipole moments, and the interaction energy near a nucleus due to an orbiting electron.
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Presentation Transcript
PHY 712 Electrodynamics 12-12:50 AM MWF Olin 103 Plan for Lecture 12: Continue reading Chapter 5 A. Examples of magnetostatic fields B. Magnetic dipoles C. Hyperfine interaction 02/10/2020 PHY 712 Spring 2020 -- Lecture 12 1
Various forms of Amperes law : ( ) r ( ) r = B J 0 ( ) : J ( ) r = = B r A Vector potential : ( ) r A Coulomb For gauge = 0 ( ) ( ) density r J 2 A r r 0 confined For current : ( ) ' ( ) r = 3 A 0 ' d r r r 4 ' 02/10/2020 PHY 712 Spring 2020 -- Lecture 12 3
Other examples of current density sources: expression mechanical Quantum current for density r particle a for of mass M charge and e and of probabilit amplitude y : ( ) ( ) ) r e = * * J r r r r ( ) ( ) ( ) ( ) ( 2 Mi For an electron in a spherical potential (such as in an atom): R = ( ) r ( ) r ( ) r Y ( ) r nlm nl lm l l ( ) ( ) r * r Y Y 1 e Mi ( ) J r ( ) r ( ) r ( ) r 2 lm lm = * R Y Y l l nl lm lm 2 sin r l l m e M r ( ) r 2 = l nlm sin l z r sin r = + = y x Note that: sin cos m e M r 2 ( ) J r ( ) ( r ) = z r l nlm 2 2 sin l 02/10/2020 PHY 712 Spring 2020 -- Lecture 12 4
Details of the electron orbital magnetic dipole moment m e m r ( ) r 2 = ( ) J r l n l m = sin l e + x y Note that: Magnetic dipole moment: 1 = 2 sin cos r e m ( ) r 2 = 3 3 m r ( ) J r d r d r l n lm 2 sin ' m r l e ' e m r ( ) r 2 = 3 d r l nlm 2 s i n ' m r l e Note that: z e m ( ) r 2 = 3 m d r l nl m 2 m m l e e z = l 2 m e 02/10/2020 PHY 712 Spring 2020 -- Lecture 12 5
Summary of magnetic field generated by point magnetic dipole moment: ( ) r r 3 8 = + e e B ( ) r ( ) r 0 e 3 4 3 r e Magnetic field near nucleus due to orbiting electron: 1 ( ) 4 "Hyperfine" interaction energy: eL m = B r z 0 O z 3 r e ( ( ) = + H B ( ) r B r ( ) HF N O e )( ) r r 3 L 8 e + + N e r N e =4 ( ) r 0 N N e 3 3 3 m r e 02/10/2020 PHY 712 Spring 2020 -- Lecture 12 6
( )( ) r r 3 L 8 e + + N e r N e H = ( ) r 0 N HF N e 3 3 4 3 m r e e e N N 02/10/2020 PHY 712 Spring 2020 -- Lecture 12 7
