Magnetostatics: Comparison and Alternative Forms of Equations
Explore the concepts of magnetostatics including magnetic flux density, current densities, and differential equations. Delve into the comparison with electrostatics and understand the implications of Ampere's law in magnetostatics. Discover the alternative forms of magnetostatic equations and the absence of magnetic monopoles in this study.
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PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 11: Start reading Chapter 5 A. Magnetostatics B. Vector potential C. Example: current loop 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 1
02/09/2015 PHY 712 Spring 2015 -- Lecture 11 2
Magnetostatics Magnetic flux density or magnetic induction field B Steady state (time constant) current density J y ( ) i = 3 J r v r r ( ) iq i i qi vi ri x = J Note that " statics" implies that . 0 continuity the from follows This J t equation : + = 0 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 3
Comparison of electrostatics and magnetostatics ( ) r charge to due field tic Electrosta density : r r 1 ' ( ) r ( ) r = ' 3 E ' d r 3 4 r r ' 0 ( ) r J Magnetosta current to due field tic density : r r ' ( ) r ( ) r = 3 B J 0 ' ' d r 3 4 r r ' 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 4
Alternative forms magnetostatic equations ( ) J r Magnetostatic field due to current density : r r r r r r r r ' 1 ( ) ( ) J r ( ) J r = = 3 3 B r ' ' ' ' 0 0 d r d r 3 r r 4 4 ' ' 1 ' = Note that: 3 r r ' ' ( ) ( ) ( ) r V r J r r ( ) r ( ) ( ) r ( ) = + V r V r Also note that: s s s ( ) ' ( ) = 3 B r ' 0 d r = V r V r ( ) J r In this case ( so that ) r 4 ' ( ) 0 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 5
Alternative forms magnetostatic equations -- continued ( ) ' 4 = r J r B ( ) J r ' = 3 B r 0 ' d r r r ( ) ( ) of B r 0 = magnetic No monopoles ( ) Ampere' law s 0 " Proof" Ampere' magnetosta for law s J r d system tic : ( ) r ' ( ) r = 3 B 0 ' r r 4 ' ( ) = 2 V V V Note that : 1 ( ) ( ) r = = 2 3 r r J Recall that : 4 ' and 0 r PHY 712 Spring 2015 -- Lecture 11 ' r 02/09/2015 6
Differential forms of magnetostatic equations: ( ) ( ) 0 r B = = B r 0 J magnetic No monopoles ( ) r Ampere' law s Magnetostatic vector potential ( ) ( ) r B = 4 ( ) r = B r A ( ) r ' J ' 3 0 ' d r r r ' ( ) r J ( ) r ( ) r = + 3 A 0 ' d r s r r 4 ' 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 7
Non uniqueness of the magnetostatic vector potential ( ) ( ) r ( ) r ( ) r = = B r A A Note that : = ' ( ) r ( ) r + A A if ' s ( ) r = = B z Example : for B 0 ( ) ( ) ( ) r ( y y ) A r y x B x y 1 0 2 B = A r or x 0 B = A x or 0 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 8
Differential form of Amperes law in terms of vector potential: ( ) ( ( ) ( ) r A ( ) ( ) gauge) = = B r A r J r 0 ( ) ( ) r ) = 2 A r J r 0 ( ) r ( ) r = = 2 A A J If (Coulomb 0 0 ( ) r J ' ( ) r = 3 A 0 ' d r r r 4 ' 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 9
Magnetostatics example: current loop z y x I ( ) r ( ) ( )( ) = + J x y ' ' ' sin ' cos ' ' sin cos r a a ( ) r J ' ( ) r = 3 A 0 ' d r r r 4 ' 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 10
Magnetostatics example: current loop -- continued ( ) ( ) ( )( ( ) 2 0 ' ' cos ' ' 4 r a I ) = ' ' + J r x y ' ' ' sin cos ' sin cos r a a J r ' ( ) r = 3 A 0 ' d r 4 r r ' ( ) ( )( ) 2 rr cos sin sin + ' x y ' ' sin r ' cos ' ' sin cos I r a ( ) r = d A r dr d ( ) ( ( ) ) / 1 2 + + ' 2 2 ' ' cos cos ' Completing integratio over n ' and : ' r ( ) 2 + 2 x y ' ' sin cos Ia ( ) r 0 = d A 0 ' ( ) ( ( ) ) 4 / 1 2 a + 2 2 2 sin cos ' r a ra ' sin Let ' ( ) = ' = sin sin cos cos sin ( non ) trivial = - = terms + cos cos cos cos sin sin Remaining 2 sin cos ra Ia ( ) r ( ) 0 = d A x y 0 sin cos ( ) ( ) 4 / 1 2 + 2 2 2 cos r a 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 11
Magnetostatics example: current loop -- continued 2 sin cos ra Ia ( ) r ( ) 0 = d A x y 0 sin cos ( ) ( ) 4 / 1 2 + 2 2 2 cos r a Elliptic integrals : / 2 du 0 = ( ) K m ( 1 ) / 1 2 2 sin m u / 2 ( 1 ) / 1 2 0 = 2 ( ) sin E m m u du ( ) ( + ) 2 2 x + y sin a cos ra 2 ( k ) 2 ( ) k K k E k ( ) r = A 0 4 Ia ( ) 4 / 1 2 2 2 2 sin r ra 4 a sin ar 2 where : k + + 2 2 2 sin r ( ) r ( ) r = B A 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 12
Magnetostatics example: current loop -- continued ( ( sin 2 ra a r + + y r x 0 , sin : 0 For = = = ( ) ) 2 x y + sin a + cos ra 2 ( k ) 2 ( ) k K k E k ( ) r = A 0 4 Ia ) 4 / 1 2 2 2 2 2 sin r 4 sin ar 2 where : k 2 2 ( ) 2 1 2 ( ) 2 ( ) k K k 2 E k ( ) r = = A y y ) 0 ( , 4 A x z Ia ( ) y / 1 2 4 k + + + 2 2 2 2 x z a ax 4 ax + 2 where : k + + 2 2 2 2 x z a ax Ay(x,z) 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 13
Magnetostatics example: current loop -- continued ( ) ( + ) 2 x + y sin a cos ra 2 ( k ) 2 ( ) k K k E k ( ) r = A 0 4 Ia ( ) / 1 2 2 4 2 2 2 sin r ra 4 a sin + ar where : k + 2 2 2 sin r ( ) r spherical for that Note = A ( ) r = B A = x y polar coordinate : s sin cos ( ) r ( ) r A ( ) 2 4 2 ( k ) 2 ( ) Ia k K k E k ( ) r A = 0 where ( ) r / 1 2 2 4 + + 2 2 2 sin rA r a ra ( ) ( ) ( ) r A ( ) r sin 1 1 r ( ) r = r B sin r For : r ( ) r 2 I a ( ) r + r B 0 2 cos sin 3 4 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 14
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 atom) an in as (such potential spherical a in electron an For Y R : ( ) r ( ) r ( ) r ( ) r = nlm nl lm l l ( ) r ( ) r * Y Y 1 e ( ) r ( ) r ( ) r ( ) r 2 * lm lm = J R Y Y l l nl lm lm 2 sin Mi r l l m e ( ) r 2 = l nlm sin M r l z sin r r = + = x y Note that : sin cos m e ( ) r ( ) ( r ) r 2 = J z l nlm 2 2 sin M r l 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 15
Magnetic vector potential for this case: ( ) r J ' ( ) r = 3 A 0 ' d r r r 4 ' m e ( ) r ( ) ( ' r ) 2 = J z r ' ' l nlm 2 2 ' sin ' M r l ( ) r 2 ( ) ' ' z r r ' e m ( ) r nlm = 3 A 0 ' l l d r 2 2 r 4 ' sin ' M r = example For electron : in the 211 state of H : nlm l 2 1 ' r ( ) r 2 = / ' r 2 a ' sin ' e 211 3 64 a a ( ) r 2 3 z r e r r r ( ) r = + + + / r a A 0 1 1 e 3 2 3 8 2 8 M a a a 02/09/2015 PHY 712 Spring 2015 -- Lecture 11 16
