Lyu SP 1 Lecture 1 B-dipole

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Space Physics (I) [AP-3044]  Lecture 1  by Ling-Hsiao Ly u Oct. 2011 "#" Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines 1.1. Dipole Magnetic Field Since ! " B = 0  we can define B = ! " A  (1.1) where A  is called the vector potential. We use the bold face fon t to denote vector. For static magnetic field, we have ! " B  = µ 0 J  (1.2) Substituting Eq. (1.1) into Eq. (1.2) to eliminate B , it yields ! " (! " A) = #! 2 A + !(! $ A) = µ 0 J  (1.3) We choose the Coulomb gauge, ! " A = 0 , the equation (1.3) can be rewritten as ! 2 A  = " µ 0 J  (1.4)  Note that: If we choose the Coulomb gauge: ! " A = 0 , the scalar potential will contain no electromagnetic component. If we choose the Lorentz gauge:  1 c  2 !" (x, t ) ! t + # $ A (x, t ) = 0 , the scalar potential will contain an electromagnetic component. Eq. (1.4) is similar to the Poisson equation of the electrostatic potential ! 2 "  = #  $ c % 0  (1.5) General solution of Eq. (1.5) can be written as ! (r) = " c (r ') 4#$ 0  |  r %  r ' | &  d r '  (1.6) Special Case: The scalar potential create by a point charge q  is !  = q 4"# 0 r  Likewise, the general solution of Eq. (1.4) can be written as A(r ) = µ 0 J(r ') 4!  | r "  r ' | #  d r '  (1.7)

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