By A.L. Septier
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2 Concept of Bound Current 0 The magnetization pm makes a contribution to the current density j, known as the magnetization current or bound current. 47) and using the relations ðð ððð r Â Gdt ¼ À G Â ^ n dS; r Â ðuFÞ ¼ ur Â F À F Â ru and t S we can obtain the magnetic vector potential generated by magnetization: m ¼ 0 4p m ¼ 0 4p ððð Mðrq Þ Â rq tq ððð tq ! 1 rp À rq dtq rq Â Mðrq Þ m0 rp À rq dt q À 4p ððð rq Â tq ! Mðrq Þ rp À rq dtq: ððð rq Â tq ð2:50Þ ! ðð Mðrq Þ Mðrq Þ Â ^ n m dtq þ 0 r p À r q rp À rq dSq 4p 0 m Am ðrp Þ ¼ 0 4p Sq Defining the body bound current density: jmt ðrq Þ ¼ rq Â Mðrq Þ ð2:51Þ jms ðrq Þ ¼ Mðrq Þ Â ^ n; ð2:52Þ and surface bound current density: we obtain the vector potential and magnetic induction intensity m0 4p ððð j ðr Þ m mt q dtq þ 0 rp À rq 4p ðð j ðr Þ ms q dSq r p À r q 0 Am ðrp Þ ¼ tq Sq ð2:53Þ Magnetostatic Equations for the Magnet Structure 31 and !
It establishes an interpolation algorithm and transforms the extremal problem of this function into a set of algebraic equations to be solved. Moreover, the FEM can deal with the boundary problems of complex structures and with nonlinear media. Due to the recent strong advances in numerical processing and the progress in mesh generated techniques, the FEM plays a leading role in electromagnetic engineering. This chapter introduces the FEM in electromagnetic analysis and practical engineering applications.
For the first kind of boundary conditions, the expression is DA ¼ 0 (at boundary G2). For homogeneous boundary conditions of the second kind, the expression at the boundary G2 is: vz @A 0 @A 0 n þ vr n ¼ 0: @r r @z z ð3:15Þ The interfacial boundary condition is: @A1 0 @A1 0 @A2 0 @A2 0 n þ vr1 n À vz2 n þ vr2 n ¼ 0: vz1 @r r @z z @r r @z z Ð L Á Á Á dl and the third term @ vz @A @ vr @A þ ¼ 0: @r r @r @z r @z Ð 0 Therefore, in the variation equation, the first term necessary condition for DF(A) ¼ 0 is ð3:16Þ G Á Á Á dl vanish.
Applied Charged Particle Optics by A.L. Septier