Corrientes de Magnetizacion



τ 

S  τ 

 n

S 

 r  M(r )

dτ   r

dτ  dm = M(r )dτ 

dm P 

r

µ0 dm × (r − r ) µ0 M(r ) × (r − r )  dA(r) = = dτ  4π |r − r|3 4π |r − r |3 P 



dτ 

τ 

µ0   M(r ) × (r − r )  dτ  A(r) =  dA(r) = τ  4π τ  |r − r |3

 



r−r |r−r |3

= ∇





1

|r−r |

 M(r )

× (r − |r − r |3

  r)

  = M(r ) × ∇ 

− ∇ × (f A) ∇ × (f A) = f (∇ × A) − A × ∇f  f  = f (r )  f  = |r−1r| M(r ) A) 

   M(r ) × ∇ 

  

1 |r − r |

A

× ∇f  = f (∇ ×

A

= A(r ) A

        = ∇ × M(r ) − ∇ ×  M(r )    

1 |r − r |

|r − r |

|r − r |

A(r)

µ0 A(r) = 4π



 M(r )

∇ × µ0  dτ  − τ  |r − r| 4π

 

   M(r )       ∇ × dτ   τ 

 

|r − r |

=



   ∇ × F dτ  = −   F × dS τ 

F

S 

τ  S  dS S 

S 

   µ M(r )       ∇ × dτ  =   τ  4π |r − r | µ   M(r ) µ    0

−

  0

4π

S  |r

−

r |

× dS = −

0

4π

S 

 M(r )

× n  dS   |r − r | P 

µ0   ∇ × M(r )  µ0   M(r ) × n  dτ  + dS  A(r) =     τ  S  4π |r − r | 4π |r − r |  KM (r )  KM (r )

= M(r ) × n

 JM (r )  JM (r )

= ∇  × M(r)

S   r

∈ S 

τ  r



∈ τ 

µ0   KM (r )  µ0   JM (r )  dS  + dτ  A(r) = 4π S  |r − r| 4π τ  |r − r |



 KM (r )

S   JM (r )

τ  S   KM (r ) 

τ   JM (r )  KM (r )

 JM (r )  KM (r )  JM (r )

P 

µ0   KM (r) × (r − r)  µ0   JM (r ) × (r − r )  dS  + dτ  B(r) =     3 3 S  τ  4π |r − r | 4π |r − r |



 KM (r )  JM (r )

 KM (r )

a×b×c

M

= M 0uz 

KM |z =0

= M × n = M 0uz  × (−uz ) = 0 KM |z =c = M × n = M 0 uz  × (+uz ) = 0

 JM (r )



KM |x=0

= M × n = M 0uz  × (−ux ) = −M 0 uy KM |x=a = M × n = M 0 uz  × (+ux ) = +M 0 uy KM |y =0 = M × n = M 0 uz  × (−uy ) = +M 0 ux KM |y=b = M × n = M 0 uz  × (+uy ) = −M 0 ux

JM 

= ∇ × M = ∇ × (M 0uz ) =

∂M 0 ∂M 0 ux − uy = 0 ∂y ∂x



KM |x=0 KM |x=a KM |y=0

M

= M z (x)uz 

KM |y =b

M z (x)

x

dM z  0 JM  = ∇ × M = − uy = dx

x



y

JM 



z  a M

l

= M 0uz 

KM |z =0

= M × n = M 0uz  × (−uz ) = 0 KM |z =l = M × n = M 0 uz  × (+uz ) = 0

KM |ρ=a

= M × n = M 0uz  × uρ) = M 0 uϕ

∇ × (M 0uz ) = 0



M 0

NI/l N  l

NI/l = M 0

I 



