Corrientes de magnetizacion PDF

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°

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 |r − r′|3 4π 4π |r − r′|3 P

°

dτ ′

τ′

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

r−r′ |r−r′ |3

′

=∇

µ

1 |r−r′ |

¶

M(r′) × (r − r′) 1  ′ ′   = M(r ) × ∇ |r − r′|3 |r − r′| 



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

A = A(r′) A =

M(r′)  1  ∇′ × M(r′) ′  ′ ′  =  −∇ × M(r ) × ∇  |r − r′| |r − r′| |r − r′| 







A(r) ′ µ0 Z ∇′ × M(r′) ′ µ0 Z  M(r )  ′ ′   dτ A(r) = dτ − ′ ′∇ × ′ τ ′ τ 4π |r − r | 4π |r − r | 



°

∇ × F dτ = − τ S dS R

τ

H

S

F × dS

S

F

S

′ µ0 Z  M(r )  ′ ′  dτ =  ∇ × ′ 4π τ |r − r′| µ0 I M(r′) µ0 I M(r′) × n′ ′ ′ − × dS = − dS 4π S ′ |r − r′| 4π S ′ |r − r′| 



P µ0 Z ∇′ × M(r′) ′ µ0 I M(r′) × n′ ′ A(r) = dτ + dS 4π S ′ |r − r′| 4π τ ′ |r − r′| KM (r′) KM (r′) = M(r′) × n′ JM (r′) JM (r′) = ∇′ × M(r′)

S′ r′ ∈ S ′ τ′ r′ ∈ τ ′

µ0 I KM (r′) ′ µ0 Z 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 I KM (r′) × (r − r′) ′ µ0 Z JM (r′) × (r − r′) ′ dS + dτ B(r) = |r − r′|3 |r − r′|3 4π S ′ 4π τ ′

°

KM (r′) JM (r′)

KM (r′)

a×b×c

M = M0uz

KM |z=0 = M × n = M0uz × (−uz ) = 0 KM |z=c = M × n = M0uz × (+uz ) = 0

JM (r′)

°

KM |x=0 = M × n = M0uz × (−ux) = −M0uy KM |x=a = M × n = M0uz × (+ux) = +M0uy KM |y=0 = M × n = M0uz × (−uy ) = +M0ux KM |y=b = M × n = M0uz × (+uy ) = −M0ux

JM = ∇ × M = ∇ × (M0uz ) =

∂M0 ∂M0 ux − uy = 0 ∂y ∂x

°

KM |x=0 KM |x=a KM |y=0

M = Mz (x)uz

KM |y=b

Mz (x)

x

x

JM = ∇ × M = −

dMz uy 6= 0 dx

°

y

JM

°

z a

l

M = M0uz

KM |z=0 = M × n = M0uz × (−uz ) = 0 KM |z=l = M × n = M0uz × (+uz ) = 0

KM |ρ=a = M × n = M0uz × uρ) = M0uϕ

∇ × (M0uz ) = 0

°

M0

N I/l N l

N I/l = M0

I

°

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