electro-magnetisme de ecuacions de fluxe-zero

Principi:

E_{e}(x,y,z) = qk_{e}·(1/r^{2})·( < x,y,z >/r )

B_{e}(x,y,z) = (-1)·qk_{e}·(1/r^{2})·( < d_{t}[x],d_{t}[y],d_{t}[z] >/r )

Lley: [ de anti-gravetat ]

(-r)·(2pi/T) = pq·( k_{e}/m )·(1/r^{2})·( T/(2pi) )

r·(2pi/T) = (-1)·pq·( k_{e}/m )·(1/r^{2})·( T/(2pi) )

T = [ segon ]·[ Radiá ]

Deducció:

d_{tt}^{2}[ r·cos(vt) ] = (-r)·cos(vt)·v^{2}

vT = 2·pi

d_{tt}[r·cos(ut)·sin(vt)] = 0

d_{tt}[r·sin(ut)·sin(vt)] = 0

u = 0

Ecuacions de Maxwell de Fluxe-Zero:

Lley: [ de Maxwell-Coulomb en forma integral ]

anti-potencial[ rot[ E_{e}(r·f(t),r·g(t),r·h(t)) ] ] = ...

... qk_{e}+...

... (1/3)·anti-potencial[ ( 1/(f(t)·g(t)·h(t)) )·int[ B_{e}(r·f(t),r·g(t),r·h(t)) ]d[t] ]

Lley: [ de Maxwell-Ampere en forma integral ]

anti-potencial[ rot[ B_{e}(r·f(t),r·g(t),r·h(t)) ] ] = ...

... d_{t}[q(t)]·k_{e}+...

... (-1)·(1/3)·anti-potencial[ ( 1/(f(t)·g(t)·h(t)) )·d_{t}[ E_{e}(r·f(t),r·g(t),r·h(t),q(t)) ] ]+...

... (-1)·(1/3)·anti-potencial[ ( 1/(f(t)·g(t)·h(t)) )·B_{e}(r·f(t),r·g(t),r·h(t),q(t)) ]

Lley: [ de Maxwell-Coulomb en forma diferencial ]

rot[ E_{e}(r·f(t),r·g(t),r·h(t)) ] = ...

... H_{e}(r·f(t),r·g(t),r·h(t))+(1/3)·( 1/(f(t)·g(t)·h(t)) )·int[ B_{e}(r·f(t),r·g(t),r·h(t)) ]d[t]

Lley: [ de Maxwell-Ampere en forma diferencial ]

rot[ B_{e}(r·f(t),r·g(t),r·h(t)) ] = ...

... J_{e}(r·f(t),r·g(t),r·h(t),q(t))+...

... (-1)·(1/3)·( 1/(f(t)·g(t)·h(t)) )·d_{t}[ E_{e}(r·f(t),r·g(t),r·h(t),q(t)) ]+...

... (-1)·(1/3)·( 1/(f(t)·g(t)·h(t)) )·B_{e}(r·f(t),r·g(t),r·h(t),q(t))

Lley: [ de Maxwell-Coulomb de l'inducció eléctrica ]

H_{e}(r·f(t),r·g(t),r·h(t)) = ...

... rot[ E_{e}(r·f(t),r·g(t),r·h(t)) ]+...

... (1/3)·qk_{e}(1/r^{2})·( 1/(f(t)·g(t)·h(t)) )·< f(t),g(t),h(t) >

Lley: [ de Maxwell-Ampere de l'inducció magnética ]

J_{e}(r·f(t),r·g(t),r·h(t),q(t)) = ...

... rot[ B_{e}(r·f(t),r·g(t),r·h(t)) ]+...

... (1/3)·d_{t}[q(t)]·k_{e}(1/r^{2})·( 1/(f(t)·g(t)·h(t)) )·< f(t),g(t),h(t) >

Lley: [ de Coulomb de l'inducció eléctrica ]

anti-potencial[ H_{e}(r·f(t),r·g(t),r·h(t)) ] = qk_{e}

Lley: [ de Ampere de l'inducció magnética ]

anti-potencial[ J_{e}(r·f(t),r·g(t),r·h(t),q(t)) ] = d_{t}[q(t)]·k_{e}

Lley: [ de Gauss en forma integral ]

anti-potencial[ E_{e}(r·f(t),r·g(t),r·h(t)) ] = ...

... 3qk_{e}·f(t)·g(t)·h(t)

anti-potencial[ B_{e}(r·f(t),r·g(t),r·h(t)) ] = ...

... (-1)·qk_{e}·( d_{t}[f(t)]·g(t)·h(t)+f(t)·d_{t}[g(t)]·h(t)+f(t)·g(t)·d_{t}[h(t)] )

Lley: [ de Gauss en forma diferencial ]

div[ E_{e}(r·f(t),r·g(t),r·h(t)) ] = ...

... 3qk_{e}·(1/r^{3})

div[ B_{e}(r·f(t),r·g(t),r·h(t)) ] = ...

... (-1)·qk_{e}·(1/r^{3})·( ...

... ( d_{tt}^{2}[f(t)]/d_{t}[f(t)] )+...

... ( d_{tt}^{2}[g(t)]/d_{t}[g(t)] )+...

... ( d_{tt}^{2}[h(t)]/d_{t}[h(t)] ) ...

... )

Deducció:

int-int-int[ ( d_{tt}^{2}[x]/d_{t}[x] ) ]( d_{t}[x]·d[t] )d[y]d[z] = d_{t}[x]·yz

Ecucions de Maxwell-Gauss originals:

Lley: [ de Gauss integral ]

anti-potencial[ E_{e}(r,r,r) ] = 3q(r^{s})·k_{e}

anti-potencial[ B_{e}(r,r,r) ] = 0

Lley: [ de Gauss diferencial ]

div[ E_{e}(r,r,r) ] = 3·d_{rrr}^{3}[q(r^{s})·(a/b)]·k_{e}

div[ B_{e}(r,r,r) ] = 0

Deducció:

div[ E_{e}(r,r,r) ] = 3·q(r^{s})·k_{e}(1/r^{3})

div[ E_{e}(r,r,r) ] = 3·q(r^{s})·k_{e}·d_{111}^{3}[(a/b)]·(1/r^{3})

div[ E_{e}(r,r,r) ] = 3·d_{rrr}^{3}[q(r^{s})·(a/b)]·k_{e}

Lley: [ de Maxwell en forma integral ]

anti-potencial[ rot[ E_{e}(r,r,r) ] ] = ...

... qk_{e}+(1/3)·anti-potencial[ int[ B_{e}(r,r,r) ]d[t] ]

anti-potencial[ rot[ B_{e}(r,r,r) ] ] = ...

... d_{t}[q(t)]·k_{e}+(-1)·(1/3)·anti-potencial[ d_{t}[ E_{e}(r,r,r,q(t)) ] ]

Lley: [ de Maxwell en forma diferencial ]

rot[ E_{e}(r,r,r) ] = H_{e}(r,r,r)+(1/3)·int[ B_{e}(r,r,r) ]d[t]

rot[ B_{e}(r,r,r) ] = J_{e}(r,r,r)+(-1)·(1/3)·d_{t}[ E_{e}(r,r,r) ]

Densitats de carga:

Lley:

anti-potencial[ E_{e}(r,r,r) ] = 3q·(a_{1}·r)·k_{e}

div[ E_{e}(r,r,r) ] = 3·d_{rrr}^{3}[q·(a_{1}·r)]·k_{e}

Lley:

anti-potencial[ E_{e}(r,r,r) ] = 3q·(a_{2}·r^{2})·k_{e}

div[ E_{e}(r,r,r) ] = 3·d_{rrr}^{3}[q·(a_{2}·r^{2})·(1/2)]·k_{e}

Lley:

n >] 3

anti-potencial[ E_{e}(r,r,r) ] = 3q·(a_{n}·r^{n})·k_{e}

div[ E_{e}(r,r,r) ] = 3·d_{rrr}^{3}[q·(a_{n}·r^{n})·(1/(n·(n+(-1))·(n+(-2))))]·k_{e}

Lley:

anti-potencial[ E_{e}(r,r,r) ] = 3q·(a_{n}·r^{(-n)})·k_{e}

div[ E_{e}(r,r,r) ] = 3·d_{rrr}^{3}[q·(a_{n}·r^{(-n)})·(1/((-n)·(n+1)·(n+2)))]·k_{e}

Ecuacions de fluxe-zero para un rectangle cúbic:

Lley:

anti-potencial[ rot[ E_{e}(ra,rb,rc) ] ] = ...

... qk_{e}+...

... (1/3)·anti-potencial[ ( 1/(abc) )·int[ B_{e}(ra,rb,rc) ]d[t] ]

anti-potencial[ rot[ B_{e}(ra,rb,rc) ] ] = ...

... d_{t}[q(t)]·k_{e}+...

... (-1)·(1/3)·anti-potencial[ ( 1/(abc) )·d_{t}[ E_{e}(ra,rb,rc,q(t)) ] ]+...

... (-1)·(1/3)·anti-potencial[ ( 1/(abc) )·B_{e}(ra,rb,rc,q(t)) ]

Lley:

rot[ E_{e}(ra,rb,rc) ] = ...

... H_{e}(ra,rb,rc)+(1/3)·( 1/(abc) )·int[ B_{e}(ra,rb,rc) ]d[t]

rot[ B_{e}(ra,rb,rc) ] = ...

... J_{e}(ra,rb,rc,q(t))+...

... (-1)·(1/3)·( 1/(abc) )·d_{t}[ E_{e}(ra,rb,rc,q(t)) ]+...

... (-1)·(1/3)·( 1/(abc) )·B_{e}(ra,rb,rc,q(t))

Lley:

H_{e}(ra,rb,rc) = ...

... rot[ E_{e}(ra,rb,rc) ]+(1/3)·qk_{e}·(1/r^{2})·( 1/(abc) )·< a,b,c >

J_{e}(ra,rb,rc,q(t)) = ...

... rot[ B_{e}(ra,rb,rc) ]+(1/3)·d_{t}[q(t)]·k_{e}·(1/r^{2})·( 1/(abc) )·< a,b,c >

Lley:

Vectors de inducció en un cub:

H_{e}(rd,rd,rd) = ...

... rot[ E_{e}(rd,rd,rd) ]+(1/3)·qk_{e}·(1/r^{2})·( 1/d^{2} )·< 1,1,1 >

J_{e}(rd,rd,rd,q(t)) = ...

... rot[ B_{e}(rd,rd,rd) ]+(1/3)·d_{t}[q(t)]·k_{e}·(1/r^{2})·( 1/d^{2} )·< 1,1,1 >

Ecuacions de fluxe-zero para un cilindre:

Lley:

anti-potencial[ rot[ E_{e}(r·cos(s),r·sin(s),r·h) ] ] = ...

... qk_{e}+...

... (1/3)·anti-potencial[ ( 1/(cos(s)·sin(s)·h) )·int[ B_{e}(r·cos(s),r·sin(s),r·h) ]d[t] ]

anti-potencial[ rot[ B_{e}(r·cos(s),r·sin(s),r·h) ] ] = ...

... d_{t}[q(t)]·k_{e}+...

... (-1)·(1/3)·anti-potencial[ (1/(cos(s)·sin(s)·h))·d_{t}[ E_{e}(r·cos(s),r·sin(s),r·h,q(t)) ] ]+...

... (-1)·(1/3)·anti-potencial[ (1/(cos(s)·sin(s)·h))·B_{e}(r·cos(s),r·sin(s),r·h,q(t)) ]

Lley:

rot[ E_{e}(r·cos(s),r·sin(s),r·h) ] = ...

... H_{e}(r·cos(s),r·sin(s),r·h)+...

... (1/3)·( 1/(cos(s)·sin(s)·h) )·int[ B_{e}(r·cos(s),r·sin(s),r·h) ]d[t]

rot[ B_{e}(r·cos(s),r·sin(s),r·h) ] = ...

... J_{e}(r·cos(s),r·sin(s),r·h,q(t))+...

... (-1)·(1/3)·( 1/(cos(s)·sin(s)·h) )·d_{t}[ E_{e}(r·cos(s),r·sin(s),r·h,q(t)) ]+...

... (-1)·(1/3)·( 1/(cos(s)·sin(s)·h) )·B_{e}(r·cos(s),r·sin(s),r·h,q(t))

Lley:

Vectors de inducció en els eishos de coordenades de un cilindre que no existeishen:

s = 0+2pi·k <==> ( [1] & [2] )

[1] H_{e}(r,0,r·h) = ...

... rot[ E_{e}(r,0,r·h) ]+(1/3)·qk_{e}·(1/r^{2})·(1/h)·< oo,1,oo·h >

[2] J_{e}(r,0,r·h,q(t)) = ...

... rot[ B_{e}(r,0,r·h) ]+(1/3)·d_{t}[q(t)]·k_{e}·(1/r^{2})·(1/h)·< oo,1,oo·h >

s = pi+2pi·k <==> ( [1] & [2] )

[1] H_{e}((-r),0,r·h) = ...

... rot[ E_{e}((-r),0,r·h) ]+(-1)·(1/3)·qk_{e}·(1/r^{2})·(1/h)·< (-oo),1,oo·h >

[2] J_{e}((-r),0,r·h,q(t)) = ...

... rot[ B_{e}((-r),0,r·h) ]+(-1)·(1/3)·d_{t}[q(t)]·k_{e}·(1/r^{2})·(1/h)·< (-oo),1,oo·h >

s = (pi/2)+2pi·k <==> ( [1] & [2] )

[1] H_{e}(0,r,r·h) = ...

... rot[ E_{e}(0,r,r·h) ]+(1/3)·qk_{e}·(1/r^{2})·(1/h)·< 1,oo,oo·h >

[2] J_{e}(0,r,r·h,q(t)) = ...

... rot[ B_{e}(0,r,r·h) ]+(1/3)·d_{t}[q(t)]·k_{e}·(1/r^{2})·(1/h)·< 1,oo,oo·h >

s = (-1)·(pi/2)+2pi·k <==> ( [1] & [2] )

[1] H_{e}(0,(-r),r·h) = ...

... rot[ E_{e}(0,(-r),r·h) ]+(-1)·(1/3)·qk_{e}·(1/r^{2})·(1/h)·< 1,(-oo),oo·h >

[2] J_{e}(0,(-r),r·h,q(t)) = ...

... rot[ B_{e}(0,(-r),r·h) ]+(-1)·(1/3)·d_{t}[q(t)]·k_{e}·(1/r^{2})·(1/h)·< 1,(-oo),oo·h >