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Charge and current elements for 1-, 2- and 3-dimensional integration © Frits F.M. de Mul
Basic symmetries planar cylindrical spherical
Coordinate systems e  z r r e  Z e   Z Y X e z e y e x e z e r Z e    e r (x,y,z) (r,  ,z) (r,  ,  ) e z e r e  e z e y e x e r e r
Basic symmetries for Gauss’ Law  extending plane  long cylinder sphere Gauss “pill box” Height  0 Gauss cylinder, Radius r (r<R or r>R), length L Gauss sphere, Radius r (r<R or r>R).
Basic symmetries for Ampere’s Law  extending plane  long solenoide  long cylinder toroïde with/without gap Ampère-circuit Length L; Height h  0 Ampère-circle, Radius r (r<R of r>R) Ampère-circuit, Mean circumference line , length L,
Charge elements: Thin wire dz z O dQ=   dz Thin wire Charge distribution:   (z)  [C/m] If homogeneous:    = const dQ
Charge elements: Thin ring d   R Thin ring   (thickness <<R ) Charge distribution :  (  )   [C/m] If homogeneous:    = const. dQ=   R d 
Charge elements: Flat surface X Y dA=dx dy Flat surface (in general) (also for rectangles) Charge distribution:   (x,y)   [C/m 2 ] dQ=  (x,y) dx dy
Charge elements: Thin circular disk d   r R dr Thin circular disk Charge distribution:  (r,  )   [C/m 2 ] dA = R d   dr dQ=  (r,  ).r d  .dr If    = const: dQ=  .2   r.dr
Charge elements: Cylindrical surface dz R d  Cylinder surface Charge distribution:   (  ,z)  [C/m 2 ];  Radius R dA=R d   dz dQ=  (  ,z) R d   dz If   =const: dQ=   2  R dz
Charge elements: Spherical surface R Z d    Spherical surface: Radius:  R Charge distribution:   (  ,  )  [C/m 2 ] dA=(Rd  )(Rsin   d  ) d Q=   (R d  )(R sin   d  ) d  R sin 
Charge elements: Spatial distribution V dV=dx dy dz Z Y X General spatial charge distribution :  (x,y,z)   [C/m 3 ] dQ=  (x,y,z).dxdydz
Charge elements: Cylindrical volume Z dz r dr d  dQ=   rd   dr dz If    independent of    : dQ=   2  r dr dz R dV=rd   dr dz Cylindrical spatial charge distribution:  (r,  ,z)   [C/m 3 ]
Charge elements: spherical distribution r Z d    d  Spherical charge distribution  (r,  ,  )   [C/m 3 ] dQ=   (rd  )(rsin   d  ) dr If    independent of    and    : dQ=   4  r 2  dr dV=(rd  )(r sin   d  ) dr r.sin  dr
Current elements: Flat surface dI X Y dy J (x,y) General flat surface with current density: J  (x,y) , in [A/m] Contribution to current element  dI  through strip  dy  in +X-direction: dI = ( J    e x )   dy = J x  .dy J x   stroomdichtheid [A/m]
Current elements: solenoid surface J  X Solenoid surface current N  windings, length  L ; Current densities : (1):  j    tangential [in A per m length] (2):  j x   parallel to X-as  [in A per m circumference length] X R R dI = J   dx = NI dx/L dI = J x  R.d  dx R d  (1) (2) d  J x
Current elements: General current tube General current tube: J (x,y,z)  : [A/m 2 ] =  volume current  through material,  current element   dI z  = contribution to current  in Z-direction : J (x,y,z) X Z Y dI z  =  J (x,y,z)    e z  dxdy = J z  dxdy
Current elements: Thick wire R d  r dr J (r,  ) Cylinder Current density [A/m 2 ] through material, parallel to  symmetry axis dI = J(r,  ).rd  . dr dA =rd   dr

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Integration elements

  • 1. Charge and current elements for 1-, 2- and 3-dimensional integration © Frits F.M. de Mul
  • 2. Basic symmetries planar cylindrical spherical
  • 3. Coordinate systems e  z r r e  Z e   Z Y X e z e y e x e z e r Z e    e r (x,y,z) (r,  ,z) (r,  ,  ) e z e r e  e z e y e x e r e r
  • 4. Basic symmetries for Gauss’ Law  extending plane  long cylinder sphere Gauss “pill box” Height  0 Gauss cylinder, Radius r (r<R or r>R), length L Gauss sphere, Radius r (r<R or r>R).
  • 5. Basic symmetries for Ampere’s Law  extending plane  long solenoide  long cylinder toroïde with/without gap Ampère-circuit Length L; Height h  0 Ampère-circle, Radius r (r<R of r>R) Ampère-circuit, Mean circumference line , length L,
  • 6. Charge elements: Thin wire dz z O dQ=  dz Thin wire Charge distribution:  (z) [C/m] If homogeneous:  = const dQ
  • 7. Charge elements: Thin ring d   R Thin ring (thickness <<R ) Charge distribution :  (  ) [C/m] If homogeneous:  = const. dQ=  R d 
  • 8. Charge elements: Flat surface X Y dA=dx dy Flat surface (in general) (also for rectangles) Charge distribution:  (x,y) [C/m 2 ] dQ=  (x,y) dx dy
  • 9. Charge elements: Thin circular disk d   r R dr Thin circular disk Charge distribution:  (r,  ) [C/m 2 ] dA = R d  dr dQ=  (r,  ).r d  .dr If  = const: dQ=  .2  r.dr
  • 10. Charge elements: Cylindrical surface dz R d  Cylinder surface Charge distribution:  (  ,z) [C/m 2 ]; Radius R dA=R d  dz dQ=  (  ,z) R d  dz If  =const: dQ=  2  R dz
  • 11. Charge elements: Spherical surface R Z d    Spherical surface: Radius: R Charge distribution:  (  ,  ) [C/m 2 ] dA=(Rd  )(Rsin  d  ) d Q=  (R d  )(R sin  d  ) d  R sin 
  • 12. Charge elements: Spatial distribution V dV=dx dy dz Z Y X General spatial charge distribution :  (x,y,z) [C/m 3 ] dQ=  (x,y,z).dxdydz
  • 13. Charge elements: Cylindrical volume Z dz r dr d  dQ=  rd  dr dz If  independent of  : dQ=  2  r dr dz R dV=rd  dr dz Cylindrical spatial charge distribution:  (r,  ,z) [C/m 3 ]
  • 14. Charge elements: spherical distribution r Z d    d  Spherical charge distribution  (r,  ,  ) [C/m 3 ] dQ=  (rd  )(rsin  d  ) dr If  independent of  and  : dQ=  4  r 2 dr dV=(rd  )(r sin  d  ) dr r.sin  dr
  • 15. Current elements: Flat surface dI X Y dy J (x,y) General flat surface with current density: J (x,y) , in [A/m] Contribution to current element dI through strip dy in +X-direction: dI = ( J  e x ) dy = J x .dy J x stroomdichtheid [A/m]
  • 16. Current elements: solenoid surface J  X Solenoid surface current N windings, length L ; Current densities : (1): j  tangential [in A per m length] (2): j x parallel to X-as [in A per m circumference length] X R R dI = J  dx = NI dx/L dI = J x R.d  dx R d  (1) (2) d  J x
  • 17. Current elements: General current tube General current tube: J (x,y,z) : [A/m 2 ] = volume current through material, current element dI z = contribution to current in Z-direction : J (x,y,z) X Z Y dI z = J (x,y,z)  e z dxdy = J z dxdy
  • 18. Current elements: Thick wire R d  r dr J (r,  ) Cylinder Current density [A/m 2 ] through material, parallel to symmetry axis dI = J(r,  ).rd  . dr dA =rd  dr