University Electromagnetism: How to integrate over 2D and 3D distributions (e.g. charges and currents)

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