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