Charge and current elements

Transcription

1 Charge and current elements for 1-, 2- and 3-dimensional integration Frits F.M. de Mul

2 Presentations: Electromagnetism: History Electromagnetism: Electr. topics Electromagnetism: Magn. topics Electromagnetism: Waves topics Capacitor filling (complete) Capacitor filling (partial) Divergence Theorem E-field of a thin long charged wire E-field of a charged disk E-field of a dipole E-field of a line of dipoles E-field of a charged sphere E-field of a polarized object E-field: field energy Electromagnetism: integrations Electromagnetism: integration elements Gauss Law for a cylindrical charge Gauss Law for a charged plane Laplace s and Poisson s Law B-field of a thin long wire carrying a current B-field of a conducting charged sphere B-field of a homogeneously charged sphere

3 Charge and current elements for 1-, 2- and 3-dimensional integration To perform integrations: always rewrite charge and current elements (dq, di) in the form of coordinate elements (dx,dy,dz or other), using charge and current densities (λ, σ, ρ and j). Basic symmetries planar cylindrical spherical

4 Coordinate systems Z e z e x e z ey Z z r e e X e x e y e r Y e z e r e z e r e Z e r r e e r (x,y,z) (r,,z) (r,, ) e

5 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).

6 Basic symmetries for Ampere s Law = current direction 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

7 Charge elements: Thin wire dq z O dz Thin wire Charge distribution: (z) [C/m] If homogeneous: = const dq= dz To perform integrations: always rewrite charge and current elements (dq, di) in the form of coordinate elements (dx,dy,dz or other), using charge and current densities (λ, σ, ρ and j).

8 Charge elements: Thin ring R Thin ring (thickness<<r) Charge distribution: ( ) [C/m] If homogeneous: = const. dq= R

9 Charge elements: Flat surface Y Flat surface (in general) (also for rectangles) Charge distribution: (x,y) [C/m 2 ] X da=dx dy dq= (x,y) dx dy

10 Charge elements: Thin circular disk Thin circular disk Charge distribution: (r, ) [C/m 2 ] R r dq= (r, ).r.dr dr da = R dr If = const: dq=.2 r.dr

11 Charge elements: Cylindrical surface Cylinder surface Charge distribution: (,z) [C/m 2 ]; Radius R dz R da=r dz dq= (,z) R dz If =const: dq= 2 R dz

12 Charge elements: Spherical surface Z Surface element da=(r)(rsin ) R Charge element dq= (R )(R sin ) R sin Part of: Spherical surface: Radius: R Charge distribution: (, ) [C/m 2 ]

13 Charge elements: Spatial distribution V Z dv=dx dy dz General spatial charge distribution: (x,y,z) [C/m 3 ] Y Charge element dq= (x,y,z).dxdydz X

14 Charge elements: Cylindrical volume Z R Cylindrical spatial charge distribution: (r,,z) [C/m 3 ] r dr Charge element dq= r dr dz dz If independent of : dq= 2 r dr dz dv=r dr dz

15 Charge elements: spherical distribution Z r dr Spherical charge distribution (r,, ) [C/m 3 ] Volume element dv=(r)(r sin ) dr Charge element dq= (r)(rsin ) dr If independent of and : r.sin dq= 4 r 2 dr

16 Current elements: Flat surface dy Y General flat surface with current density: J (x,y), in [A/m] J(x,y) X di Contribution to current element di through strip dy in +X-direction: Current element di = (J e x ) dy = J x.dy J x current density [A/m]

17 Current elements: solenoid surface (1) dx R Solenoid surface current N windings, length L; J (2) J x R X X Current densities: (1): j tangential [in A per m length] di = J dx = NI dx/l (2): j x parallel to X-as [in A per m circumference length] R di = J x R.

18 Current elements: General current tube Z General current tube: J(x,y,z) : [A/m 2 ] = volume current through material, X J(x,y,z) Y current element di z = contribution to current in Z-direction : di z = J(x,y,z) e z dxdy = J z dxdy

19 Current elements: Thick wire Cylinder Current density [A/m 2 ] through material, parallel to symmetry axis surface element: ring element da =r dr R dr r J(r, ) Current density di = J(r, ).r. dr the end