integration elements
DESCRIPTION
University Electromagnetism:How to integrate over 2D and 3D distributions (e.g. charges and currents)TRANSCRIPT
Charge and current elementsCharge and current elements
for 1-, 2- and 3-dimensional integrationfor 1-, 2- and 3-dimensional integration
© Frits F.M. de Mul
Basic symmetriesBasic symmetries
planar cylindrical spherical
Coordinate systems
eϕ
zr
r
eθ
Z
eϕϕ
Z
Y
X
ez
eyex
ez
er
Z
eϕϕ
θ
er
(x,y,z) (r,ϕ,z) (r,θ,ϕ)
ez
er
eϕ
ez
eyex
er
er
Basic symmetries for Gauss’ LawBasic 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 LawBasic symmetries for Ampere’s Law∞ extending plane∞ long solenoide
∞ long cylinder toroïdewith/without gap
Ampère-circuitLength L; Height h → 0
Ampère-circle,Radius r (r<R of r>R)
Ampère-circuit,Mean circumferenceline , length L,
Charge elements: Thin wireCharge elements: Thin wire
dz
z
O dQ=λ dz
Thin wireCharge distribution: λ(z) [C/m]If homogeneous: λ = const
dQ
Charge elements: Thin ringCharge elements: Thin ring
dϕ
ϕR
Thin ring (thickness<<R)Charge distribution: λ(ϕ) [C/m]If homogeneous: λ = const.
dQ=λ R dϕ
Charge elements: Flat surfaceCharge elements: Flat surface
X
Y
dA=dx dy
Flat surface (in general)(also for rectangles)Charge distribution: σ(x,y) [C/m2]
dQ=σ(x,y) dx dy
Charge elements: Thin circular diskCharge elements: Thin circular disk
dϕ
ϕrR
dr
Thin circular diskCharge distribution: σ(r,ϕ) [C/m2]
dA = R dϕ dr
dQ=σ(r,ϕ).r dϕ.dr
If σ = const:
dQ=σ.2π r.dr
Charge elements: Cylindrical surfaceCharge elements: Cylindrical surface
dz R
dϕ
Cylinder surfaceCharge distribution: σ(ϕ,z) [C/m2]; Radius R
dA=R dϕ dz
dQ=σ(ϕ,z) R dϕ dz
If σ=const:
dQ=σ 2πR dz
Charge elements: Spherical surfaceCharge elements: Spherical surface
R
Z
dϕϕ
θ
Spherical surface:Radius: RCharge distribution: σ(θ,ϕ) [C/m2]
dA=(Rdθ)(Rsinθ dϕ)
dQ=σ (R dθ)(R sinθ dϕ)
dθ
R sinθ
Charge elements: Spatial distributionCharge elements: Spatial distribution
V
dV=dx dy dzZ
Y
X
General spatial charge distribution:ρ(x,y,z) [C/m3]
dQ=ρ(x,y,z).dxdydz
Charge elements: Cylindrical volumeCharge elements: Cylindrical volume
Z
dz
r dr
dϕ
dQ=ρ rdϕ dr dz
If ρ independent of ϕ :
dQ=ρ 2πr dr dz
R
dV=rdϕ dr dz
Cylindrical spatial charge distribution:ρ(r,ϕ,z) [C/m3]
Charge elements: spherical distributionCharge elements: spherical distribution
r
Z
dϕ
ϕ
θ
dθ
Spherical charge distributionρ(r,θ,ϕ) [C/m3]
dQ=ρ (rdθ)(rsinθ dϕ) dr
If ρ independent of θ and ϕ :
dQ=ρ 4πr2 dr
dV=(rdθ)(r sinθ dϕ) dr
r.sinθ
dr
Current elements: Flat surfaceCurrent elements: Flat surface
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 • ex) dy = Jx .dy
Jx stroomdichtheid [A/m]
dI
Current elements: solenoid surfaceCurrent elements: solenoid surface
Jϕ
X
Solenoid surface currentN windings, length L;
Current densities:(1): jϕ tangential[in A per m length]
(2): jx parallel to X-as [in A per m circumference length]
X
R
R
dI = Jϕ dx = NI dx/L
dI = Jx R.dϕ
dx
R dϕ
(1)
(2)
dϕ
Jx
Current elements: General current tubeCurrent elements: General current tube
General current tube:J(x,y,z) : [A/m2] = volume current through material,
current element dIz =contribution to current in Z-direction :
J(x,y,z)
X
Z
Y dIz = J(x,y,z)• ez dxdy = Jz dxdy
Current elements: Thick wireCurrent elements: Thick wire
R
dϕ
r
drJ(r,ϕ)
CylinderCurrent density [A/m2]through material, parallel to symmetry axis
dI = J(r,ϕ).rdϕ. dr
dA =rdϕ dr