Integration elements

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