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Magnetic Equations

Definitions - UNDER CONSTRUCTION

 

Units of measure fro Wikipedia Page: http://en.wikipedia.org/wiki/SI_electromagnetism_units

Symbol[1]Name of QuantityDerived UnitsUnitBase Units
I Electric current ampere (SI base unit) A A (= W/V = C/s)
Q Electric charge coulomb C A·s
U, ΔV, Δφ; E Potential difference; Electromotive force volt V J/C = kg·m2·s−3·A−1
R; Z; X Electric resistance; Impedance; Reactance ohm Ω V/A = kg·m2·s−3·A−2
ρ Resistivity ohm metre Ω·m kg·m3·s−3·A−2
P Electric power watt W V·A = kg·m2·s−3
C Capacitance farad F C/V = kg−1·m−2·A2·s4
ΦE Electric flux volt metre V·m kg·m3·s−3·A−1
E Electric field strength volt per metre V/m N/C = kg·m·A−1·s−3
D Electric displacement field coulomb per square metre C/m2 A·s·m−2
ε Permittivity farad per metre F/m kg−1·m−3·A2·s4
χe Electric susceptibility (dimensionless) - -
G; Y; B Conductance; Admittance; Susceptance siemens S Ω−1 = kg−1·m−2·s3·A2
κ, γ, σ Conductivity siemens per metre S/m kg−1·m−3·s3·A2
B Magnetic flux density, Magnetic induction tesla T Wb/m2 = kg·s−2·A−1 = N·A−1·m−1
Φ, ΦM, ΦB Magnetic flux weber Wb V·s = kg·m2·s−2·A−1
H Magnetic field strength ampere per metre A/m A·m−1
L, M Inductance henry H Wb/A = V·s/A = kg·m2·s−2·A−2
μ Permeability henry per metre H/m kg·m·s−2·A−2
χ Magnetic susceptibility (dimensionless) - -

Magnetic field - Tesla

 

Equations

 

Maxwell equations

Lorentz Force

Magnetic Field

Ampere's Law

Biot-Savart Law

 

Calculating magnetic forces

Magnetic Flux Density

Magnetic flux density for a uniformly magnetizied body can be calculated by the formula:

(1)

Formula 1 is a scalar potentiel of flux density, M is a constant vector of magnetization, ϕ is a scalar potentiel of the same body charged with unity charge density. The potential can be calculated by the formula:

(2)

calculated of the volume of body, Fig:1.

Figure 1. Potential Calculation

The volumetric integral calculation is not easy in general so we calculate magnetic flux density only for regular bodies such as blocks/cubes, discs/cylinders, rings and spheres.

Magnetic flux density generated by steady magnets can be calculated by simple formulae only for spherical magnets or for z axis for cylindrical and rectangular magnets.

Blocks/Cubes

For a block or cube, Fig. 2;

Figure 2. Block/Cube

The calculation gives the formulae

(3)

Examples of graphs of magnetic flux density for a rectangle are on Fig.2. The graph is plotted for M=100,a=5mm, b=10mm, d=5mm, range of z is [0.01,10]mm.

Figure 3. Example magnetic flux density with distance

Disc/Cylinder and Ring

For a disc or cylinder calculation of integrals gives explicit formulae only for z axis, Fig. 4.

Figure 4. Disc/Cylinder and Ring  (for disc/cylinder ignore r)

 

For disk magnetic flux density on z-axis is

(4)

 

For a ring magnetic flux density on z-axis is

(5)

 

Examples of graphs of magnetic flux density for a ring (red) and a disk (bleu) are on Fig:4. The graph is plotted for M=100,diameter=10mm, height=5mm, range of z is [0.01,10]mm. For the ring an interior diameter=4mm.