The fundamental equations magnetostatics are linear equations,
∇∙B = 0, ∇×B = μ0j = j/(ε0c2) (SI units).
The principle of superposition holds.
The magnetostatic force on a particle with charge q is F = qv × B.
Drift velocity: <v> = (1/N)∑ivi, N = number of charge carriers.
Current density: j = nq<v>, n = N/V.
Current: I = dQ/dt = ∫Aj∙n dA.
The continuity equation is ∇∙j = -∂ρ/∂t. In statics ∂ρ/∂t = 0 --> ∇∙j = 0.
I = ∫j∙dA or I = dQ/dt.
Resistance: R = ΔV/I.
Resistance of a straight wire: R = ρl/A.
Power: P = IΔV = I2R = (ΔV)2/R.
Resistors in series: R = R1 + R2 + R3 + ... .
Parallel Resistors: 1/R = (1/R1) + (1/R2) + (1/R3) + ... .
Kirchhoff's first rule : (Junction rule)
At any junction point in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction. (This is a consequence of charge conservation.)
Kirchhoff's second rule : (Loop rule)
When any closed circuit loop is traversed, the algebraic sum of the changes in the potential must equal zero. (This is a consequence of conservation of energy.)
In situations with enough symmetry, Ampere's law alone can be used to find the magnitude of B.
The flux of B through any closed surface is zero. ∫closed surface B·dA = 0.
B(r) = (μ0/(4π))∫ dV' j(r')×(r-r')/|r-r'|3.
For filamentary currents we have B(r) = (μ0/(4π))∫ I dl'×(r-r')/|r-r'|3.
∇∙B = 0 --> B = ∇×A.
A is not unique. A' = A + ∇ψ + C, with ψ an arbitrary scalar field and C an arbitrary constant vector is also a vector potential for the same field.
In magnetostatics we choose ∇∙A = 0.
Then ∇2A = -μ0j, A(r) = (μ0/(4π)) ∫V'dV' j(r')/|r - r'|.
The uniqueness theorem:
If if the current density j is specified throughout a volume V and
A or its normal derivatives are specified at the boundaries of a volume V,
then a unique solution exists for A inside V.
Or, if the current density j is specified throughout a volume V and and either A or B are specified at the boundaries of a volume V, then a unique solution exists for B inside V.
(B2 - B1)∙n2
(B2 - B1)∙t = μ0k∙n.
A is continuous across the boundary.
F = ∫V j(r) × B(r)
For filamentary currents we have F = ∫V I dl × B(r).
m = IAn = ½∫V r×j(r) dV.
The vector potential of a magnetic dipole at the origin is A(r) = (μ0/4π)m×r/r3.
The magnetic field of a magnetic dipole at the origin is B(r) = (μ0/4π)(3(m∙r)r/r5 - m/r3).
The energy of a magnetic dipole in an external magnetic field is
Umech = -m∙B.
This is the mechanical work done to bring the dipole from infinity to its present position.
The force on a dipole is F =
The torque on a dipole is τ = m × B.
The magnetization M = dm/dV is defined as the magnetic dipole moment per unit volume.
The total current density is due to free and to magnetization current
j = jf + jm, km = kf + km,
jm = ∇×M, km = M×n.
H = B/μ0 - M. (This definition is not unique.)
For linear, isotropic, homogeneous (lih) magnetic materials we have
M = ΧmH, B = μ0(H + M) = μ0(1 + Χm)H = μ0κmH = μH.
Χm < 0 for diamagnetic materials, Χm > 0 for paramagnetic materials, permanent magnets are not lih.
(H2 - H1)∙t2 = kf∙n, ∇∙H ≠ 0 in general.
The magnetostatic energy stored in a current distribution is given by
U = (2μ0)-1∫all spaceB∙B dV.
In the presence of a magnetic material, the total work done in establishing a
free current distribution is
W = ½∫all spaceB∙H dV,
or, in the presence of a lih magnetic material
U = (2μ)-1∫all spaceB∙B dV.