Magnetostatics

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.

Definitions:

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.


Currents and Circuits

Current:   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.)


Ampere's law

ΓB∙dr = μ0Ithrough Γ.
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 dA = 0.


The Biot-Savart law

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.


The magnetic vector potential

∇∙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 = -μ0jA(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.


Boundary conditions in magnetostatics

image

(B2 - B1)∙n2 = 0,  (B2 - B1)∙t = μ0kn.
A
is continuous across the boundary.


The force on a current distribution

F =  ∫V j(r) × B(r) dV.
For filamentary currents we have F =  ∫V I dl × B(r).


The magnetic dipole moment of a charge distribution

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 = (m∙B).
The torque on a dipole is  τ = m × B.


Magnetic Materials

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 densities.
j = jf + jmkm = kf + km,
jm = ×Mkm = M×n.
H
= B0 - M.  (This definition is not unique.)

For linear, isotropic, homogeneous (lih) magnetic materials we have
M = ΧmHB = μ0(H + M) = μ0(1 + Χm)H =  μ0κmH =  μH.
Χm < 0 for diamagnetic materials, Χm > 0 for paramagnetic materials, permanent magnets are not lih.


Boundary conditions for H

(H2 - H1)∙t2 = kfn,  ∇∙H ≠ 0  in general.


Energy in magnetostatics

The magnetostatic energy stored in a current distribution is given by 
U = (2μ0)-1all 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)-1all spaceB∙B dV.