Page 347 - Intro to Tensor Calculus

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ZZ
~
where J · b e n dσ is the total flux (current) passing through the surface which is created by encircling
S
some curve about the wire. Equating like terms in equation (2.6.63) gives the differential form of Ampere’s
law
~
~
∇× B = µ 0 J. (2.6.64)
Magnetostatics in Materials
Similar to what happens when charges are introduced into materials we have magnetic fields whenever
there are moving charges within materials. For example, when electrons move around an atom tiny current
loops are formed. These current loops create what are called magnetic dipole moments ~m throughout the
~
material. When a magnetic field B is applied to a material medium there is a net alignment of the magnetic
~
~
dipoles. The quantity M, called the magnetization vector is introduced. Here M is associated with a
dielectric medium and has the units [amp/m] and represents an average magnetic dipole moment per unit
~
~
volume and is analogous to the polarization vector P used in electrostatics. The magnetization vector M
~
acts a lot like the previous polarization vector in that it produces bound volume currents J b and surface
~
~
~
~
~
currents K b where ∇× M = J b is a volume current density throughout some volume and M × b e n = K b is a
surface current on the boundary of this volume.
∂ ~ E
From electrostatics note that the time derivative of 0 ∂t has the same units as current density. The
~ ~ ~ ∂ ~ E ~ ~
total current in a magnetized material is then J t = J b + J f + 0 where J b is the bound current, J f is the
∂t
∂ ~ E
free current and 0 is the induced current. Ampere’s law, equation (2.6.64), in magnetized materials then
∂t
becomes
~
~
∂E ∂E
~ ~ ~ ~ ~
∇× B = µ 0 J t = µ 0 (J b + J f + 0 )= µ 0 J + µ 0 0 (2.6.65)
∂t ∂t
~ ~ ~ ∂ ~ E
where J = J b + J f . The term 0 is referred to as a displacement current or as a Maxwell correction to
∂t
the field equation. This term implies that a changing electric field induces a magnetic field.
~
An auxiliary magnet field H defined by
1
H i = B i − M i (2.6.66)
µ 0
~
~
is introduced which relates the magnetic force vector B and magnetization vector M. This is another con-
stitutive equation which describes material properties. For an anisotropic material (crystal)
j j
B i = µ H j and M i = χ H j (2.6.67)
i i
j j
where µ is called the magnetic permeability tensor and χ is called the magnetic permeability tensor. Both
i i
of these quantities are dimensionless. For an isotropic material
j j
µ = µδ i where µ = µ 0 k m . (2.6.68)
i
2
Here µ 0 =4π × 10 −7 N/amp is the permeability of free space and k m = µ is the relative permeability
µ 0
j j
coefficient. Similarly, for an isotropic material we have χ = χ m δ where χ m is called the magnetic sus-
i i
ceptibility coefficient and is dimensionless. The magnetic susceptibility coefficient has positive values for