Page 311 - Intro to Tensor Calculus

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Figure 2.5-3. Interaction of various fields.

processes the equality sign holds. The Clausius-Duhem inequality is assumed to hold for all independent
thermodynamical processes.
If in addition there are electric and magnetic fields to consider, then these fields place additional forces
upon the material continuum and we must add all forces and moments due to these effects. In particular we
must add the following equations

1 ∂ √
~
i
Gauss’s law for magnetism ∇· B =0 √ ( gB )= 0.
g ∂x i
1 ∂ √
Gauss’s law for electricity ~ ( gD )= % e .
i
∇· D = % e √ i
g ∂x
~
∂B ∂B i
Faraday’s law ~ ijk
∇× E = − E k,j = − .
∂t ∂t
~
∂D ∂D i
i
~
Ampere’s law ∇× H = J + ijk H k,j = J + .
~
∂t ∂t
j
i
where % e is the charge density, J is the current density, D i = E j + P i is the electric displacement vector,
i
j
H i is the magnetic field, B i = µ H j + M i is the magnetic induction, E i is the electric field, M i is the
i
magnetization vector and P i is the polarization vector. Taking the divergence of Ampere’s law produces the
law of conservation of charge which requires that
∂% e ∂% e 1 ∂ √ i
~
+ ∇· J =0 + √ ( gJ )= 0.
∂t ∂t g ∂x i
The figure 2.5-3 is constructed to suggest some of the interactions that can occur between various
variables which define the continuum. Pyroelectric effects occur when a change in temperature causes
changes in the electrical properties of a material. Temperature changes can also change the mechanical
properties of materials. Similarly, piezoelectric effects occur when a change in either stress or strain causes
changes in the electrical properties of materials. Photoelectric effects are said to occur if changes in electric
or mechanical properties effect the refractive index of a material. Such changes can be studied by modifying
the constitutive equations to include the effects being considered.
From figure 2.5-3 we see that there can exist a relationship between the displacement field D i and
electric field E i . When this relationship is linear we can write D i = ji E j and E j = β jn D n ,where ji are