Also assume that the objects modeled are not moving v = 0 so that there is no contributions from Lorentz forces. These are treated later on.
where p is the air pressure,
I is the identity 3-by-3 tensor (or matrix), and
E and
B are 3-by-1 vectors. In this expression of the stress tensor, air is considered to be nonpolarizable and nonmagnetizable. When air is approximated by vacuum,
p = 0. This expression, with
p = 0, of the stress tensor is also known as the Maxwell stress tensor.
Using the fact that, for air, D =
ε0E and
B =
μ0H the expression for the stress tensor can be written as
Thus, using the same terminology as earlier, fem = 0 for air, with
σM = −pI. In the derivation of the total force on an elastic solid surrounded by vacuum or air, the approximation
∇p = 0 has been used.
is useful (and similarly for B). From the right-hand side it is clear (using Maxwell’s equations) that this is zero for stationary fields in free space.
where n1 is the surface normal, a 1-by-3 vector, pointing out from the solid. This expression can be used directly in the boundary integral of the stress tensor for calculating the total force
F on the solid.
A material that is nonpolarizable and nonmagnetizable (P = 0 and
M = 0) is called a
pure conductor. This is not necessarily equivalent to a perfect conductor, for which
E = 0, but merely a restriction on the dielectric and magnetic properties of the material. The stress tensor becomes identical to the one for air, except for
−pI being replaced by the purely mechanical stress tensor
σM:
where in σ(
E,
B) the dependence of
E and
B has not been separated out. Thus
σ is not a purely mechanical stress tensor in this general case. Different material models give different appearances of
σ(
E,
B). The electromagnetic contributions to
σ(
E,
B) typically represent pyroelectric, pyromagnetic, piezoelectric, piezomagnetic, dielectric, and magnetization effects. The expression for the stress tensor in vacuum, air, and pure conductors can be derived from this general expression by setting
M = P = 0.
T1 must be symmetric. The terms
EPT and
−MBT are symmetric in the case of a linear dielectric and magnetic material because
Here, the magnetic susceptibility χB differs slightly from the classical
χm. The other explicit terms are all symmetric, as is
σ(
E,
B). In the general case this imposes constraints on the properties of
σ(
E,
B). For a nonlinear material
σ(
E,
B) might need to include terms such as
−EPT or +
MBT to compensate for asymmetric
EPT or
−MBT.