Page 342 - Intro to Tensor Calculus

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Electrostatic Fields in Materials
When charges are introduced into materials it spreads itself throughout the material. Materials in
which the spreading occurs quickly are called conductors, while materials in which the spreading takes a
long time are called nonconductors or dielectrics. Another electrical property of materials is the ability to
hold local charges which do not come into contact with other charges. This property is called induction.
For example, consider a single atom within the material. It has a positively charged nucleus and negatively
~
charged electron cloud surrounding it. When this atom experiences an electric field E the negative cloud
~
~
~
moves opposite to E while the positively charged nucleus moves in the direction of E.If E is large enough it
can ionize the atom by pulling the electrons away from the nucleus. For moderately sized electric fields the
atom achieves an equilibrium position where the positive and negative charges are offset. In this situation
the atom is said to be polarized and have a dipole moment ~p.
~
Definition: When a pair of charges +q and −q are separated by a distance 2d the electric dipole
~
p
p
moment is defined by ~ =2dq,where ~ has dimensions of [C m].
~
~
In the special case where d has the same direction as E and the material is symmetric we say that ~
p
~
~
is proportional to E and write ~ = αE,where α is called the atomic polarizability. If in a material subject
p
to an electric field their results many such dipoles throughout the material then the dielectric is said to be
~
~
polarized. The vector quantity P is introduced to represent this effect. The vector P is called the polarization
2
vector having units of [C/m ], and represents an average dipole moment per unit volume of material. The
vectors P i and E i are related through the displacement vector D i such that
P i = D i − 0 E i . (2.6.42)
For an anisotropic material (crystal)
j j
D i = E j and P i = α E j (2.6.43)
i
i
j j
where is called the dielectric tensor and α is called the electric susceptibility tensor. Consequently,
i
i
j j j j j j j
P i = α E j = E j − 0 E i =( − 0 δ )E j so that α = − 0 δ . (2.6.44)
i i i i i i i
A dielectric material is called homogeneous if the electric force and displacement vector are the same for any
two points within the medium. This requires that the electric force and displacement vectors be constant
parallel vector fields. It is left as an exercise to show that the condition for homogeneity is that j =0.
i,k
A dielectric material is called isotropic if the electric force vector and displacement vector have the same
j i i
direction. This requires that = δ where δ is the Kronecker delta. The term = 0 K e is called the
i j j
2
2
dielectric constant of the medium. The constant 0 =8.85(10) −12 coul /N · m is the permittivity of free
space and the quantity k e = is called the relative dielectric constant (relative to 0 ). For free space k e =1.
0
j j
Similarly for an isotropic material we have α = 0 α e δ where α e is called the electric susceptibility. For a
i i
~
~ ~
linear medium the vectors P, D and E are related by
D i = 0 E i + P i = 0 E i + 0 α e E i = 0 (1 + α e )E i = 0 K e E i = E i (2.6.45)