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| Electric field to be rotated in search of maximum dielectric energy storage. |
For an ANisotropic dielectric, the amount of energy stored at a given applied electric field depends on the direction of the electric field. To keep this section simple, we suppose that the electric field has the form 
.
We seek to find the direction of the electric field (holding the magnitude of the field constant) in which the energy storage is maximum. This is a question of interest to capacitor manufacturers. The energy stored (in a unit volume) in this case (with 
) is
Note that we have economized on notation. Since the potential energy is zero when the electric field is zero, 
if we start at zero, as usual.
Suppose that the electric field makes an angle 
with the x axis, as shown in the figure. We ask for the value of c that maximizes the stored energy. In terms of and the (constant by design) magnitude of the electric field, the stored energy per unit volume is
where 
is the magnitude of the vector 
.
Electric field in a dielectric, to be rotated in search of maximum energy storage.
To find the maximum stored energy, we differentiate U with respect to and set the derivative equal to zero. The result is
where 
is the value of at which stored energy is maximum.
This is all well and good; the equation can be solved graphically, or with a computer which just tries different values of until it finds one that works. Of much more interest is a special case:
Suppose we made a "lucky" guess, and our original x axis was along the direction of maximum energy storage. Then the answer to the equation above must be
.
This means that the equation for is
This in turn requires that
.
Furthermore, since v, we have 
also.
Extending this result to include the z components gives the result that is, all components of "mixed" type, such as 
, etc. are all zero. Only 
, and 
are not zero.
That is, when we are in the "right" coordinate system, the 9 component polarizability tensor
is completely described by only 3 terms. The tensor becomes
It is said that:
In the coordinate system which maximizes the energy storage along one axis, the polarizability tensor is "diagonal."
This special coordinate system is called, in English, the "proper coordinate system." It can be found as long as the dielectric potential energy is well defined. In this coordinate system, we call the 3 non-zero components of polarizability
, and 
.
In this coordinate system the energy per unit volume is written
Note well: In any other coordinate system, the energy would have the same value, but its calculation would be more tedious, because we would have to use all 9 components of the polarizability tensor, and cross products of the electric field components.
GENERALIZATION
We have discovered an example of a quite general result: For a symmetric tensor, there is always a coordinate system in which the "mixed" terms are all zero. In this coordinate system the tensor is said to be "diagonal." The values of the "diagonal elements"
, and
are called the eigenvalues, and the problem of finding the proper coordinate system is called "the eigenvalue problem."
A useful result is that, for a cubic crystal, the proper coordinates are along the cube axes, and the three polarizability eigenvalues are all equal to each other. Thus for sodium chloride, a cubic crystalline dielectric, only one number is needed to completely specify the polarizability.
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