THE POLARIZABILITY TENSOR IS SYMMETRIC

John W. Dooley, Physics Department, Millersville University

The polarizability tensor is said to be symmetric with respect to interchange of its subscripts. This means that

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This can be regarded as a consequence of our assumption that energy can be stored in a dielectric as potential energy. From this assumption we conclude that the potential energy change which occurs when the state of the material is changed is independent of the path by which the change was accomplished.

In this context, "path" means a sequence of electric field values leading from the initial field to the final field. The path describes the detailed manner in which the field is increased to its final value. The claim is that we need only know the final electric field in order to calculate the electrical potential energy of the system. (We take potential energy to be zero when the applied electric field is zero.)

Consider building up an electric field in a particular sequence: First then , then . Looking at the integrals in

we see that the integrals involving and are zero, because when the field is increasing, the and fields are zero. Similarly, the integral is zero because when the is "turned up," the field is still zero. The potential energy is then calculated as

Now imagine repeating the experiment with one variation: Instead of "turning up" the field in the sequence: first then , then , we use the sequence: First then , then . Now is zero when is turned up, but is no longer zero when is turned up. The resulting expression for potential energy is

As long as we know that the electric force is conservative (i.e. the work by the electric force is independent of path) this can only be true if

Similar arguments can be made for the other "mixed" terms in the polarizability tensor, and in general we conclude that the polarizability tensor is symmetric:

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