Calculate the angle between the given vectors. When does the conductivity tensor component take the form [MATH]\sigma_{ab} = \bar \sigmaδ_{ab}[/MATH]?

[Karl Karlsson]

In a certain anisotropic conductive material, the relationship between the current density [MATH]\vec j[/MATH] and the electric field [MATH]\vec E[/MATH] is given by: [MATH]\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)[/MATH] where [MATH]\vec n[/MATH] is a constant unit vector.



i) Calculate the angle between the vectors [MATH]\vec j[/MATH] and [MATH]\vec E[/MATH] if the angle between [MATH]\vec E[/MATH] and [MATH]\vec n[/MATH] is α



ii) Now assume that [MATH]\vec n=\vec e_3[/MATH] and define a coordinate transformation ξ = x, η = y, ζ = γz where γ is a constant. For what value of γ does the conductivity tensor component take the form [MATH]\sigma_{ab} = \bar \sigmaδ_{ab}[/MATH] and what is the value of the constant [MATH]\bar\sigma[/MATH] in the new coordinate system?





My attempt:



I don't really know if I get it into the simplest possible form but i guess one way of solving i) would be:



[MATH]\vec E\cdot\vec j = |\vec E|\cdot|\vec j|\cdot cos(\phi)= \sigma_0\vec E^{2} + \sigma_1\vec n\cdot \vec E(\vec n\cdot\vec E) \implies \phi =arccos(\frac {\sigma_0|\vec E^{2}| + \sigma_1\cdot cos(α)\cdot|\vec E|\cdot cos(α)|\cdot|\vec E|} {|\vec E|\cdot|\vec j|})[/MATH]


Is this the best way to solve this?



On ii) i am completely lost. What do the coordinate transformations mean? x, y and z are not even in the given expression [MATH]\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)[/MATH]. I have already found a matrix [MATH]\sigma[/MATH] that transforms [MATH]\vec E[/MATH] to [MATH]\vec j[/MATH]. Do they want me to find eigenvectors and eigenvalues? Why?



Thanks in advance!