Karl Karlsson
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- Joined
- Nov 4, 2019
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- 9
In a certain anisotropic conductive material, the relationship between the current density \(\displaystyle \vec j\) and the electric field \(\displaystyle \vec E\) is given by: \(\displaystyle \vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)\) where \(\displaystyle \vec n\) is a constant unit vector.
i) Calculate the angle between the vectors \(\displaystyle \vec j\) and \(\displaystyle \vec E\) if the angle between \(\displaystyle \vec E\) and \(\displaystyle \vec n\) is α
ii) Now assume that \(\displaystyle \vec n=\vec e_3\) 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 \(\displaystyle \sigma_{ab} = \bar \sigmaδ_{ab}\) and what is the value of the constant \(\displaystyle \bar\sigma\) 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:
\(\displaystyle \vec E\cdot\vec j = |\vec E|\cdot|\vec j|\cdot cos(\phi)= \sigma_0\vec E^{2} + \sigma_1\vec n\cdot \vec E(\vec n\cdot\vec E) \implies \phi =arccos(\frac {\sigma_0|\vec E^{2}| + \sigma_1\cdot cos(α)\cdot|\vec E|\cdot cos(α)|\cdot|\vec E|} {|\vec E|\cdot|\vec j|})\)
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 \(\displaystyle \vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)\). I have already found a matrix \(\displaystyle \sigma\) that transforms \(\displaystyle \vec E\) to \(\displaystyle \vec j\). Do they want me to find eigenvectors and eigenvalues? Why?
Thanks in advance!
i) Calculate the angle between the vectors \(\displaystyle \vec j\) and \(\displaystyle \vec E\) if the angle between \(\displaystyle \vec E\) and \(\displaystyle \vec n\) is α
ii) Now assume that \(\displaystyle \vec n=\vec e_3\) 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 \(\displaystyle \sigma_{ab} = \bar \sigmaδ_{ab}\) and what is the value of the constant \(\displaystyle \bar\sigma\) 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:
\(\displaystyle \vec E\cdot\vec j = |\vec E|\cdot|\vec j|\cdot cos(\phi)= \sigma_0\vec E^{2} + \sigma_1\vec n\cdot \vec E(\vec n\cdot\vec E) \implies \phi =arccos(\frac {\sigma_0|\vec E^{2}| + \sigma_1\cdot cos(α)\cdot|\vec E|\cdot cos(α)|\cdot|\vec E|} {|\vec E|\cdot|\vec j|})\)
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 \(\displaystyle \vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)\). I have already found a matrix \(\displaystyle \sigma\) that transforms \(\displaystyle \vec E\) to \(\displaystyle \vec j\). Do they want me to find eigenvectors and eigenvalues? Why?
Thanks in advance!