 #1
Karl Karlsson
 104
 11
 Homework Statement:

In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and
the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector.
i) Calculate the angle between the vectors ##\vec j## and ##\vec E## if the angle between ##\vec E## and ##\vec n## is α
ii) Now assume that ##\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 ##\sigma_{ab} = \bar \sigmaδ_{ab}## and what is the value of the constant ##\bar\sigma## in the new coordinate system?
 Relevant Equations:

##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)##
ξ = x, η = y, ζ = γz
In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and
the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector.
i) Calculate the angle between the vectors ##\vec j## and ##\vec E## if the angle between ##\vec E## and ##\vec n## is α
ii) Now assume that ##\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 ##\sigma_{ab} = \bar \sigmaδ_{ab}## and what is the value of the constant ##\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:
##\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 ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)##. I have already found a matrix ##\sigma## that transforms ##\vec E## to ##\vec j##. Do they want me to find eigenvectors and eigenvalues? Why?
Thanks in advance!
the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector.
i) Calculate the angle between the vectors ##\vec j## and ##\vec E## if the angle between ##\vec E## and ##\vec n## is α
ii) Now assume that ##\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 ##\sigma_{ab} = \bar \sigmaδ_{ab}## and what is the value of the constant ##\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:
##\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 ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)##. I have already found a matrix ##\sigma## that transforms ##\vec E## to ##\vec j##. Do they want me to find eigenvectors and eigenvalues? Why?
Thanks in advance!
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