when electrical conductivity varies with direction

anisotropic-conductivity.md

:uid: anisotropic-conductivity :title: Anisotropic Conductivity :description: when electrical conductivity varies with direction :tooltip: None :tag: conductivity,anisotropy,dc,em,physical properties :group: simpeg :license: CC-BY-4.0 :source: https://api.github.com/gists/56c47e9a872aa908705d

When a physical property of a material varies with direction, that material is anisotropic (not to be confused with heterogeneity. Check out Matt Hall's discussion on anisotropy). Anisotropy in electrical conductivity can have a variety of causes. To name a few:

1. alignment of cracks or pores,
2. bedding / lamination,
3. alignment of mineral grains.

Any of these, or similar phenomena can create preferential directions for current to flow. To examine this a bit, lets look at the constitutive relation that relates the current density $J$ to the electric field $E$ through the electrical conductivity $\sigma$ (aka. Ohm's Law).

$$ \vec{J} = \sigma \vec{E}$$

In an isotropic medium, $\sigma = \sigma(x,y,z)$ is a scalar which may be a distributed property. If you apply a constant electric field $\vec{E}$ (which is a vector having both magnitude and direction), that exerts a force on the little charges in the conductor which is directly proportional to strength of the electric field. This will cause the charges to start to move, and the resistivity ($ \sigma^{-1}$) of the medium acts much like friction, controlling how easily the charges move. This description of the flow of charges is described by the current density, $\vec{J}$.

In an anisotropic medium, $\sigma$ is a $3\times3$ tensor, in which case, Ohm's law looks like:

$$ \left(\begin{array}{c} J_x \ J_y \ J_z \end{array} \right) = \left(\begin{array}{ccc} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{xy} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz} \end{array}\right) \left(\begin{array}{c} E_x \ E_y \ E_z \end{array} \right) $$

Lets break this down and consider a few examples. Since drawing vectors and such is easier in 2D, lets consider a 2D example, where everything is happening in the x-z plane. In the figure below, we have a "rock" that is blue and gray.

The blue regions are conductive and the gray regions are resistive. If we apply an electric field in the x-direction, the charges have a choice to either go through the resistive or conductive region, like a parallel circuit. If instead we apply an electric field in the z-direction, the charges have no choice which medium they can go through.

If we want to describe the conductivity of this rock, we use a $2x2$ diagonal tensor. In this case, where the principal axes of anisotropy align with our cartesian coordinate system, the conductivity tensor describing this system is diagonal i.e.

$$ \sigma = \left( \begin{array}{cc} \sigma_{xx} & \\ & \sigma_{zz} \end{array} \right) $$

If we excite this with an electric field oriented in the $x$- direction, then we see

$$ \vec{J}_{1}

$$

In a similar vein, if we use an electric field oriented in the $z$-direction, then

$$ \vec{J}_{2}

$$

So we can multiply a matrix and a vector! Thats not why I am showing you this... lets have a look at the magnitude of the current density in each case. For the first case, we have that

$$ |\vec{J}_{1}| = \sigma_{xx} E_{x} $$

and in the second case,

$$ |\vec{J}_{2}| = \sigma_{zz} E_{z} $$

For the example here, we know that \sigma_{xx} > \sigma_{zz}. So if the magnitude of the inducing field is identical in both cases, the resulting current density magnitude is different between the two, namely $|\vec{J}{1}| > |\vec{J}2|$. So if we wanted the current density to be same, we need to apply a bigger push in the $z$-direction; in particular, we would need: $E{z} = \sigma{xx}/\sigma_{zz} E_x$.

In this example, everything lined up with our coordinate system, so the various components didn't talk to each other. Lets change that.

I started planning on writing this about the behavior of anisotropy in cylindrically symmetric settings, but got a bit sidetracked. Anyways, that will come at another time!

References

http://www.agilegeoscience.com/blog/2015/2/9/what-is-anisotropy