# Can resistance be directional?

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## Solution 1

Definitely! Consider Ohm's law, $\vec j=\sigma\vec{E}$, where $\vec j$ is the current density, $\sigma$ is conductivity (the inverse of resistivity) and $\vec E$ is the electric field. Anisotropic conductivity corresponds to turning $\sigma$ into a tensor-valued quantity, a $3\times3$ matrix. Ohm's law is then given by

$$j_i=\sigma_{ij}E_j,$$

where $\sigma_{ij}$ is now the conductivity matrix, where different components $i,j\in x,y,z$ may have different entries, depending on the material in question. Note that restivity is now also a matrix, which can be acquired by inverting conductivity, i.e. $\rho_{ij}=\sigma_{ij}^{-1}$.

Physical systems with anisotropic restivities include various metals, crystals and the concept also plays a role in geophysics, in the analysis of sediments and oil fields. An example of a system that exhibits this property would be one consisting of layers of several materials, each with a different resistivity. Currents flowing along one specific layer will face different resistivity than those which flow perpendicularly.

## Solution 2

To a degree this is matter of terminology. Resistance is a scalar quantity, but it is derived from the resistivity, which is a second rank tensor and is anisotropic in many materials.

For an isotropic material we have the usual formula:

$$R = \rho\frac{\ell}{A}$$

and the usual:

$$V = IR$$

where $\rho$ is the (isotropic) resistivity, $\ell$ is the length and $A$ the area of the conducting region. In an isotropic material life gets considerably more complicated as you need to treat the current density as a vector field, and multiplying this by the resistivity tensor gives the electric field:

$${\bf E}(\vec{r}) = {\bf \rho}(\vec{r}){\bf J}(\vec{r})$$

where all the quantities are functions of the position $\vec{r}$. Resistance isn't a terribly useful concept in this case as the electric field and the current are not necessarily in the same direction so they can't simply be related by a scalar.

However we can often choose our axes so that the resistivity tensor can be written as a diagonal matrix:

$$\rho = \left( \begin{matrix} \rho_x & 0 & 0 \\ 0 & \rho_y & 0 \\ 0 & 0 & \rho_z \\ \end{matrix} \right)$$

And in that case the resistance for a current flowing along these axes can simply be written as:

$$R_x = \rho_x\frac{\Delta x}{A_x}$$

and likewise for the $y$ and $z$ axes. So the resistance does depend on direction. However it is still a scalar - it's just that the value of that scalar depends on direction.

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### Fraïssé

Updated on September 07, 2020

When we think of resistance, we always think of a scalar value associated with a piece of a material. After all, resistance is but resistivity times surface geometry.

But can resistance be directional meaning that it is stronger in one direction but weaker in another?

The point is that $i$ and $j$ can be x,y or z. Maybe the way you wrote it makes it clearer.