Why isn't Resistance a Vector?

[ecoli]

Logically, I equate resistance with a frictional force inherent in a conductor. So, I would expect that resistance should be a vector quantity, antiparallel to the direction of the current. I liken this to the the frictional forces acting antiparallel to a force that pushes an object across a floor.

 

However, my professor tells me that resistance is a scalar quantity. Why is this?

[timo]

Conductivity in general can be/is a tensor relating electrical field and current: [math]\vec j = \sigma \vec E[/math]. In the case of (simple?) cubic crystals (due to symmetry) and poly-crystalline matter (due to statistics) you can describe it as a scalar number. I'd assume you only define resistance in this case, but that's just a guess. For many practical applications (electrical circuits), the direction should be a given, anyways.

[ecoli]

Conductivity in general can be/is a tensor relating electrical field and current: [math]\vec j = \sigma \vec E[/math]. In the case of (simple?) cubic crystals (due to symmetry) and poly-crystalline matter (due to statistics) you can describe it as a scalar number. I'd assume you only define resistance in this case, but that's just a guess. For many practical applications (electrical circuits), the direction should be a given, anyways.

This is just a simple algebra based physics class, so we only deal with metal conductors, typically. I asked my prof about it, he said it's not a vector quantity... shouldn't that imply that there is no direction (note that I initially disagreed with him).

[timo]

Ok, above comment was a comment from solid-state physics, where you try calculating properties of materials from their sub-microscopic structure (but often have to assume idealized forms of solid states called "ideal crystal"). So you possibly don't need to bother.

 

What would you think the equations would look like if resistance was a vector? Particularly, what would happen if you reverse the direction of the current? Just play around with the thought, see if you get any sensible results and post them here. My guess would be that as soon as you relate resistance to some frictional force in some sensible way, then the "vector part" is taken by the current, leaving resistance as a simple pre-factor.

[swansont]

How would V = IR or P = I^2R work if R was a vector?

 

The parallel you make isn't quite apropos. The coefficient of friction is a scalar, and you get the magnitude of the force of friction by multiplying by the magnitude of the normal force, which is perpendicular to the motion. The direction is tacked on at the end, when we want to put it in a vector equation.

[John Cuthber]

Resistivity may be a tensor, but resistance isn't the same thing. One is the property of a material and the other is a property of an object.