I was reading resistivity values for different materials on the wikipedia page. They report the values for standard temperature, but they give no information about the cross-section or length of the material/resistor. Are this values independent of the cross-section or the length of the resistor?
Resistivity is a property of a material. Resistance is a property of an item that is made up of a material.
All copper wires, irrespective of their shape and size, have approximately the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
The basic formula for resistance of an object is as follows:
- \$R\$ is resistance
- \$\rho\$ is the resistivivity of the material
- \$l\$ is the length of the object
- \$A\$ is the cross-sectional area of the object
The resistivity of annealed copper per ASTM B3 is 0.15328 Ω·g/m² at 20°C. This works out to a resistance of 0.15328 Ω for a wire of length 1 m and mass 1 g (diameter 378.45 µm). A wire of the same diameter of length 2 m will have a resistance of 0.30656 Ω.
Let's start with a cube of material (exact size not important for now), with two opposing faces coated with a conductive material of zero resistance. You with me? Now you can apply a voltage across the two electrode faces, measure the current, and derive the resistance using Ohm's Law.
Now stack 4 of theses cubes together face-to-face, making a block 4 units long. The resistance is clearly 4 times the resistance of 1 cube. In other words, the resistance will be proportional to the length of the assembly.
Now try connecting 4 of these in parallel. The resistance will be 1/4 of a single cube. So the resistance is inversely proportional to the area of the assembly.
This says that, using a unit cube as a starting point, and calling its resistance a reference point, the resistance of any block of the material can be expressed as the resistance of a unit cube multiplied by the length and divided by the area (assuming the block as a rectangular prism).
This in turn says that we can talk about the resistance of a unit cube as being a material-dependent quality (which we call resistivity) multiplied by unit length and divided by unit area, which means that the resistivity will have the units of ohms times length - you can do the math to see how the units cancel. Since you can do this for a unit cube, then scale that to any desired size, you can use that resistivity for any size conductor.