If Resistivity = $\frac{RA}{L}$, why does it not depend on dimension?

The formula for resistivity is: $$\rho = \frac{RA}{L}$$ where $\rho$ is resistivity, $R$ is resistance, $A$ is cross-sectional area, and $L$ is the length of the conductor.

We can see from the formula that $A$ and $L$ are involved, why then does resistivity not depend on dimensions?

• Well, $L = RA/\rho$, so shouldn't L depend on the resistance by this logic? – Abhijeet Melkani Oct 17 '17 at 14:53
• The resistivity is the resistance between opposite faces of a unit cube of the material – Farcher Oct 17 '17 at 15:21

It's because resistance ($R$) is also a function of size.

A short and wide section of a material will have lower resistance than a long thin section of the same material. Larger cross sections have less resistance, and longer conductors have more resistance.

Therefore, by multiplying resistance by area and dividing by length, you get a value for a material property (resistivity $\rho$) that doesn't depend on the size of the conductor.

That is the point of resistivity, to be applicable to a material over various cross sections and lengths.

True, $A$ and $L$ are involved. But resistivity is a constant.

In the equation $$\rho = \frac{R A}{L}$$ if you change the resistance and length of conductor and then experimentally measure the resistance, and put it in the formula, you'll find that the resistivity has remained a constant for all different $A$ and $L$, because $R$ gets suitable modified to keep resistivity constant.

This is an experimental result. I remember trying this out myself, and the results showed that resistivity is a constant.

For another example, consider this formula for Coloumb's law: $$F = \frac{1}{4\pi \epsilon _0} \frac{q_1 q_2}{r^2}$$ where $\epsilon _0$ is the permittivity of free space. Now, you can aptly write $$\epsilon _0 = \frac{1}{4\pi F} \frac{q_1 q_2}{r^2}$$ and say that it is not a constant. But experiments show that for different values of $q_1$, $q_2$ and $r$, the force $F$ varies such that permittivity $\epsilon _0$ remains a constant for a particular material, at a particular temperature.

• Just a clarification for the OP: resistivity is constant for different shapes and sizes of a given material. It of course varies with the material (that's what makes it useful to characterize different materials), but for a given material, it also varies with e.g. temperature. – NickD Oct 17 '17 at 17:16
• @Nick I expected the OP to know that, but I've added a small bit. – Wrichik Basu Oct 17 '17 at 18:16

Resistivity is a property of a material. But, because we have to take measurements of resistance on a particular specimen, with a particular size and shape, we need to use those dimensions in the calculations so we can cancel them out. Resistivity is not (normally) measured directly, but resistance is because you can use any specimen of any size and shape to take a measurement on, and then do the math to get resistivity. Otherwise you'd have to use a specimen of unit size.