Esmeralda Lane

2022-07-13

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?

Jayvion Mclaughlin

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
$\rho =\frac{RA}{L}$
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:
$F=\frac{1}{4\pi {ϵ}_{0}}\frac{{q}_{1}{q}_{2}}{{r}^{2}}$
where ${ϵ}_{0}$ is the permittivity of free space. Now, you can aptly write
${ϵ}_{0}=\frac{1}{4\pi F}\frac{{q}_{1}{q}_{2}}{{r}^{2}}$
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 ${ϵ}_{0}$ remains a constant for a particular material, at a particular temperature.

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