# Resistance and resistivity: which one is the intrinsic and which is the geometric property? Why?

The electrical resistance $R$ and electrical resistivity $\rho$ of a metal wire are related by $$\rho=\frac{RA}{l}$$ where $l$ is the length and $A$ is the cross-sectional area of the wire. One could also have written $$R=\frac{\rho l}{A}.$$ From the first relation, it implies that resistivity is a geometric property of the conductor while the second relation implies that resistance is a geometric property. However, I know that resistance is a geometric property while resistivity is an intrinsic property. See here. But it's not clear to me why.

• Re, "...the first relation...the second relation..." Why do you think those are two different relations? Either one can be trivially derived from the other. You isolate a variable (e.g., $\rho$ or $R$) when you think it is more important. But isolating it doesn't make it more important. – Solomon Slow Apr 30 '18 at 17:26
• @jameslarge No. I didn't mean they are different relations. But from the first, it seems that $\rho$ depends on $R$, $A$ and $l$. Written differently, it seems that $R$ depends on $\rho$, $A$ and $l$. – mithusengupta123 Apr 30 '18 at 18:59
• @mithusengupta123 The second equation is more meaningful to me where $\rho$ is just a constant of proportionality. For example, in Newton's law of gravitation $F=Gm_1m_2/r^2$, $F$ depends on $m_1,m_2$ and $r$. It's wrong to say that $G$ depends on $m_1,m_2, F$ and $r$ because $G=Fr^2/m_1m_2$. Similarly, for a wire of a given material, two wires with different lengths and cross-section the ratio $R_1A_1/l_1=R_2A_2/l_2$ where $R_1,R_2$ have to found from Ohm's law for a fixed $V$ and dividing by the measured currents $I_1,I_2$. – SRS Apr 30 '18 at 19:10