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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.

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    $\begingroup$ 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. $\endgroup$ – Solomon Slow Apr 30 '18 at 17:26
  • $\begingroup$ @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 $l$. $\endgroup$ – mithusengupta123 Apr 30 '18 at 18:59
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    $\begingroup$ @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 $I_1,I_2$. $\endgroup$ – SRS Apr 30 '18 at 19:10
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Resistivity is the resistance of a given material when the material is of unit length and unit area. So, resistivity is an intrinsic property. Resistance changes with the material geometry, for example, the resistance of the material is doubled when length of the material is doubled and halved when area of cross-section of the material is doubled. However, in both the above cases, the resistivity of the material remains the same because it is still calculated over a unit length and a unit area of the material.

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Resistivity is an intensive property. Resistance is neither extensive or intensive. However for the simple case of resistors wired in series, the resistance behaves extensively. Resistivity is regarded as a material property (1 over the dc conductivity), while resistance is a cumulative or bulk effect. If you had a wire say, whose resistivity varied along its length, you would need to integrate this along the length of the wire to get the resistance. Or, put another way, the resistance of a wire increases with length, while the resistivity does not (assuming a homogeneous wire).

The first formula you quote from wikipedia, is for a specific model: "uniform cross section with a uniform flow of electric current, and are made of one material." So not surprisingly if you measure the resistance and assume the resistivity is constant, then you can get average resistivity as shown. But this is not a general result.

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    $\begingroup$ If there is an error point it out, don't just downvote. That serves no educational purpose. $\endgroup$ – JohnS Apr 30 '18 at 19:11
  • $\begingroup$ I didn't downvote it, but I think it got downvoted because you are discussing intensive and extensive properties when the question asks about intrinsic and extrinsic properties. $\endgroup$ – Anurag B. May 18 '18 at 7:08
  • $\begingroup$ A doubt – how exactly is resistance neither extensive nor intensive? I'm tempted to think it is an extensive property since it changes with amount of wire. $\endgroup$ – Anurag B. May 18 '18 at 7:10
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The resistivity is a material property, thus it's intrinsic. Depending on the shape of the piece of the material you are using, the resistance will change accordingly.

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