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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. $\endgroup$ Commented Apr 30, 2018 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 $l$. $\endgroup$ Commented Apr 30, 2018 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
    Commented Apr 30, 2018 at 19:10
  • $\begingroup$ Because that's how we define resistivity. Resistance offered by something with a unit length and unit cross sectional area. You can't just plug values in that area and length variable according on your own when you just defined resistivity as Resistance of a material having a unit length and unit cross sectional area. $\endgroup$ Commented Jan 16, 2022 at 5:54

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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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  • $\begingroup$ If it is "resistance" then the units doesn't work out. Numerical value equality doesn't indicate equality of physical quantities. $\endgroup$ Commented Dec 5, 2020 at 21:38
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Let me remark, that from microscopic point of view it is more common to talk about the conductance and conductivity, which are inverse to the resistance and resistivity. Thus, I might use below these terms interchangeably.

Resistivity is a property of a material
Within classical electrodynamics (i.e., when the averaging over a macroscopic volume is implied) resistivity is determined by the intrinsic factors, such as the properties of the material and temperature. Resistivity can be expressed in terms underlying physical processes, such as collisions of electrons with impurities, photons, electron-electron scattering, etcs. Drude formula famously expresses resistivity/conductivity in terms of the scattering time, resulting from all these processes.

As long as we can ignore the boundary effects (i.e., the material is macroscopic) none of these depends on the size of the conductor. The total current flowing through the conductor however depends on its geometric properties:

  • the dependence on the cross-sectional area allows more current to pass through - the analogy with a wider pipe is nearly literal here
  • the longer conductor means that the electrons experience more scattering events while travelling from one end to the other.

Resistance and conductance on microscopic scale
On microscopic scale, e.g., when dealing with nanostructures, one often cannot neglect the fact that the size of the conductor is comparable to the mean free path of electrons. In this case the simple formulas relating the resistance do not apply anymore, and one often has to resort to discussing lobal quantities such as conductance and resistance. The numerous associated effects are: ballistic conductance, Anderson localization, weak localization, quantum Hall effect, etc.

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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
    Commented Apr 30, 2018 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. $\endgroup$ Commented May 18, 2018 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. $\endgroup$ Commented May 18, 2018 at 7:10
  • $\begingroup$ This is a perfectly legit (and good) answer. The resistivity is not extensive because two resistors in parallel behave differently than two resistors in series (there is no obvious additivity property, which is the property of extensive variables). The link with thermodynamics siggested in the above answer is interestig (afrer all the resistivity is a thermodynamic property around equilibrium). $\endgroup$
    – Quillo
    Commented Apr 12, 2021 at 7:53
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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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