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I was wondering which interpretation could we find for the resistivity, what image correspond to the concept.

Furthermore, how to interpret its unit $\Omega \:\rm m$ . Why is it more logical than a $ \Omega /m$ ? (I'm not seeking a proof by formula but a physical sense)

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  • $\begingroup$ It is roughly the electric field needed per current density $\endgroup$ – Mauricio Jun 20 '18 at 12:30
  • $\begingroup$ Hi @Mauricio thanks for answering. I want to try to see what it can clearly modelize and mean . $\endgroup$ – Pagode Jun 20 '18 at 12:38
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Fifty or sixty years ago, it was quite common for textbooks to define the resistivity of a material as the resistance between opposite faces of a unit cube of the material. Ignoring units, that is literally correct, and it may or may not be of help in grasping the concept. But this so-called definition has a number of faults…

  1. To measure the resistance between opposite faces of a $1\text{m}^3$ cube, it would be no good touching your ohm-meter probes on to points in the middle of the faces. You'd need to make contact across the whole of these faces. That means you'd need connecting leads of cross-section greater than that of a cube face. In other words you can't measure the required resistance!

  2. The 'definition' based on a unit cube gives you no clue as to how resistivity enables you to find the resistance of a conductor. Indeed you might be misled into thinking that the more metal there is in your conductor, the more, necessarily, its resistance. So the modern definition is as the constant $\rho$ in$$R=\rho\frac{L}{A}.$$in which $R$ is the resistance of a wire of length$L$ and cross-sectional area $A$.

  3. This equation immediately gives the units of resistivity as ${\Omega \text m}$. The unit cube gives them simply as $\Omega$.

To sum up, I'd urge you to ponder the role of $\rho$ in the equation above. It's the factor that takes account of the effect of the material on the resistance of a conductor.

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I assume you have an understanding of what resistance is.

Let's say you have some wire. It has a certain electrical resistance (measured in $\Omega$). If you have an identical wire that is twice as long, it will have twice the resistance. If you have an identical wire that has twice the cross section it will have half the resistance of the original wire.

Now, physicists are usually not interested in a particular wire, but in the general case. If you are going to make a theory for current through a metal wire, you are not going to make that for say a $1~m\times 1~mm^2$ wire, because that theory would be somewhat narrow, limited to one specific case. (What if my wire is 1.5 m long?)

In order to get a more general theory you get rid of the length and cross section dependence, simply by studying the resistivity, $\rho$, which is defined through the equation

$$R=\rho\frac{l}{A}$$

By this procedure you get rid of the size dependence and $\rho$ will depend basically on the material only (and temperature, etc). Also it explains the unit $\Omega m$.

A procedure like this is very common in physics and usually denoted by the word specific, i.e. you could call $\rho$ the specific (electrical) resistance.

Another example from physics is the "specific heat capacity", which is the heat capacity per kg. Just like the resistivity, the specific heat capacity depends on the material only unlike the heat capacity which is proportional to the mass.

Or an example from everyday life: If you go to a shop to buy 0.5 kg of apples you pay a price for them. If your friend goes to the same shop to buy 2 kg of apples he will pay a different price (four times as much as you), simply because he bought a different amount of apples. In order to avoid having to display prices for 0.5 kg, 2 kg, and all kind of other amounts, the shop just displays the price per kilogram (in physics language, the "specific price" of apples). So, while there is some interest in the price of 2 kg/0.5 kg of apples in certain situation (namely for you and your friend), for the general case, the "specific price" is much more convenient.

To answer your question, whether to use resistance or resistivity is basically just a question of context. If you are interested in a particular piece of resistor, you are likely concerned with its resistance. If you are studying more general things, resistivity is your quantity of choice.

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  • $\begingroup$ A longer wire will have a higher resistance than a shorter one , not logical, is it ? $\endgroup$ – Pagode Jun 20 '18 at 15:53
  • $\begingroup$ I mean why it is more logical to define $ \Omega.m$ than $\Omega /m$ for an intrinsic property (semi-intrinsic according to temperature dependance) $\endgroup$ – Pagode Jun 20 '18 at 16:27
  • $\begingroup$ @Pagode: Well, actually it is $\Omega \frac{m^2}{m}=\Omega m$, where the $m^2$ comes from making it intrinsic with respect to the cross section and the $m$ from making it intrinsic with respect to the length. $\endgroup$ – user1583209 Jun 20 '18 at 16:29
  • $\begingroup$ There isn't in fact proper meaning for the $m$ because of it results from a combination of $S$ and $l$ . That's clear. Moreover , is it possible to interpret that , I mean a strict physical meaning of $\Omega.m$ because if the lenght grows the resistance decreases , unlogical ... $\endgroup$ – Pagode Jun 20 '18 at 16:36
  • $\begingroup$ As I wrote above, the $m$ in $\Omega m$ results from cancelling $\frac{m^2}{m}$. (ie.. area divided by length) It does not make sense trying to interpret it as corresponding to the length. What is your question about the total R? $\endgroup$ – user1583209 Jun 20 '18 at 16:53

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