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I am really sorry if this question is inappropriate or wrong. But this is something that I can never perfectly agree with, it just keeps on striking my mind when I am studying something new in Physics. So, my question is:

Whenever developing any new theory or formulating any law, why have scientist always considered the constant of proportionality to be constant for every case?

Here is my question in depth:

Lets take the case of resistivity:

  • We say: $R \varpropto L$
  • And, $R \varpropto 1/A$
  • So, $R \varpropto L/A$
  • And hence, $R = \rho L/A$

And this works in the real world very well. Now, have a look at this example and my question. Why is the $\rho$ constant considered to be constant for every case? I mean I understand that if $R$ increases, $L/A$ will definitely increase(which is intuitive enough), but the thing which I am not able to understand is why, for every $k$ times increase or reduction in $R$, why does $L/A$ also gets effected by the same $k$? Is this not more guess work or making assumptions instead of precision?

I have another question: Suppose in the above example, we noted that $R \varpropto L$.

So, could we not make another constant, like $R = kL$? Will this constant not help us to determine $L$ without knowing what is $A$? Why do we combine all variables into one equation while formulating laws?

Note: I request the readers to give a more general answer. This was just an example I quoted, as we all know most of the physics laws involve these constants.

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  • $\begingroup$ Because usually observable objects are large. The difference between a wire of lengths $L_1$ and $L_2$ are the relative importance of the "caps". Lets assume we have a bulk property $\alpha$ with $R=\alpha V$ for a uniform material. on the caps the system is different, so lets say the property is then $\beta$ on a length $\delta$ from the cap. the we have $R\propto 2 \beta \delta + \alpha (L-\delta)$. $\delta$ is given by microscopic lengthscales (and possibly the radius) all of which are very small compared to a typically used length, so that the observed $\rho$ is length independent $\endgroup$ – Bort Feb 8 '16 at 15:54
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    $\begingroup$ Physical constants are not considered constant by physicists. We keep measuring them all the time and we keep looking for deviations from the assumption that they are constant. What you got there in your example is not even a physical constant but it's a material constant and that one isn't even close to being "constant". $\endgroup$ – CuriousOne Feb 8 '16 at 16:04
  • $\begingroup$ @Siddhant , the resistivity "constant" in your resistance equation is actually dependent on temperature, so it isn't really constant after all. $\endgroup$ – David White Aug 9 '16 at 16:13
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why have scientist always considered the constant of proportionality to be constant for every case

Because it has been experimentally verified many, many, many times. When a certain correlation like your example is found in one case, then it is tested in many other cases. In many, many, many other cases. Eventually you believe that it is indeed a correlation that holds for every case. You can't prove it, just like Newton's laws of motion can't be proven, but you might be able to falsify it if it is wrong if you manage to find a counter-example.

If a correlation becomes very established and has never shown any inconsistencies, we believe it enough to call it a law of nature.

BUT then people might still find out that some correlations that have become established do not work in every instance. But maybe this is not because the correlation or involved constants are wrong - maybe they just do not apply in certain circumstances.

Like measuring the gravitational constant $g=9.82\;\mathrm{m/s^2}$. All experiments would show (around) the same value. But then suddenly someone meaures it on the Moon, and all is screwed...

Then you apply a boundary. A limit of validity. A range within which, the law or correlation or constant is fruitful to use, while it will not be in the extreme cases. This happens for example with relatively when velocity is extremely large or with quantum mechanics when things are extremely small, where many very simple and intuitive expressions and formulae suddenly do not work.

We still trust them but just remember to only use them in the ranges, where we know they work. While a new expression is made for the extreme cases. And everything is fine again.

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