What is the real derivation of Ohm's law?
Solution 1
The thing we call "resistance" is a constant of proportionality for the relationship between $V$ and $I$. Because of the word, we like to think that "more resistance" should mean "less current".
Simply stating "current is proportional to voltage" leads to an expression of the general form
$$I = \alpha V$$
Now we need to decide the relationship between $\alpha$ and the thing we want to call "resistance", $R$. If resistance increases, we want current to decrease, so apparently
$$\alpha = \frac{1}{R}$$
Now we rearrange, and we obtain the familiar
$$V = I\cdot R$$
Solution 2
Maybe you haven't got it right.
When we say that Ohm's law is about proportionality between $I$ and $V$ we mean that when you double $I$ then $V$ doubles as well (and vice versa, double $V$ and you'll get double $I$), and so on. We don't mean which is the constant of proportionality.
One formulation of Ohm's law is $V = R \cdot I $, and here the constant is the resistance $R$. But when you invert that formula, you get another formulation of Ohm's law, namely $I = \dfrac{V}{R} = \dfrac{1}{R} \cdot V$.
In this latter case the constant of proportionality is $\dfrac{1}{R}$, which is sometimes called conductance and represented with a $G$ letter, so that Ohm's law can also be rewritten as $I=G \cdot V$.
In both cases you have a proportionality relationship, i.e. a linear relationship between $I$ and $V$. They are not different laws, it's the same physical law expressed mathematically with different formulas.
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Comments

Usman Siddiq almost 4 years
$I$ is proportional to $V$. But then how it is that I directly got $V=IR$? It looks impossible to handle this thing! If $I$ is proportional to $V$, then it must be $I=RV$. Why it is $V=IR$? The problem is of dependence of $V$ on $I$.

ACuriousMind almost 9 yearsI have no idea what you are asking.

Usman Siddiq almost 9 yearsI is propotional to V right.What next

Involute almost 9 yearsI believe the proportionality of voltage to current was determined experimentally, and it only applies to particular components, namely the resistor (Ohm's law does not apply to components such as transistors, op amps, etc.). Once you find a proportionality between two variables, it's always nice to determine a constant of proportionality. The resistance is exactly that, and is defined as the ratio of voltage to current (however, it is not the ratio of current to voltage. That would be one divided by the resistance.) The resistance is obtained by experiment.


jabirali almost 9 yearsUsman, if this post helped you understand what you were trying to figure out, then you should accept this answer :)