# What is the real derivation of Ohm's law?

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## 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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### Usman Siddiq

I am student

Updated on January 16, 2020

$I$ 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 $I$.