How can one derive Ohm's Law?

Question

I am looking for the derivation of Ohm's Law, i.e., $V$ is directly proportional to $I$. Can someone help me with it?

Answer 1

Ohm's Law is not a construct which can be derived. It is essentially a generalized observation. It is only useful for a few materials (conductors and medium resistivity), and even then virtually all of those materials show deviations from the ideal, such as temperature coefficients and breakdown voltage limits.

Rather, Ohm's Law is an idealization of the observed behavior of these materials. As the saying goes, "All models are wrong. Some models are useful." In this case, Ohm's Law is extraordinarily useful, but that doesn't make it universal. Semiconductors, for instance, do not follow Ohm's Law in any large sense, and look how widespread their use is.

As originally discovered and formulated, there was a great deal of wishful thinking involved. There was no understanding of the forces involved, and there was no real definition, for instance, of voltage or current. Nonetheless, it was determined that a self-consistent set of values was possible (you can define different battery chemistries as producing specific voltages, and get consistent behavior of galvanometers - as long as you're willing to accept experimental error). Over time, standards were set and more objective measures discovered, such as the quantity of electrons in a coulomb, so that a current of 1 amp can be unambiguously measured) Eventually, a very good understanding of the behavior of electrons (and holes) in conductors was reached, and that understanding is generally, for a wide range of useful conditions, expressible as Ohm's Law.

But it is not derived.

Answer 2

You could start from Drude in zero magnetic field, that equates the derivative of the momentum $\vec p$ by the electrostatic force $\vec F_{el} = q \vec E$ as a product of charge $q$ and electric field $\vec E$ minus a scattering term (with time constant $\tau$; compared to Newtons second law that does not feature the latter, crystal term):

$~~~~~~\dot {\vec p} = q {\vec E} - \cfrac{\vec p}{\tau}$

The stationary solution ($\dot {\vec p} = 0$) utilizing the current density $\vec j$

$~~~~~~\vec j = n q \vec v$

as the product of carrier denstiy $n$, charge $q$ and carrier velocity $\vec v$ entails

$~~~~~~\vec j = \cfrac{q^2}{m} \tau n \vec E$

which is nothing but the linear relationship between current (density) "$j = \cfrac{I}{A}$" and electric field (potential gradient) "$E = \cfrac{U}{d}$" that is stated by Ohm.

Answer 3

In my opinion, the mathematical equation we call Ohm's Law is best understood not as a “law”, a fact about the universe, but as the definition of the quantity “resistance”.

$$R \overset{\mathrm{def}}{=} \frac{V}{I}$$

Given this definition of $R$, we can then make (as other answers have mentioned) the empirical observation that many materials have approximately constant $R$ (which we call being ohmic) and therefore $R$ is a useful quantity to have defined.

But if there is something to derive, then it is the answer to “why do we observe approximately constant $R$?”, not $V = IR$.

Answer 4

Ohm's law isn't fundamental and holds true only under certain conditions, like constant temperature for example. However, there is a simple way to think about it. Imagine the flow of massive objects through a wide water pipe. This is like a current. The water pressure causes the objects to flow quickly, that's your voltage. If the pipe is narrow then the objects can't flow as quickly. Also, the objects can be slowed down as they hit the sides of the pipe which isn't perfectly smooth. This is the resistance. Now if you take the water pressure (voltage) and divide by a resistance to flow, you get the rate at which objects flow through the pipe. Dividing by a large resistance means less flow.

Think of it as V/R = I instead of V = IR Division is easier to grasp mentally. As the resistance gets smaller the current goes up and vice versa.