# Ohm’s Law

How did anyone ever figure anything out about anything?

Simple! Two ways: 1. By thinking real hard about it (Aristotle and co. were all about that), 2. By looking real hard at what it is you’re lookin at (Me at age 5), 3. A combination of both (Science!)

Einstein mostly thought a real hard, whereas most of the other chumps – Ohm included – had to scrape by on experimental observations and thinking real hard.

Ohm’s Law describes how Electric Currents form across Electric Potential Differences in Conductors (under constraints – the conductors need to be “Ohmic materials” a.k.a. “follow Ohm’s Law“). It relates Voltage with the current, and something called Resistance – a property of materials to resist electric currents flowing.

How’d Ohm figure it all out? Let’s have a think.

It’s been mentioned before, but the Drift Velocity of electrons in an Electric Field permeating throughout a conductive material is given by: $v_{drift} = v_{average} + {(\frac{qE}{m})}{\tau}$  – and Electric Current, by definition, is the amount of electrons flowing through a unit cross section of whatever, per unit time.

This can be formulated as follows: $dQ = qnAv_{drift} dt$ -> $I = \frac{dQ}{dt} = qnAv_{drift} = JA$ where $J = qnv_{drift}$ is the Current Density of a material. It depends on the material properties “n” and “τ”, the amount of electrons per volume, and the Mean Free Time respectively.

Ohm thought (I think): $J=\frac{I}{A} = qnv_{drift} = q^2n\frac{E}{m}$ (assuming no average velocity) And so he put ${\sigma} = \frac{q^2n}{m}$ making $J={\sigma}E$ Where that $\sigma$ is the Conductivity of the conductor (different for different materials).

That’s Ohm’s Law believe it or not! To get the form we all know and love (if you’ve ever done electronics ever) we just find that $V = EL$ where “V” is the voltage across a distance of length “L” with electric field “E” – and also that $I=JA$ from before.

$E = \frac{V}{L} = \frac{I}{{\sigma}A}$—>$V = \frac{IL}{{\sigma}A} = I\frac{{\rho}L}{A} = IR$ where $\rho = \frac{1}{\sigma}$ is the Resistivity (reciprocal of the conductivity) of the material, and $R = {\rho}\frac{L}{A}$ is the Resistance across a potential difference of “V” with current “I”.

Don’t forget that “E” is a vector quantity, so that means “J” is too!