Foundations — Ohm's law — microscopic origin, resistivity

This page assumes nothing. Before you read Ohm's law — microscopic origin, resistivity, you must be fluent in every letter it writes. Below, each symbol gets three things: a plain-words meaning, the picture it stands for, and why the topic needs it. They are ordered so each one only uses ideas already built.

0. The scene: what is inside a wire?

Picture a copper wire zoomed in a billion times. You would see a neat grid of heavy atoms locked in place — the lattice — and a swarm of light, loose electrons wandering freely between them, like flies in a warehouse of shelves.

Two facts to hold onto:

• The atoms are fixed (they only jiggle in place — more jiggle when hot).
• The electrons are free to roam — these are the only things that actually move down the wire to make a current.

1. Charge, and the electron's charge $e$

Plain words: Charge is the property that makes particles feel electric pushes and pulls. We measure it in coulombs (C).

The picture: think of it as an invisible "tag" on a particle that says how strongly the electric world grabs this thing.

Why the topic needs it: every force on an electron and every bit of current is counted in multiples of $e$. Without it we can't turn "how many electrons moved" into "how much current."

2. Electric field $\vec{E}$ — the push, drawn as arrows

Plain words: the electric field at a point tells you the force a unit of positive charge would feel there. It has a size and a direction, so we draw it as an arrow and write it with a little arrow on top: $\vec{E}$.

The picture: imagine an arrow at every point in the wire, all pointing the same way (down the wire). A positive charge is pushed along the arrow; the negative electron is pushed opposite.

Why the topic needs it: $\vec{E}$ is the push. See Electric field inside a conductor for why a steady $\vec{E}$ can even exist inside a wire.

3. Voltage $V$ — the push, counted as a total

Plain words: voltage (or potential difference) is the total "electric shove" between two ends of the wire. If the field is the push per metre, voltage is the push added up over the whole length.

The picture: a hill. The field is the steepness at each spot; the voltage is the total drop from top to bottom. A ball rolls down because of the steepness; the total distance it can fall is the voltage.

Why the topic needs it: the macroscopic law is $V=IR$. To connect it to what electrons feel ($\vec{E}$), we need this bridge $E=V/L$.

4. Speed, velocity $\vec{v}$, and acceleration $\vec{a}$

Plain words:

• Speed = how fast (a plain number).
• Velocity $\vec{v}$ = how fast and which way (an arrow).
• Acceleration $\vec{a}$ = how fast the velocity is changing (also an arrow).

The picture: a dot moving along the wire. Its velocity is an arrow showing its motion right now; if that arrow is growing, it is accelerating.

Why the topic needs it: step 1 of the whole derivation is force → acceleration via $\vec{a}=\vec{F}/m$. This is the honest starting point for "what does one electron do."

5. Drift velocity ${\vec{v}}_d$ — the slow crawl, not the fast wiggle

Here is the subtle one. The electrons are already flying around fast and randomly from heat — but that random motion goes every which way and cancels out: no net travel down the wire.

The picture (figure above): the left panel shows one electron's real path — a jagged random walk with a slight lean downstream. The right panel strips away the wiggle to show only the lean: that lean is ${\vec{v}}_d$.

Why the topic needs it: current is net charge flow. Only the drift ${\vec{v}}_d$ is net, so it is the star of every current formula.

6. Relaxation time $\tau$ — the average gap between bumps

Plain words: $\tau$ (Greek letter "tau") is the average time an electron flies freely between two collisions with the jiggling lattice.

The picture: in the random-walk figure, $\tau$ is the average length of one straight segment (in time) before the next kink.

Why the topic needs it: $\tau$ is what turns "accelerates forever" into "reaches a steady drift." It sits at the heart of the Drude model of conduction and inside $\sigma =n{e}^2\tau /m$.

7. Number density $n$ — how crowded the electrons are

Plain words: $n$ is the number of free electrons packed into each cubic metre of the metal.

The picture: count the roaming dots inside a $1\text{m}\times 1\text{m}\times 1\text{m}$ box. For copper that count is a staggering $n\approx 8.5\times 10^{28}$ per m³.

Why the topic needs it: more carriers → more charge sweeping past each second → more current. $n$ appears in both $J=ne{v}_d$ and $\sigma =n{e}^2\tau /m$.

8. Cross-sectional area $A$ and length $L$ — the wire's shape

Plain words:

• $A$ = area of the wire's circular face (m²) — how fat it is.
• $L$ = length of the wire (m) — how long it is.

The picture: slice the wire straight across; the flat disc you expose has area $A$. Stretch it end to end; that distance is $L$.

Why the topic needs it: geometry ($L$, $A$) is what turns the material property $\rho$ into a real resistance $R=\rho L/A$. See Resistors in series and parallel.

9. Current $I$ and current density $\vec{J}$

Plain words:

• Current $I$ = charge flowing past a point per second (units: amperes, A = C/s).
• Current density $\vec{J}$ = current per unit area (A/m²), an arrow pointing the way charge flows.

The picture: $I$ is the total river flow; $\vec{J}$ is how hard the water rushes at each spot in the riverbed.

Why the topic needs it: these convert the microscopic drift into the measurable current. Deep dive: Electric current and current density.

10. Conductivity $\sigma$, resistivity $\rho$, resistance $R$

Now the payoff symbols — all just names for "how easily current flows."

The picture: $\rho$ is the "muddiness" of the material; $R$ is the total struggle to cross this specific muddy field, which also depends on how long and narrow the path is.

Why the topic needs it: $\sigma$, $\rho$, $R$ are the destination. The entire derivation exists to compute them from $n,e,\tau ,m$.

The map: how these feed the topic

Equipment checklist

Test yourself — cover the right side and answer aloud.

Connections

• Parent topic (Hinglish)
• Electric current and current density
• Electric field inside a conductor
• Drude model of conduction
• Drift and diffusion of carriers
• Resistors in series and parallel
• Temperature dependence of resistance