Drift velocity, mobility, conductivity

Intuition The big picture (WHY this exists)

Inside a metal wire, electrons are always zooming around at huge thermal speeds ( $10^5$ m/s) in random directions — like a swarm of bees in a jar. With no battery, all that motion cancels out: net flow = zero, no current. When you apply a field $E$ , you add a tiny, steady sideways nudge on top of the chaos. The electrons still mostly zoom randomly, but their average sags slightly in one direction. That tiny average sag is the drift velocity $v_d$ — and it is shockingly slow ( $10^{-4}$ m/s, slower than a snail!). The puzzle this resolves: if electrons crawl, why does a bulb light instantly? Because the field travels near light speed and pushes all electrons at once — like a tube already full of water; push one end, water exits the other end immediately even though each drop barely moves.

1. Drift velocity — derived from scratch

HOW we derive it (Newton + collisions):

WHAT happens between collisions? A free electron feels force $\vec F = -e\vec E$ , so acceleration: $\vec a = \frac{\vec F}{m} = \frac{-e\vec E}{m}$

WHY introduce a time $\tau$ ? Electrons collide with vibrating ions. After each collision, direction is randomized, so the gained velocity is wiped. Let $\tau$ = relaxation time = average time between collisions.

Average drift = (acceleration) × (average time gaining speed since last collision): $\boxed{\vec v_d = \vec a\,\tau = \frac{-e\vec E}{m}\,\tau}$

Why this step? Right after a collision the gained velocity is ~0; just before the next it is $a\tau$ . Averaging over many electrons → factor of one $\tau$ (the mean free time).

Magnitude: $v_d = \frac{eE\tau}{m}$

2. Current in terms of drift velocity

Number of electrons in that cylinder = (number density $n$ ) × (volume) = $n \cdot A\,v_d\Delta t$ .

Charge through = $\Delta Q = e\, n A v_d \Delta t$ .

$I = \frac{\Delta Q}{\Delta t} = neAv_d$

Current density $J = I/A$ : $J = nev_d$

3. Ohm's law, conductivity & mobility — emerge naturally

Substitute $v_d = \dfrac{eE\tau}{m}$ into $J=nev_d$ :

$J = ne\cdot\frac{eE\tau}{m} = \underbrace{\frac{ne^2\tau}{m}}_{\sigma}\,E$

This is microscopic Ohm's law $J = \sigma E$ , with:

Connect them: since $\sigma = ne^2\tau/m = ne\cdot(e\tau/m)$ : $\boxed{\sigma = ne\mu}$

Figure — Drift velocity, mobility, conductivity

4. Worked examples

5. Steel-manned mistakes

6. Active recall

Recall Self-test (cover the answers)

Q: Why is current essentially zero with no field despite fast electrons? A: Random thermal velocities cancel; net average = 0.
Q: Where does the single factor of $\tau$ come from? A: Average gaining time between collisions equals the mean free time $\tau$ .
Q: Why does resistance rise with temperature? A: More ion vibration → smaller $\tau$ → smaller $\sigma$ , larger $\rho$ .
Q: Derive $\sigma=ne\mu$ . A: $J=nev_d=ne\mu E=\sigma E\Rightarrow\sigma=ne\mu$ .

Recall Feynman: explain to a 12-year-old

Imagine a long pipe already packed full of marbles. Electrons are the marbles. They're always jiggling around super fast but going nowhere on average. When you tilt the pipe a tiny bit (the battery), all the marbles slowly creep one way — that creep is so slow you'd barely see it. But because the pipe is already full, the instant you tilt it, a marble pops out the far end right away. That's why the light turns on instantly even though each electron is a lazy crawler!

Flashcards

Define drift velocity

Average velocity of free electrons due to applied field, on top of random thermal motion;

v_d=eE\tau/m

Formula linking current and drift velocity

I=neAv_d

What is relaxation time

\tau

Average time between successive collisions of an electron with ions.

Expression for conductivity

\sigma=ne^2\tau/m = ne\mu

Define mobility and its unit

\mu=v_d/E=e\tau/m

; unit

\text{m}^2\text{V}^{-1}\text{s}^{-1}

Why does resistivity increase with temperature?

Ions vibrate more → more collisions →

\tau

decreases →

\sigma

falls,

\rho

rises.

Typical magnitude of drift speed in copper

10^{-4}

m/s (slower than a snail).

Why does a bulb light instantly despite slow drift?

The electric field propagates near light speed, pushing all electrons simultaneously.

Microscopic form of Ohm's law

J=\sigma E

Relate

\tau

and mobility

\tau=\mu m/e

Connections

Ohm's Law — macroscopic $V=IR$ is $J=\sigma E$ in disguise.
Resistivity and temperature dependence — $\rho(T)$ via $\tau(T)$ .
Current density — $\vec J = ne\vec v_d$ .
Free electron model of metals — origin of $n$ and random motion.
Electric field inside conductors — drives the drift.
Semiconductors — mobility & carrier density govern conduction there too.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek metal wire ke andar electrons hamesha bahut tezi se idhar-udhar bhaag rahe hote hain — random directions mein, jaise ek jar mein machhiyon ka jhund. Bina battery ke ye sab cancel ho jaata hai, net flow zero. Jaise hi tum field $E$ lagate ho, har electron ko ek halki si push milti hai ek hi direction mein. Is wajah se inki average velocity thodi si jhuk jaati hai — yahi hai drift velocity $v_d=eE\tau/m$ . Mazedaar baat: ye itni dheemi hoti hai (~ $10^{-4}$ m/s) ki snail bhi tez nikle!

Toh phir bulb turant kaise jalta hai? Kyunki field light speed se travel karta hai aur saare electrons ko ek saath dhakka deta hai — jaise paani se bhari pipe, ek taraf push karo to doosri taraf paani turant nikalta hai, chahe har drop khud zyada na hile.

Current ka formula bana hum simple counting se: time $\Delta t$ mein jo electrons cross-section paar karte hain unhe gino → $I=neAv_d$ . Isme $v_d$ daal do to $J=\sigma E$ aata hai, aur $\sigma=ne^2\tau/m=ne\mu$ . Yahan $\mu$ (mobility) matlab "per unit field kitni drift" milti hai. $\tau$ (relaxation time) collisions ke beech ka average time hai.

Important point exam ke liye: temperature badhne par ions zyada vibrate karte hain, collisions zyada hote hain, $\tau$ kam hota hai — isliye conductivity girti hai aur resistance badhta hai. Ye "Sister Nancy Eats Mangoes" yaad rakho → $\sigma=ne\mu$ . Bas itna samajh lo to poora topic clear hai!

1. Drift velocity — derived from scratch

2. Current in terms of drift velocity

3. Ohm's law, conductivity & mobility — emerge naturally

4. Worked examples

5. Steel-manned mistakes

6. Active recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections