Drift velocity, mobility, conductivity — Hinglish notes

1.8.15 · Physics › Electromagnetism

Intuition Badi picture (YEH kyun exist karta hai)

Ek metal wire ke andar, electrons hamesha bahut badi thermal speeds ( $1 0^{5}$ m/s) par random directions mein ghoom rahe hote hain — bilkul ek jar mein bees ke swarm ki tarah. Koi battery nahi to yeh saari motion cancel ho jaati hai: net flow = zero, koi current nahi. Jab tum field $E$ apply karte ho, tab tum chaos ke upar ek chhoti, steady sideways nudge add karte ho. Electrons ab bhi zyaadatar randomly ghoomte hain, lekin unka average thoda sa ek direction mein jhuk jaata hai. Woh chhota sa average jhukav hi drift velocity $v_{d}$ hai — aur yeh shockingly slow hoti hai ( $1 0^{- 4}$ m/s, ek snail se bhi slow!). Yeh puzzle solve karta hai: agar electrons rengta hain, to bulb instantly kyun jalta hai? Kyunki field light speed ke paas travel karti hai aur saare electrons ko ek saath push karti hai — jaise paani se bhari ek tube; ek end push karo, doosre end se paani turant nikalega chahe har drop bahut kam hila ho.

1. Drift velocity — scratch se derive karo

HOW hum ise derive karte hain (Newton + collisions):

Collisions ke beech WHAT hota hai? Ek free electron force $F = - e E$ feel karta hai, to acceleration: $a = \frac{F}{m} = \frac{- e E}{m}$

WHY time $τ$ introduce karte hain? Electrons vibrating ions se collide karte hain. Har collision ke baad, direction randomize ho jaati hai, to gain ki gayi velocity wipe ho jaati hai. Maano $τ$ = relaxation time = collisions ke beech average time.

Average drift = (acceleration) × (last collision ke baad speed gain karne ka average time): $v_{d} = a τ = \frac{- e E}{m} τ$

Yeh step kyun? Ek collision ke bilkul baad gain ki gayi velocity ~0 hoti hai; agle se bilkul pehle yeh $a τ$ hoti hai. Bahut saare electrons par average karo → ek $τ$ ka factor (mean free time).

Magnitude: $v_{d} = \frac{e E τ}{m}$

2. Drift velocity ke terms mein current

Us cylinder mein electrons ki sankhya = (number density $n$ ) × (volume) = $n \cdot A v_{d} Δ t$ .

Se charge = $Δ Q = e n A v_{d} Δ t$ .

$I = \frac{Δ Q}{Δ t} = n e A v_{d}$

Current density $J = I / A$ : $J = n e v_{d}$

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

$J = n e v_{d}$ mein $v_{d} = \frac{e E τ}{m}$ substitute karo:

$J = n e \cdot \frac{e E τ}{m} = σ \frac{n e ^{2} τ}{m} E$

Yeh microscopic Ohm's law $J = σ E$ hai, jisme:

Inhe connect karo: kyunki $σ = n e^{2} τ / m = n e \cdot (e τ / m)$ : $σ = n e μ$

Figure — Drift velocity, mobility, conductivity

4. Worked examples

5. Steel-manned mistakes

6. Active recall

Recall Self-test (answers cover karo)

Q: Bina field ke, fast electrons ke bawajood current essentially zero kyun hoti hai? A: Random thermal velocities cancel ho jaati hain; net average = 0.
Q: $τ$ ka single factor kahaan se aata hai? A: Collisions ke beech average gaining time mean free time $τ$ ke barabar hoti hai.
Q: Temperature ke saath resistance kyun badhta hai? A: Zyada ion vibration → chhota $τ$ → chhota $σ$ , bada $ρ$ .
Q: $σ = n e μ$ derive karo. A: $J = n e v_{d} = n e μ E = σ E \Rightarrow σ = n e μ$ .

Recall Feynman: ek 12-saal ke bache ko explain karo

Socho ek lambi pipe pehle se marbles se bhari hui hai. Electrons woh marbles hain. Woh hamesha super fast hil rahe hain lekin average mein kahin nahi ja rahe. Jab tum pipe ko thoda sa tilt karte ho (battery), saare marbles dheere dheere ek taraf creep karte hain — yeh creep itni slow hai ki tum barely dekh pao. Lekin kyunki pipe pehle se bhari hai, jaise hi tum tilt karte ho, ek marble doosre end se turant nikalti hai. Isliye light turant jal jaati hai chahe har electron ek lazy crawler ho!

Flashcards

Define drift velocity

Applied field ki wajah se free electrons ki average velocity, random thermal motion ke upar;

v_{d} = e E τ / m

Formula linking current and drift velocity

I = n e A v_{d}

What is relaxation time

τ

Ek electron ki ions se successive collisions ke beech average time.

Expression for conductivity

σ = n e^{2} τ / m = n e μ

Define mobility and its unit

μ = v_{d} / E = e τ / m

; unit

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

Why does resistivity increase with temperature?

Ions zyada vibrate karte hain → zyada collisions →

τ

ghatta hai →

σ

girti hai,

ρ

badhti hai.

Typical magnitude of drift speed in copper

1 0^{- 4}

m/s (snail se bhi slow).

Why does a bulb light instantly despite slow drift?

Electric field light speed ke paas propagate karti hai, saare electrons ko simultaneously push karti hai.

Microscopic form of Ohm's law

J = σ E

Relate

τ

and mobility

τ = μ m / e

Connections

Ohm's Law — macroscopic $V = I R$ asal mein $J = σ E$ hi hai disguise mein.
Resistivity and temperature dependence — $ρ (T)$ , $τ (T)$ ke zariye.
Current density — $J = n e v_{d}$ .
Free electron model of metals — $n$ aur random motion ka origin.
Electric field inside conductors — drift ko drive karta hai.
Semiconductors — wahan bhi mobility & carrier density conduction govern karti hai.