Drift velocity, mobility, conductivity: Visual walkthrough

1.8.15 · D2 · Physics › Electromagnetism › Drift velocity, mobility, conductivity

Intuition Yeh page kya karta hai

Parent note ne tumhe punchline bata di thi: tez, random electrons ki bheed par ek chota sa sideways dhakka ek slow average creep produce karta hai jise drift kehte hain, aur uss ek idea se sab kuch — current, current density, Ohm's law, conductivity, mobility — nikal aata hai. Yahan hum uss story ko draw karte hain, bilkul bare essentials se shuru karke — koi formula assume nahi, har symbol khud kamaya hua. Nau steps, nau pictures. Koi symbol tab tak nahi aata jab tak tum use dekh na lo. End tak tum $σ = \frac{n e ^{2} τ}{m}$ ko page par khud assemble hote dekh chuke hoge. (Temperature ke baare mein note: "from scratch" yahan ka matlab hai pedagogically kuch nahi se — not absolute zero par. Electrons ordinary room temperature par hain aur pehle se $\sim 1 0^{5}$ m/s par zoom kar rahe hain. Temperature sirf Step 9 mein, bilkul end mein aata hai.)

Hum build karenge, is order mein: random swarm → nudge → ek electron ka zig-zag → average creep $v_{d}$ → ek slice se electrons ginना → current $I$ → current density $J$ → conductivity $σ$ aur mobility $μ$ → temperature edge case.

Step 1 — Swarm bina battery ke (starting picture)

KYA HAI. Ek metal ek fixed lattice hai positive ions ka, saath mein ek gas hai free electrons ki — wo electrons jo kisi ek atom se tied nahi hain, poore wire mein roam karne ke liye free hain. Bina battery ke, har electron ek straight line mein udta hai, kisi ion se takrata hai, ek random nayi direction mein ricochets leta hai, phir udta hai. Isse thermal motion kehte hain, aur yeh tez hoti hai: lagbhag $1 0^{5}$ metre har second.

YEH YAHAN SE KYU SHURU KAREIN. Kyunki is poore topic ka surprise yeh hai ki current electrons ka chalna shuru karna nahi hai — woh pehle se pagalon ki tarah move kar rahe hain. Current ek bias hai jo upar se laaya jaata hai. Bias dekhne ke liye pehle us chaos ko dekhna padega jis par woh sawaar hota hai.

PICTURE. Neeche, har grey dot ek ion hai; har coloured arrow ek electron ki velocity hai. Arrows sab jagah point kar rahe hain. Inhe tip-to-tail add karo (dashed black arrow) aur tum wahan pahunch jaate ho jahan se shuru kiya tha: vector sum zero hai.

Figure — Drift velocity, mobility, conductivity

Step 2 — Field on karo: ek universal nudge

KYA HAI. Battery connect karo. Isse wire ke along ek electric field $E$ set up hota hai. Field hai "force-per-charge waiting to happen": usme ek charge $q$ daalo aur woh $F = q E$ feel karta hai. Hamara charge electron hai, $q = - e$ , jahan $e = 1.6 \times 1 0^{- 19}$ coulombs electron ke charge ka (positive) size hai.

MINUS SIGN KYU MATTER KARTA HAI. Kyunki $q = - e$ negative hai, force $F = - e E$ $E$ ke opposite point karta hai. Field ek taraf point karta hai (hamari chosen positive direction); electrons doosri taraf dhakele jaate hain. Yeh soch ke raho — yahi woh minus sign hai jo baad mein "current runs with $E$ " ke saath cancel ho jaata hai, magnitude formulas ko clean chhodta hua.

PICTURE. Field wire ko bhar rahi faint lavender arrows hain (har jagah same → "uniform"). Ek electron par force coral mein draw ki hai: woh field ke against point karti hai.

Step 3 — EK electron par zoom in karo: zig-zag

KYA HAI. Ek single electron ko follow karo. Do collisions ke beech mein woh ek saath do kaam karta hai: apni purani random thermal velocity rakhta hai aur constant acceleration $a$ ki wajah se dheere dheere muda jaata hai. Toh uska path thoda curved dart hota hai. Phir — bang — kisi ion se takrata hai, aur collision uski direction dobara randomise kar deti hai, woh jo bhi sideways speed $a$ ne build ki thi use mita deti hai.

COLLISIONS HI POORA TRICK HAIN. Bina collisions ke, $a$ electron ko hamesha ke liye speed up karta rehta ( $v = a t$ bina kisi ceiling ke) aur current bound ke bina badhta rehta. Collisions ek reset button ki tarah kaam karte hain: har ek field-gained velocity throw away karta hai, isliye average par electron sirf ek free flight ka gain kabhi accumulate karta hai. Yahi drift ko small aur steady rakhta hai.

PICTURE. Zig-zag: roughly straight darts (har ek force direction ki taraf thoda sa bend karta hua) jo collision points (coral dots) se punctuate hain. Notice karo ki har segment pure randomness se thoda zyada downfield lean karta hai — woh lean drift ka seed hai.

Step 4 — Darts ko average karo: drift velocity paida hoti hai

KYA HAI. Ek flight mein field-gained velocity ko average karo. Yeh linearly $0$ se $a τ$ tak ramp karta hai, isliye apni flights mein random points par bahut saare electrons ka average $a$ ka ek factor of $τ$ times hoga: $v_{d} = a τ .$

SIRF EK $τ$ KYU. Kyunki har collision counter ko zero par throw back kar deti hai. Electron kabhi "ek flight se zyada" field velocity build up nahi kar paata — mean flight $τ$ lastti hai. Woh ek, humble $τ$ hi poori memory hai jo metal field ki rakhta hai.

PICTURE. Step 1 jaisi wahi swarm, lekin ab arrows bilkul perfectly random nahi hain — har ek mein $E$ ke against ek tiny extra component push kiya hua hai. Inhe tip-to-tail sum karo: woh almost cancel ho jaate hain, lekin jo bacha hai (thick mint arrow) woh start par wapas nahi aata. Woh chota sa bacha hua, $N$ se divide kiya hua, $v_{d}$ hai.

Step 5 — Ek line cross karne wale electrons gino: current

KYA HAI. Wire mein ek flat slice lo, area $A$ . Poocho: $Δ t$ time mein kitna charge usse cross karta hai? Jo bhi electron slice ke $v_{d} Δ t$ peeche ke andar hai woh us time mein wahan pahunch jaayega (woh $v_{d}$ par creep karta hai aage). Toh jo electrons cross karte hain woh length $v_{d} Δ t$ aur base $A$ ke ek imaginary cylinder mein bharte hain.

CYLINDER KYU. Yeh simply "distance = speed × time" ko volume mein badalna hai. Length $v_{d} Δ t$ hai ki drift $Δ t$ mein ek electron ko kitna carry karta hai; slice area $A$ se multiply karo volume milta hai jo empty hota hai. (Hum yahan $v_{d}$ ko positive magnitude ke roop mein use karte hain — creep ki speed; direction Step 4 mein fix ho gayi thi.)

PICTURE. Wire, slice (mint), aur uske peeche shaded cylinder of length $v_{d} Δ t$ . Cylinder ke andar har dot cross karta hai; bahar wale nahi karte.

Count aur charge: $electrons crossing N_{cross} = per m^{3} n \times cylinder volume A v_{d} Δ t, Δ Q = e N_{cross} = e n A v_{d} Δ t .$ Yahan $e$ positive charge magnitude hai aur $Δ Q$ positive amount of charge hai jo $Δ t$ mein forward deliver hota hai.

Step 6 — Master current relation assemble karo

KYA HAI. Current hai charge-per-time. $Δ Q$ ko $Δ t$ se divide karo aur time cancel ho jaata hai: $I = \frac{Δ Q}{Δ t} = \frac{e n A v _{d} Δ t}{Δ t} .$

$Δ t$ KYU CANCEL HOTA HAI. Kyunki current depend nahi karta kitni der hum dekhte hain — ek steady stream ek fixed rate par charge deliver karta hai. Dono baar utna time, dono baar utna charge; ratio fixed hai.

PICTURE. Ek "term ledger": $I = n e A v_{d}$ ka har factor ek knob ki tarah draw kiya hua jo dikhata hai ki physically kya badal raha hai.

Wire ki thickness hatao $A$ se divide karke current density $J$ pao (current per unit area): $J = \frac{I}{A} = n e v_{d} .$ Yeh material aur field ki property hai, is baat ki nahi ki wire kitni moti hai. Poore vector ke roop mein, $J = n e v_{d} = - n e v_{d} \hat{E}$ , $E$ ke along point karta hai kyunki $v_{d}$ khud uske against point karta hai — phir woh do signs cancel ho kar $J$ ko $E$ ke along de dete hain.

Step 7 — Drift ko $J$ mein substitute karo: conductivity appear hoti hai

KYA HAI. Hamare paas do facts hain (dono $+ E$ direction mein positive magnitudes ke roop mein): $J = n e v_{d} (Step 6) aur v_{d} = \frac{e E τ}{m} (Step 4) .$ Doosre ko pehle mein daalo: $J = n e \cdot \frac{e E τ}{m} = (\frac{n e ^{2} τ}{m}) E .$

YEH PAYOFF KYU HAI. Bracket mein sab kuch — $n, e, τ, m$ — metal ki ek fixed property hai, $E$ se independent (wahi assumption thi jo Step 3 mein flag ki gayi thi). Toh $J$ simply $E$ ke proportional hai. Wahi proportionality exactly wahi hai jo Ohm's law claim karta hai. Humne Ohm's law assume nahi kiya; woh emerge hua.

PICTURE. Substitution dikhata hua jaise $v_{d}$ (mint) $J$ expression mein slot ho raha hai, constant lump ko circle karke $σ$ rename kiya ja raha hai.

Step 8 — Mobility: wahi physics, re-packaged

KYA HAI. Mobility $μ$ ko define karo as "per unit field kitna drift milta hai": $μ = \frac{v _{d}}{E} .$ Kyunki $v_{d} = e E τ / m$ hai, $E$ cancel ho jaata hai: $μ = \frac{e τ}{m} .$

JAB HAMARE PAAS PEHLE SE $σ$ HAI TOH KYUN BOTHER KARNA? Kyunki $μ$ ek carrier ki responsiveness isolate karta hai, alag kitne carriers hain usse. Semiconductors mein, jahan $n$ huge factors se change ho sakta hai, "kitne" ( $n$ ) aur "kitne nimble" ( $μ$ ) ko alag karna natural language hai.

PICTURE. Ek straight line: vertical axis par $v_{d}$ , horizontal par $E$ . Uska slope $μ$ hai. Steeper line = zyada mobile carrier.

Step 9 — Temperature edge case (aur absolute zero ke paas kya hota hai)

KYA HAI. Wire garam karo. Ions zyada tezi se vibrate karte hain, toh ek chalte electron ko woh zyada baar milte hain — average free flight $τ$ chhota ho jaata hai. $σ = n e^{2} τ / m$ mein aur kuch zyada change nahi hota. Toh $τ ↓\Rightarrow σ ↓\Rightarrow ρ ↑$ .

YEH EK TRAP KYU HAI JISPAR APNA STEP CHAHIYE. Thermal speed temperature ke saath badhti hai ( $\propto T$ ), jo tumhe sochne par majboor karta hai ki drift bhi badhti hai. Lekin drift $τ$ par depend karta hai, aur $τ$ heat se girta hai. Badi thermal jiggle → zyada obstacles → chhhota $τ$ → usi field ke liye chhhota drift. Do speeds temperature ke saath opposite directions mein move karte hain.

$σ$ $T \to 0$ PAR KABHI BLOW UP KYU NAHI KARTA. Cooling se ion vibration (jise phonon scattering kehte hain) fade away hoti hai, jo akele $τ \to \infty$ aur $σ \to \infty$ bhej deta — unphysical. Real metals isse bachte hain kyunki electrons impurities aur defects par bhi scatter hote hain jo freeze ho jaate hain, aur jo temperature ki parwah nahi karte. Yeh leftover scattering collisions par ek floor fix kar deta hai, isliye $T \to 0$ par resistivity ek nonzero residual resistivity $ρ_{0}$ par settle ho jaati hai, aur $σ$ finite rehta hai. (Yeh split — vibration part + impurity part — Matthiessen's rule hai.)

PICTURE. Do panels: cold lattice (ions still, lambi free flights) vs hot lattice (ions vibration se blurred, chopped-up short flights).

Ek-picture summary — nontrivial takeaway

Nau steps chain up hote hain, lekin jo insight le jaane layak hai woh chain se bhi zyada subtle hai: ek microscopic constant, $τ$ , secretly poori macroscopic story control karta hai. $τ$ badlo (garam karke, impurities daalne se) aur $v_{d}$ , $I$ , $J$ , $σ$ , $ρ$ , aur $μ$ sab lockstep mein move karte hain — is topic ka har observable actually $τ$ disguise mein hai. Neeche figure woh single lever visible banata hai: $τ$ khicho aur poori column ko respond hote dekho.

Recall Feynman retelling — plain words mein poora walkthrough

Ek jar of bees imagine karo, har ek super fast zip kar raha hai, average mein kahi nahi ja raha (Step 1). Ab jar ko thoda sa tilt karo — woh tilt electric field ka push hai (Step 2). Har bee phir bhi randomly zip karti hai, lekin har flight thodi si downhill bend karti hai, aur jab bhi koi bee wall se takrati hai woh bend bhool jaati hai aur fresh start karti hai (Step 3). Saari bees aur unke bumps ko average karo aur tumhe ek tiny, steady drift downhill milti hai (Step 4) — ek average time-between-bumps, $τ$ , se set. Yeh count karne ke liye ki kitni bees ek doorway se per second nikalty hain, woh chota cylinder sweep karo jo woh us time mein reach kar sakti hain (Step 5): woh deta hai $I = n e A v_{d}$ (Step 6). Doorway ka size hatao aur $J = n e v_{d}$ milta hai; drift plug karo aur bees ki sankhya, unka charge, unka bump-time aur weight sab ek constant mein lump ho jaate hain — conductivity $σ = n e^{2} τ / m$ , jo sirf Ohm's law hai microscope pehne (Step 7). "Ek bee ki nimbleness" ko mobility $μ$ ke roop mein relabel karo, aur $σ = n e μ$ (Step 8). Aakhir mein, jar garam karo: walls kaanpne lagti hain, bumps jaldi aate hain, $τ$ girta hai, aur poori cheez worse conduct karti hai — lekin bilkul thande jar mein bhi bees phir bhi odd stuck crumb (ek impurity) se takrati hain, toh yeh kabhi perfectly conduct nahi karta (Step 9). Ek knob, $τ$ , quietly poora show run karta hai.

Recall Quick self-test

Woh single factor of $τ$ kahan se aata hai? ::: Field-gained velocity ek flight mein 0 se $a τ$ tak ramp karti hai; averaging ek $τ$ chhodta hai. $I = n e A v_{d}$ se $Δ t$ kyun gayab ho jaata hai? ::: Current ek rate hai; cylinder ka charge aur time dono $Δ t$ ke saath scale karte hain, isliye unka ratio fixed hai. $I = n e A v_{d}$ aur $J = n e v_{d}$ sign-clean kyun hain despite electron ke $- e$ ke? ::: Electron ka negative charge aur uska backward drift do minus signs hain jo cancel ho jaate hain, toh current $E$ ke saath chalta hai aur sab magnitudes positive hain. $σ$ ko "how many" aur "how nimble" mein split karo. ::: $σ = (n e) \times μ$ — carrier count times mobility. Heating resistance kyun badhata hai agar thermal speed badhti hai? ::: Drift $τ$ par depend karta hai, thermal speed par nahi; garam ions zyada collide karte hain, toh $τ$ aur isliye $σ$ girte hain. $T \to 0$ par $σ \to \infty$ kyun nahi hota? ::: Impurity/defect scattering survive karti hai, ek residual resistivity $ρ_{0}$ chhodti hai; $σ$ finite rehta hai.

Connections

Ohm's Law — Step 7 ka $J = σ E$ microscope ke neeche $V = I R$ hi hai.
Current density — Step 6 ka $J = n e v_{d}$ .
Free electron model of metals — $n$ aur Step 1 ki random swarm supply karta hai.
Electric field inside conductors — Step 2 ka $E$ .
Resistivity and temperature dependence — Step 9 ka $τ (T)$ aur residual resistivity.
Semiconductors — jahan Step 8 ka $n$ vs $μ$ split kaam aata hai.