Drift velocity, mobility, conductivity: Foundations

1.8.15 · D1 · Physics › Electromagnetism › Drift velocity, mobility, conductivity

Is page par kuch bhi assume nahi kiya gaya. Isse pehle ki tum $v_{d}$ , $τ$ , $σ$ , ya $μ$ se parent topic par milo, hum har letter, arrow aur picture ko zero se build karte hain. Upar se neeche padho; agle symbol se pehle current symbol earn kar liya jaata hai.

0. "Free electron" kya hota hai? (the actors)

Ek metal ek positive ions ka lattice hota hai — atoms jinhoone apna outermost electron de diya aur ek grid mein freeze ho gaye. Woh diye gaye electrons poore block mein freely ghoomte hain; hum unhe free electrons kehte hain. Current carry karne ke liye actually sirf yahi cheezein move karti hain.

Figure — Drift velocity, mobility, conductivity

Figure dekho: grid mein pinned violet dots ions hain; unke beech zig-zag karte magenta dots free electrons hain. Neeche har symbol inhi electrons ko describe karta hai. Dekhein Free electron model of metals ki yeh simple picture itni achhi kyun kaam karti hai.

1. Charge $e$ — ek electron ki "electric weight" ki size

Picture: $e$ ko ek "electric weight" samjho jo har electron carry karta hai — har electron ke liye same, ek fixed pebble of charge.

Topic ko iski zaroorat kyun hai: current charge hai jo per second flow karta hai. Yeh count karne ke liye ki ek line par kitna charge cross karta hai, hum electrons ki sankhya ko har electron ke charge $e$ se multiply karte hain. $e$ ke bina hum electrons count kar sakte the lekin unhe amperes mein convert nahi kar sakte the.

Minus sign kyun matter karta hai: kyunki electron ka charge $- e$ hai, electric field use field ke relative ulti direction mein push karta hai. Hum iska track §5 mein rakhenge.

2. Electric field $E$ — "sideways nudge"

Picture: wire mein har point par ek arrow, jab battery connected ho toh sab ek hi taraf point karte hain — metal mein ek steady hawa. $E$ (aur $F$ ) ke upar chota arrow matlab hai "is quantity ki ek direction hai", sirf size nahi.

Topic ko iski zaroorat kyun hai: koi field nahi toh crowd kahin nahi jaata. Field hi ek maatra reason hai ki net drift appear hoti hai. Yeh cause hai; drift effect hai. Yeh field kahan se aati hai aur wire mein kitni gehri pahunchti hai? Dekhein Electric field inside conductors.

3. Force $F$ , mass $m$ , acceleration $a$ — Newton enters

Kyun $F = ma$ laate hain aur kuch fancy nahi: hum jaanna chahte hain ki field kis tarah electron ko speed up karti hai. "Force cheezein accelerate karti hai" ka sabse basic law exactly Newton ka hai — koi relativity nahi, koi quantum machinery nahi, kyunki drift speeds absurdly choti hain. Toh sahi tool sabse simple wala hai.

§2 aur §3 combine karo: field mein ek electron feel karta hai $a = \frac{F}{m} = \frac{- e E}{m} .$ Yeh parent ki derivation ki bilkul pehli line hai — ab isme har symbol defined hai.

4. Thermal motion vs. drift — do speeds

Yahan woh subtle part hai jis par poora topic hinges karta hai. Ek electron ek saath do motions karta hai, aur unhe confuse karna classic galti hai.

Dono panels dekho. Left: field nahi — har red arrow alag direction mein hai; unhe tip-to-tail add karo aur tum start par wapas aa jaate ho (net = zero). Right: field on — wohi wild zig-zag, lekin har segment thoda leftward jhukta hai (electron, negative, $E$ ke against drift karta hai); unhe add karo aur tum thoda left land karte ho. Woh chota leftover displacement per second hi $v_{d}$ hai.

5. Relaxation time $τ$ — drift choti kyun rehti hai

Agar field electrons ko accelerate karte rehti hai, toh woh forever tez hote hote kyon nahi jaate? Kyunki woh vibrating ions se collide karte rehte hain, aur har collision direction randomize kar deti hai — woh speed mita deti hai jo field ne abhi build ki thi.

Sawtooth padho: collisions ke beech field drift speed linearly build karti hai (slope $= a$ ); har collision par (vertical drops) gained drift zero ki taraf wapas knock ho jaati hai. Electron kabhi bhaag nahi sakta — $τ$ time ke liye gain karta hai, khota hai, phir gain karta hai. Toh average extra velocity roughly $a τ$ hoti hai: $v_{d} \approx a τ = \frac{e E}{m} τ .$

$τ$ temperature ke liye kyun matter karta hai: metal garam karo → ions zyada hilaate hain → electrons unhe jaldi hit karte hain → $τ$ shrink hoti hai → drift shrink hoti hai → resistance badhti hai. Yeh Resistivity and temperature dependence ka seed hai.

6. Number density $n$ — wire kitni crowded hai

Picture: §0 ke metal block mein dots ki density. Ek box mein zyada dots = bada $n$ .

Topic ko iski zaroorat kyun hai: current ek line cross karte electrons count karta hai. Yeh jaanne ke liye kitne cross karte hain, tumhe pata hona chahiye ki woh kitne dense hain. Ek slow-as-a-snail drift bhi bada current move karti hai jab $n$ itna bada ho — yeh "slow electrons, big current" ka resolution hai.

7. Area $A$ aur counting cylinder — $I = n e A v_{d}$ build karna

Counting trick (cylinder kyun?): $Δ t$ time mein, slice ke peeche $v_{d} Δ t$ distance ke andar har electron use just manage cross kar lega. Woh $v_{d} Δ t$ length aur face area $A$ ka ek cylinder hai. Uska volume $A v_{d} Δ t$ hai; usme $n \times (A v_{d} Δ t)$ electrons hain; woh charge $Δ Q = e n A v_{d} Δ t$ carry karte hain. $Δ t$ se divide karo: $I = \frac{Δ Q}{Δ t} = n e A v_{d} .$

Dono sides ko $A$ se divide karo toh current density $J = I / A = n e v_{d}$ milti hai — current per unit area. Dekhein Current density.

8. Conductivity $σ$ aur mobility $μ$ — do "kitni aasani se" numbers

Do final symbols sab kuch summarise karte hain yeh poochhkar "yeh metal kitni aasani se conduct karta hai?"

Do symbols kyun? $μ$ ek electron ki agility describe karta hai; $σ$ mein kitne electrons hain ( $n$ ) aur unka charge ( $e$ ) fold hota hai poori material describe karne ke liye. Bridge $σ = n e μ$ single-particle picture ko bulk material se connect karta hai — aur $J = σ E$ exactly Ohm's Law hai microscope pehne hue. Semiconductors mein, $n$ aur $μ$ dono dramatically change hote hain, isliye woh link itna useful hai.

Prerequisite map

Equipment checklist

Symbol

e

ka matlab aur iska value kya hai?

Elementary charge, ek electron ke charge ki magnitude,

e = 1.6 \times 1 0^{- 19}

Electric field

E

kya hai?

Kisi point par unit positive charge par force; "nudge" ki direction aur strength (unit V/m).

Newton ka second law us form mein batao jo hum use karte hain.

a = F / m

; ek electron ke liye

a = - e E / m

Thermal speed aur drift velocity mein farq?

Thermal speed fast ( ~~$1 0^{5}$ m/s) random hai jiska average zero hai; drift $v_{d}$ choti (~~

1 0^{- 4}

m/s) field-induced average lean hai.

Average thermal velocity zero kyun hoti hai?

Directions scrambled hain, toh arrows tip-to-tail cancel ho jaate hain.

Relaxation time

τ

kya hai?

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

v_{d} = e E τ / m

mein sirf ek factor of

τ

kyun hai?

Drift collisions ke beech ~0 se

a τ

tak badhti hai; crowd par average ek single mean free time

τ

deta hai.

Number density

n

kya count karta hai?

Free electrons per cubic metre (m

^{- 3}

); copper ~

8.5 \times 1 0^{28}

Counting cylinder ki length

v_{d} Δ t

kyun hai?

Sirf woh electrons jo slice ke peeche us distance mein hain woh

Δ t

time mein use cross kar sakte hain.

Master current relation batao.

I = n e A v_{d}

Mobility

μ

aur conductivity

σ

define karo.

μ = v_{d} / E = e τ / m

;

σ = n e^{2} τ / m = n e μ

Connections

Free electron model of metals — $n$ aur random motion kahan se aati hai.
Electric field inside conductors — $E$ ka source jo drift drive karta hai.
Current density — $J = n e v_{d}$ , §7 mein build kiya.
Ohm's Law — $J = σ E$ iska microscopic chehra hai.
Resistivity and temperature dependence — $τ$ heat ke saath kaise shrink hoti hai.
Semiconductors — jahan $n$ aur $μ$ dono strongly vary karte hain.