Drift velocity, mobility, conductivity: Question bank

1.8.15 · D5 · Physics › Electromagnetism › Drift velocity, mobility, conductivity

Definition Is page par use hone wale har symbol ki definition (trap mein phasne se pehle define karo)

$E$ = wire ke andar electric field ki strength (units V/m) — battery jo "push" per unit charge set up karti hai.
$E$ = wahi field ek arrow ki tarah likhi gayi, taaki hum uski direction ke baare mein baat kar sakein.
$e$ = electron ke charge ki magnitude, ek positive number ( $1.6 \times 1 0^{- 19}$ C). Electron khud $- e$ (negative) carry karta hai.
$m$ = ek charge carrier ki mass (ek electron, $9.1 \times 1 0^{- 31}$ kg).
$v_{d}$ = drift velocity: poore electron swarm ki tiny average velocity, jo bade random thermal jiggle ke upar sawaar rehti hai.
$τ$ = relaxation time: ek electron ki lattice se do collisions ke beech ka average time.
$n$ = number density: metal ke ek cubic metre mein kitne free electrons hain (units m⁻³).
$A$ = wire ki cross-sectional area; $I$ = current (charge per second, amperes).
$J = I / A$ = current density (current per unit area, A/m²).
$σ$ = conductivity (material kitni aasaani se conduct karta hai); $ρ = 1/ σ$ = resistivity (yeh kitna strongly resist karta hai).
$μ = v_{d} / E$ = mobility (per unit field kitna drift speed milta hai).

True or false — justify karo

Har claim ya toh ek subtle truth hai ya ek plausible-sounding lie. Reveal karne se pehle why batao.

Jab koi battery connected nahi hai, copper wire mein current exactly zero hai even though electrons ~

1 0^{5}

m/s par move karte hain.

True — fast motion random hai, isliye har ek electron jo ek direction mein ja raha hai uske opposite direction mein doosra ja raha hai; vector average zero hai, isliye koi net charge transport nahi.

Drift velocity woh speed hai jis par ek individual electron actually wire se guzarta hai.

False — ek individual electron thermal speed par zigzag karta hai (

\sim 1 0^{5}

m/s);

v_{d}

(

\sim 1 0^{- 4}

m/s) sirf poore swarm ka tiny average drift hai, kisi ek electron ki actual speed nahi.

Electrons applied field

E

ki same direction mein drift karte hain.

False — force

- e E

hai, isliye electrons

E

ke opposite drift karte hain; conventional current

I

E

ke along point karta hai kyunki current direction positive charge ke liye define ki gayi hai.

Jab tum torch switch on karte ho, bulb jalane wale electron ko pehle poori wire ki length travel karni padti hai.

False — wire pehle se hi electrons se bhari hui hai; field almost

c

par propagate hoti hai aur har ek electron ko ek saath nudge karti hai, isliye bulb ke paas wala electron turant move karna shuru kar deta hai.

Wire ki cross-sectional area

A

ko double karne par, same current rakhte hue, drift velocity half ho jaati hai.

True —

I = n e A v_{d}

se jab

I

n

e

fixed hon,

v_{d} \propto 1/ A

, isliye double area ka matlab hai half drift speed.

Metal ka temperature badhane par fixed applied field ke liye drift velocity badhti hai.

False — zyada

T

ka matlab hai ions zyada vibrate karte hain, isliye collisions jaldi aate hain,

τ

kam ho jaata hai, aur kyunki

v_{d} = e E τ / m

hai, drift girti hai (resistance badhti hai).

Mobility

μ

material ki property hai, field kitni strong hai uski nahi.

True —

μ = e τ / m

sirf charge, mass aur

τ

(fixed

T

par ek material property) par depend karta hai; isme

E

nahi hai, isliye field change karne par yeh fixed rehta hai.

Conductivity

σ = n e^{2} τ / m

badhti hai agar tum ek aisa metal use karo jisme zyada free electrons per cubic metre hon.

True —

σ \propto n

, isliye per unit volume zyada carriers ka matlab zyada conductivity hai, baaki sab barabar hone par (yahi ek reason hai silver/copper itna accha conduct karta hai).

σ = n e μ

relation mein charge

e

sirf ek baar appear karta hai, isliye ek sign error conductivity ko negative kar dega.

False —

σ

intrinsically positive hai; yahan

e

charge ki magnitude hai, aur do sign-flips (negative charge, field ke opposite drift) cancel ho ke ek positive

σ

dete hain.

Spot the error

Har line mein reasoning ya algebra mein ek galti hai. Use identify karo.

"Kyunki

v_{d} = e E τ / m

hai aur drift slow hai, field

E

bahut tiny hogi."

Error yeh hai: drift mainly isliye slow hai kyunki

τ

sirf femtoseconds hai (aur

e

tiny hai); mass

m

denominator mein hai, isliye chota

m

actually

v_{d}

badhata hai. Toh

v_{d}

ki smallness

τ

aur

e

se aati hai,

m

se nahi — aur ek modest

E

bhi ek crawl deta hai, isliye

v_{d}

ka chota hona imply nahi karta ki

E

chota hai.

I = n e A v_{d}

hai, isliye zero resistance wale superconductor mein infinite

v_{d}

hona chahiye."

Error yeh hai: zero resistance infinite drift force nahi karta; finite

n, A

ke saath finite current ke liye sirf finite

v_{d}

chahiye. Formula

v_{d} = I / (n e A)

deta hai, jo ek finite number hai.

"Kyunki

μ = v_{d} / E

hai, field badhate rahne par mobility bina limit ke badhti rehti hai."

Error yeh hai:

v_{d}

E

ke saath proportionally badhta hai (linear regime mein), isliye unka ratio

μ

constant rehta hai — yeh field ke saath nahi badhta.

"Resistivity

ρ = m / (n e^{2} τ)

hai, isliye

τ

badhane ke liye metal ko cool karne par

ρ

bada ho jaata hai."

Error yeh hai:

ρ \propto 1/ τ

hai, isliye bada

τ

ρ

ko chota karta hai; cooling vibration kam karti hai,

τ

badhata hai, aur resistivity kam hoti hai.

"Drift velocity

\propto T

hai, bilkul thermal (rms) speed ki tarah."

Error yeh hai: thermal speed

T

ki tarah scale karti hai, lekin drift

τ

ki tarah scale karta hai, jo

T

ke saath decrease karta hai; dono temperature dependences opposite direction mein jaati hain.

"Current density

J = n e v_{d}

mein velocity aur charge ke units hain, isliye yeh m/s mein measure hota hai."

Error yeh hai:

J = n e v_{d}

ke units

(m^{- 3}) (C) (m s^{- 1}) = C s^{- 1} m^{- 2} = A m^{- 2}

hain, jo ek current per area hai, speed nahi.

"Kyunki electrons continuously collide karte rehte hain, woh collisions ke beech forever speed up karte rehte hain jab tak relativistic na ho jaayein."

Error yeh hai: har collision direction randomise kar deta hai aur gained drift wipe kar deta hai; electron sirf ek short

τ

ke liye accelerate karta hai phir reset ho jaata hai, isliye yeh kabhi high speed tak nahi jaata.

Why questions

Mechanism se answer karo, formula se nahi.

v_{d} = e E τ / m

mein exactly ek factor of

τ

(na

τ^{2}

τ /2

) kyun appear karta hai?

Ek collision ke turant baad drift ~0 hai aur agli se pehle yeh

a τ

hai; population ke mean free time par us linear ramp ka average karne par ek factor of

τ

milta hai (saw-tooth figure dekho) — acceleration

a = e E / m

times mean gaining time

τ

Bulb instantly kyun jalta hai even though har electron

1 0^{- 4}

m/s par crawl karta hai?

Electric field wire mein almost light speed par spread hoti hai aur har ek electron ko simultaneously push karti hai, bilkul us paani ki tarah jo ek bhare hue pipe mein pehle se bhara hota hai — jab near end push karo, far end se turant nikalna shuru ho jaata hai.

Metal mein temperature badhne par resistance kyun badhti hai?

Garam hone se lattice ions zyada vibrate karte hain, isliye electron kuch jaldi cheez se takraata hai;

τ

shrink hota hai,

σ = n e^{2} τ / m

girti hai aur

ρ

badhti hai. Resistivity and temperature dependence dekho.

Macroscopic law

V = I R

aslmein microscopic law

J = σ E

kyun hai chhupe hue?

J = σ E

kehta hai current density har point par field ke proportional hai; ek uniform wire ki length aur area par integrate karne par

E

V

ban jaata hai aur

J

I

ban jaata hai, jo

V = I R

deta hai. Ohm's Law dekho.

Hum

μ

ko alag kyun define karte hain jab

σ

pehle se exist karta hai?

μ = v_{d} / E

ek carrier field ke liye kitna responsive hai yeh isolate karta hai, carriers ki sankhya se independent;

σ = n e μ

phir "kitne" (

n

) aur "kitne mobile" (

μ

) ko cleanly alag karta hai — Semiconductors mein zaroori.

Electron ke charge ka sign conductivity ko negative kyun nahi banata?

Electron ka negative charge use

E

ke against backward drift karata hai, lekin negative charge ki woh backward motion ek forward conventional current hai; dono minus signs cancel ho jaate hain, aur

σ > 0

rehta hai.

Current-carrying wire ke andar electric field nonzero kyun hai, unlike electrostatic equilibrium mein?

Steady current ko har collision par lost drift ko replace karte rehne ke liye ek continuous push chahiye; woh push battery dwara maintain kiya gaya nonzero

E

hai. Electric field inside conductors dekho.

Number density

n

badhane par current kyun badhti hai even if har electron ka

v_{d}

unchanged ho?

Current charge crossing per second count karta hai; zyada electrons per cubic metre ka matlab hai same area cross karne wale zyada carriers, isliye

I = n e A v_{d}

fixed drift par bhi badhti hai.

Edge cases

Ideas ko unki limits aur degenerate inputs tak push karo.

Jis instant field switch off ho jaati hai (

E \to 0

) us waqt

v_{d}

ka kya hota hai?

Driving force disappear ho jaati hai, isliye koi fresh drift add nahi hoti; kuch collision times ke andar ordered drift zero tak decay ho jaati hai aur sirf random thermal motion bachti hai — current khatam ho jaati hai.

v_{d} = e E τ / m

kya predict karta hai jab

τ \to 0

(ek bahut "dirty" ya hot metal)?

v_{d} \to 0

aur

σ \to 0

: agar collisions instantaneous hain, toh unke beech koi drift build nahi ho sakti, isliye material barely conduct karta hai.

Agar

n = 0

(koi free carriers nahi, jaise ek perfect insulator), toh

I = n e A v_{d}

kya deta hai?

Kisi bhi field ke liye

I = 0

— free electrons ke bina move karne ke liye koi conduction current nahi hogi chahe

E

kitni bhi badi ho.

Fixed current ke liye, thinner wire (chota

A

) electrons ko faster drift karaata hai ya slower?

Faster —

v_{d} = I / (n e A)

se chota

A

v_{d}

badhata hai; same charge per second ko kam area mein se squeeze hona padta hai, isliye yeh zyada tezi se move karta hai.

Zero applied field ki limit mein drift velocity kya hai, aur yeh electrons ke abhi bhi move karne ke saath consistent kyun hai?

v_{d} = 0

kyunki ordered average zero hai; electrons abhi bhi full thermal speed par move karte hain, lekin woh random velocities average mein cancel ho jaati hain. Free electron model of metals dekho.

Semiconductor mein dono electrons aur holes drift karte hain — kya holes se current, electron current se subtract hoti hai?

Nahi, woh add hoti hain: holes (positive)

E

ke along drift karte hain aur electrons uske against, lekin dono ek hi conventional-current direction mein charge transport karte hain, isliye unke contributions sum ho jaate hain. Semiconductors dekho.

Kya koi real field strength hai jis par

v_{d} = e E τ / m

eventually break down ho jaati hai?

Haan — bahut high fields par

τ

khud drift energy par depend karne lagta hai, isliye drift

E

ke saath linearly nahi badhti aur simple constant-

μ

picture fail ho jaati hai (velocity saturation).

Connections

Ohm's Law — in traps ka macroscopic chehra.
Current density — jahan $J = n e v_{d}$ build hoti hai.
Resistivity and temperature dependence — $τ (T)$ traps yahan rehte hain.
Free electron model of metals — random motion aur $n$ ka source.
Electric field inside conductors — current ke dauran $E \neq = 0$ kyun.
Semiconductors — in questions ke two-carrier extensions.