Ohm's law — microscopic origin, resistivity: Question bank

1.8.16 · D5 · Physics › Electromagnetism › Ohm's law — microscopic origin, resistivity

Shuru karne se pehle, teen words jo tumhare paas hone chahiye (neeche har question inhi par tikaa hai):

True or false — justify karo

Ohm's law

V = I R

ek fundamental law of nature hai jaise Newton's laws.

False. Ye ek approximation hai jo sirf tab hold karta hai jab

σ

E

se independent ho; diodes, plasmas aur hot filaments non-ohmic hain aur ise tod dete hain.

Agar ek ohmic wire ke across voltage double karo, to drift velocity bhi double ho jaati hai.

True.

v_{d} = e E τ / m

aur

E = V / L

, isliye

v_{d} \propto V

— voltage se drift tak ka poora chain linear hai (jab tak

τ

fixed rahe).

Ek copper sample ki resistivity

ρ

is baat par depend karti hai ki wire kitni lambi aur kitni moti hai.

False.

ρ = m / (n e^{2} τ)

sirf material par depend karta hai (

n

τ

T

). Length aur area resistance mein aate hain

R = ρ L / A

ke zariye, kabhi

ρ

mein nahi.

Electrons ki random thermal motion current mein contribute karti hai.

False. Thermal velocities har direction mein point karti hain aur net flow zero average deti hain; sirf chhota sa drift

v_{d}

hi current carry karta hai.

Steady state mein, wire ke electrons constant acceleration se move karte hain.

False. Woh constant average velocity

v_{d}

se move karte hain. Collisions ke beech woh accelerate karte hain, lekin har collision gain ko reset kar deta hai, isliye drift steady hai, acceleration nahi.

Metal wire ko cool karna uski resistance kam karta hai.

True. Thanda lattice kam vibrate karta hai → kam collisions →

τ

badhta hai →

ρ = m / (n e^{2} τ)

girta hai, isliye

R

bhi girta hai.

Semiconductors mein, heating hamesha metals ki tarah resistance badhata hai.

False. Semiconductors mein heating bahut zyada carriers free karta hai, isliye

n

tezi se badhta hai aur

ρ

girta hai — bilkul ulta trend (dekho Semiconductors vs metals).

Lamp isliye instantly jalta hai kyunki electrons switch se bulb tak bahut tezi se travel karte hain.

False. Drift speed ~

1 0^{- 4}

m/s hai (ghonghey jitni slow). Woh field/signal jo har electron ko drift karana shuru karta hai, light speed ke paas propagate karta hai, isliye sab electrons ek saath start karte hain.

Conductivity

σ

badhti hai agar tum zyada strong electric field apply karo.

False.

σ = n e^{2} τ / m

mein koi

E

nahi hai — yahi

E

-independence bilkul wahi cheez hai jo material ko ohmic banati hai.

Error dhundo

"Electron par constant force act karti hai, isliye

a = F / m

se uski speed bina limit ke badhti jaayegi, isliye current hamesha ke liye badhti rahegi."

Electron free nahi hai — har

τ

par collisions accumulated velocity erase kar dete hain. Field-gain aur collision-loss ke beech balance ek steady

v_{d}

deta hai, isliye current constant hai, runaway nahi.

"Kyunki

J = n e v_{d}

aur

v_{d}

tiny hai, isliye household wire mein current bhi tiny honi chahiye."

Carrier density

n \sim 1 0^{28}

–

1 0^{29} m^{- 3}

bahut enormous hai, aur woh

v_{d}

ko multiply karta hai. Bahut saari slow carriers phir bhi amperes deliver karti hain.

ρ = m / (n e^{2} τ)

, isliye resistivity electron ke charge ke inversely proportional hai."

Charge squared form mein appear karta hai (

e^{2}

), pehli power mein nahi — isliye

ρ \propto 1/ e^{2}

. Square isliye aata hai kyunki

e

ek baar force mein aata hai aur ek baar current density mein.

"Temperature badhne par Resistance isliye badhti hai kyunki garam hone par electrons khud bhaari ho jaate hain."

Mass

m

fixed hai.

τ

shrink karta hai: garam ions zyada vibrate karte hain, collisions zyada baar hoti hain, isliye

τ

girta hai aur

ρ

badhta hai.

"Series mein do identical wires ki resistance ek wire ki adhi hoti hai, kyunki charge ke paas do paths hain."

Series wires ek hi path share karte hain, isliye lengths add hoti hain — resistance double hoti hai. Do paths (adha karna) parallel arrangement hai (dekho Resistors in series and parallel).

v_{d} = - \frac{e τ}{m} E

mein minus sign ka matlab hai ki current

E

ke opposite flow karta hai."

Minus sign ka matlab hai ki electrons

E

ke opposite drift karte hain (woh negative hain). Conventional current

J = n e v_{d}

(jahan

e > 0

) phir bhi $E$ ke along point karta hai — current mein dono minus signs cancel ho jaate hain.

Why questions

Microscopic terms mein, lambi wire ki resistance zyada kyun hoti hai?

Lambi wire ka matlab hai ki ek electron ko apne raaste mein zyada lattice cross karna padta hai, proportionally zyada collision-causing scattering "in series" encounter karta hua —

R = ρ L / A

L

ke saath scale karta hai.

Drift velocity har ek free flight ke dauran climb karne ki bajay constant kyun rehti hai?

Ek single free flight ke dauran woh climb karti hai, lekin hum bahut saari flights ka average lete hain: har collision electron ko zero drift par wapas drop kar deta hai, isliye ensemble average fixed

v_{d} = e E τ / m

hota hai.

Crucial fact "

σ

E

se independent hai" kyun hai, sirf "

J

aur

E

dono appear karte hain" kyun nahi?

Agar

σ

E

par depend karta, to

J = σ (E) E

nonlinear hota aur

V

I

ke proportional nahi hota.

σ

ki constancy hi relation ko linear banati hai — yaani Ohm's law.

Hum keh sakte hain ki collisions velocity ko "randomize" karte hain "stop" karne ki jagah, kyun?

Ek collision electron ko ek random direction mein scatter karta hai, isliye uska drift contribution baad mein zero average ho jaata hai — lekin uski fast thermal speed unchanged rehti hai. Woh field se gain ki velocity bhool jaata hai, apni saari energy nahi.

Current-carrying wire ke andar field non-zero kyun rehti hai jabki woh conductor hai?

Ek static conductor ka internal field zero hota hai, lekin ek current-carrying conductor ko

E = J / σ

sustain karna padta hai taaki drifting electrons ko collisions ke against push karta rahe (dekho Electric field inside a conductor).

Edge cases

Jab area

A \to \infty

(infinitely moti wire) ho to

R

ka kya hoga?

R = ρ L / A \to 0

: infinitely many parallel paths, isliye koi resistance nahi — wire perfect connection ki tarah behave karta hai.

Field switch on karne se bilkul pehle (

E = 0

) drift velocity kya hogi?

v_{d} = e E τ / m = 0

: koi field nahi to koi net drift nahi, lekin electrons phir bhi thermally zero average ke saath tezi se ghoomte rehte hain.

Agar relaxation time

τ \to 0

(infinitely frequent collisions), to

ρ

ka kya hoga?

ρ = m / (n e^{2} τ) \to \infty

: constant scattering kabhi koi drift build up nahi hone deti, isliye material current ke liye perfect insulator ban jaata hai.

Agar free-carrier density

n \to 0

ho, to kya material conduct karega?

Nahi:

σ = n e^{2} τ / m \to 0

, isliye

J = σ E \to 0

. Koi carriers nahi to drift karne ke liye kuch nahi, field chahe jo ho.

Fixed voltage par, ek real filament jab glow kare tab current ko kya limit karta hai?

Temperature badhna

τ

ko short karta hai,

ρ

aur isliye

R

badhata hai; isliye current garam hone par girta hai — yahi

E

-dependent behaviour bilkul wahi reason hai ki lamp filament non-ohmic hai.

Superconductor ke liye (idealized

τ \to \infty

, no scattering),

ρ

kya hoga?

ρ = m / (n e^{2} τ) \to 0

: koi collisions nahi to drift kabhi reset nahi hota, isliye current zero resistance ke saath flow karta hai — lekin ye ordinary Drude picture ke bahar hai.

Recall One-line survival kit

Constant force but constant velocity (not acceleration) kyunki collisions har $τ$ par brake karte hain ::: yes $ρ$ = sirf material; $R = ρ L / A$ geometry add karta hai ::: yes Ohm's law = " $σ$ mein koi $E$ nahi" = linearity ::: yes Metals: heat up → $τ$ down → $ρ$ up. Semiconductors: heat up → $n$ up → $ρ$ down ::: yes

Connections

Parent: Ohm's law microscopic origin
Drude model of conduction
Electric current and current density
Electric field inside a conductor
Temperature dependence of resistance
Semiconductors vs metals
Resistors in series and parallel
Drift and diffusion of carriers
Joule heating — power dissipation