Ohm's law — microscopic origin, resistivity

1.8.16 · Physics › Electromagnetism

Current flow kyun karta hai? (first principles)

KYA chahiye: explain karna kyun $V = I R$ (macroscopic law) aata hai jo individual electrons karte hain usse.

YEH obvious kyun nahi hai: agar ek constant force ek free electron par act karti, toh woh forever accelerate karta ( $a = F / m$ ), aur current bina kisi limit ke badhti rehti. Lekin experiments ek constant voltage ke liye steady, constant current dikhate hain. Kuch toh is runaway ko rok raha hai. Woh "kuch" hai collisions.

Field steady drift kaise produce karta hai (derivation)

Step 1 — Ek electron par force. Ek electron (charge $- e$ ) field $E$ mein feel karta hai $F = - e E, a = \frac{F}{m} = - \frac{e E}{m} .$ Yeh step kyun? Newton ka 2nd law — "electron kya karta hai" ke liye sabse honest starting point.

Step 2 — Collisions velocity reset kar dete hain. Average mein ek electron $τ$ time tak travel karta hai (yeh relaxation time hai, collisions ke beech ka mean time) phir ek collision uski velocity ko randomize kar deta hai. Collision ke theek baad uska drift contribution average mein zero hota hai (collisions use random directions mein scatter kar dete hain).

Yeh step kyun? Collision hi "brake" hai. Har bump ke baad electron woh velocity bhool jaata hai jo usne build up ki thi — isliye velocity forever accumulate nahi ho sakti.

Step 3 — Collisions ke beech gain hui velocity ko average karo. Zero drift se shuru karke, time $t$ ke baad electron ke paas $v = a t$ hota hai. Last collision ke baad ke time ko average karo (jo average hokar $τ$ ho jaata hai): $v_{d} = a τ = - \frac{e τ}{m} E$ Yeh step kyun? Yeh steady-state balance hai: ek free flight $τ$ mein field se gain, aur har collision mein loss. Result time mein constant hai — isliye current steady hoti hai.

Step 4 — Drift se current density tak. Maano $n$ = free electrons ki sankhya per unit volume. Time $d t$ mein cross-section $A$ ke $v_{d} d t$ distance ke andar ke saare electrons isse cross karte hain. Jo charge cross karta hai woh hai $d Q = (n) (A v_{d} d t) (e)$ , toh $I = \frac{d Q}{d t} = n e A v_{d}, J \equiv \frac{I}{A} = n e v_{d} .$ Yeh step kyun? Current = charge per second; count karo kitne carriers sweep through karte hain.

Step 5 — Combine karo. $v_{d} = \frac{e τ}{m} E$ (magnitudes) ko $J = n e v_{d}$ mein daalo: $J = \frac{n e ^{2} τ}{m} E = σ E$

Macroscopic $V = I R$ kaise nikalta hai

Ek wire lo, length $L$ , area $A$ , uniform field. Toh $E = V / L$ aur $I = J A = σ E A$ : $I = σ \frac{V}{L} A \Rightarrow V = R \frac{L}{σ A} I = \frac{ρ L}{A} I .$

Figure — Ohm's law — microscopic origin, resistivity

Metals mein temperature badhne se resistance kyun badhta hai

$ρ = m / (n e^{2} τ)$ . Metal ko heat karo → atoms zyada vibrate karte hain → electrons zyada baar collide karte hain → $τ$ kam hota hai → $ρ$ badhta hai. (Semiconductors mein aksar ulta hota hai: $n$ $T$ ke saath tezi se badhta hai, toh $ρ$ girta hai.)

Common mistakes (Steel-man karo unhe)

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

Socho ek shopping cart ko ek bheed bhari hallway mein push kar rahe ho. Tum steadily push karte ho, lekin tum logon se takraate rehte ho. Tum kabhi forever speed up nahi karte — tum bas ek slow, steady walk par chalte rehte ho. Jitna zyada push karo (bada voltage), utna tezi chalo (zyada current). Zyada logon se bhari ek sankri hallway mein ghusna mushkil hota hai — yahi zyada resistance hai. Cheezein heat karne se sab zyada wiggle karte hain aur zyada bump karte hain, toh push karna aur bhi mushkil ho jaata hai.

Active Recall

Wire mein electrons forever accelerate karne se kya rokta hai?

Vibrating lattice ions ke saath collisions, jo drift velocity ko roughly har relaxation time

τ

mein reset karte hain.

Drift velocity define karo.

Electron sea ki

E

ke along chhoti average velocity, random thermal motion ke upar superpose hui (jo koi net current carry nahi karti).

v_{d}

E, τ

ke terms mein derive karo.

a = e E / m

, free time

τ

par average karke:

v_{d} = e E τ / m

Current density ko drift velocity se relate karo.

J = n e v_{d}

Ohm's law ka microscopic form?

J = σ E

jahan

σ = n e^{2} τ / m

Yeh "Ohm's law" kyun hai?

Kyunki

σ

E

se independent hai, toh

J \propto E

(aur isliye

I \propto V

) — ek linear relation.

Resistivity ka formula?

ρ = m / (n e^{2} τ)

Geometry se resistance?

R = ρ L / A

Temperature badhne par metal ka resistance kyun badhta hai?

Hotter lattice zyada vibrate karta hai → zyada collisions →

τ

girta hai →

ρ = m / (n e^{2} τ)

badhta hai.

Drift speed fast hai ya slow?

Bahut slow (~

1 0^{- 4}

m/s), lekin field/signal lagbhag light speed par propagate hoti hai, toh current turant shuru ho jaati hai.

ρ

aur

R

mein kya fark hai?

ρ

ek material property hai;

R = ρ L / A

mein geometry bhi shamil hai.

Ek non-ohmic device ka naam batao.

Diode (aur bhi: high T par filament lamp, gas/plasma).

Connections

Electric current and current density
Electric field inside a conductor
Drude model of conduction
Resistors in series and parallel
Temperature dependence of resistance
Joule heating — power dissipation
Semiconductors vs metals
Drift and diffusion of carriers