Visual walkthrough — Ohm's law — microscopic origin, resistivity

1.8.16 · D2 · Physics › Electromagnetism › Ohm's law — microscopic origin, resistivity

Shuru karne ke liye hume sirf teen seedhi ideas chahiye, toh pehle inhe pictures mein naam dete hain.

Step 1 — Ek electron ko field mein daalo

KYA. Ek single electron ko us region mein daalo jahan field $\vec{E}$ dayi taraf point kar rahi hai. Uspe ek force lagti hai.

YEH STEP KYUN. Physics mein jo bhi cheez move karti hai woh force se shuru hoti hai. Hum "current" (ek bheed) ki baat tab tak nahi kar sakte jab tak hum yeh na jaanein ki ek electron kya karta hai. Toh hum sabse simple aur honest sawaal se shuru karte hain: yahan kaun si force kaam kar rahi hai?

PICTURE. Figure dekho. Orange arrows field $\vec{E}$ hain jo dayi taraf point kar rahe hain. Electron (teal dot) ko force $\vec{F}$ bayi taraf feel hoti hai — arrows ke opposite — kyunki uska charge negative hai.

$\vec{F}=\underset{\text{charge (negative)}}{\underbrace{-e}}\underset{\text{push}}{\underbrace{\vec{E}}}$

• $-e$ : electron ka charge. Minus sign hi wajah hai ki force $\vec{E}$ ke relative backward point karti hai.
• $\vec{E}$ : field arrows. Bada $E$ → badi force. Simple proportionality.

Yeh sirf "force = charge × field" ka rule hai, wohi rule jo Electric field inside a conductor se aata hai.

Step 2 — Force acceleration ban jaati hai

KYA. Us force ko Newton's second law use karke motion mein change mein badlo.

YEH STEP KYUN. Force akele kuch nahi batati speed ke baare mein jab tak hum yeh na jaanein ki woh kis mass par kaam kar rahi hai. Newton's law $\vec{F}=m\vec{a}$ "push" se "speed pakadne" tak ka bridge hai.

PICTURE. Wohi electron, ab ek green acceleration arrow ke saath. Yeh force ke same direction mein point karta hai (bayi taraf) aur iski length force divided by electron ka mass $m$ hai.

$\vec{a}=\frac{\vec{F}}{m}=\underset{\text{kitna strongly respond karta hai}}{\underbrace{-\frac{e}{m}}}\vec{E}$

• $m$ : electron ka mass, $9.1\times 10^{-31}$ kg. Bahut chhota mass hai, toh acceleration bahut badi hoti hai.
• $\vec{a}$ : velocity kitni tezi se badhti hai. Agar kuch rokta nahi, toh electron forever tezi se tezi aur tezi hota jaata.

Step 3 — Collision ka brake

KYA. Crashes ke beech electron speed up karta hai; phir woh ek lattice atom se takraata hai aur uski velocity randomly scramble ho jaati hai — average par, drift wapas zero ho jaati hai.

YEH STEP KYUN. Yeh poori story ka dil hai. Field baar baar velocity build karne ki koshish karti hai; crashes baar baar use mita dete hain. Is brake ke bina koi steady state nahi aur Ohm's law bhi nahi.

PICTURE. Zig-zag path. Har seedha segment ek "free flight" hai jahan field path ko thoda mod deti hai (bayi taraf curve karti hui). Har red dot ek collision hai jo direction reset kar deti hai. Ek free flight ka average time $\tau$ kehlata hai — relaxation time.

Step 4 — Ek flight mein gain hua drift average karo

KYA. Ek collision ke turant baad zero drift se shuru karke, electron agle crash se pehle $\tau$ time tak accelerate karta hai. Uski typical extra velocity $\vec{a}$ times $\tau$ hoti hai.

YEH STEP KYUN. Hum crowd ki steady average velocity chahte hain, na ki pal pal badlti instant speed. "Ek flight mein gain, phir reset" ka average karne se ek clean number milta hai jo time ke saath nahi badlta — steady state ki definition yahi hai.

PICTURE. Velocity-vs-time sawtooth: velocity har flight mein field ke saath chadhti hai, har crash par zero ho jaati hai. Dashed horizontal line average hai — woh hai ${\vec{v}}_d$.

$\boxed{{\vec{v}}_d=\vec{a}\tau =\underset{\text{drift per unit field}}{\underbrace{-\frac{e\tau }{m}}}\vec{E}}$

• ${\vec{v}}_d$ : drift velocity — pure electron sea ka chhota sa steady average (dekho Drift and diffusion of carriers).
• $\tau$ : crashes ke beech zyada time → speed up karne ka zyada time → bada drift.
• Note karo: ${\vec{v}}_d$ time mein constant hai. Isliye current steady hoti hai, runaway nahi.

Yeh drift tezi random thermal jiggling ke upar sawaar hoti hai, lekin jiggling average par zero hoti hai aur koi net current carry nahi karti — sirf ${\vec{v}}_d$ karta hai. Yeh Drude model of conduction ki poori logic hai.

Step 5 — Charge count karo jo cross karta hai (drift → current)

KYA. "Har electron $v_d$ par drift karta hai" ko "har second ek cross-section se kitna charge guzarta hai" mein badlo.

YEH STEP KYUN. Current ek surface se charge per second hai. Hume carriers count karne hain, ek ko track nahi karna. Toh hum wire ko slice karte hain aur sabko count karte hain jo sweep through karte hain.

PICTURE. Wire ka ek cylinder, cross-section area $A$. Time $dt$ mein face se $v_{d}dt$ distance ke andar har electron use cross kar lega. Yeh volume $A{v}_{d}dt$ ka ek slab hai.

$I=\frac{dQ}{dt},dQ=\underset{\text{electrons per }{\text{m}}^3}{\underbrace{n}}\underset{\text{slab volume}}{\underbrace{(A{v}_{d}dt)}}\underset{\text{charge each}}{\underbrace{e}}$

Charge ko time aur area se divide karo:

$\boxed{J\equiv \frac{I}{A}=ne{v}_d}$

• $n$ : ek cubic metre mein kitne free electrons hain (copper ke liye, $8.5\times 10^{28}$).
• $J$ : current density — current per unit area, current ka local, geometry-free version (dekho Electric current and current density).

Step 6 — Substitute karo aur Ohm's law reveal karo

KYA. Step 4 ka drift Step 5 ki current density mein daalo.

YEH STEP KYUN. Steps 4 aur 5 do alag facts hain: ek batata hai electrons kitni tezi se drift karte hain, doosra batata hai drift current kaise banata hai. Inhe combine karne se $v_d$ hat jaata hai aur current density sirf field ke terms mein aa jaati hai.

PICTURE. Do boxed results puzzle pieces ki tarah snap ho rahe hain, bahar aata hai ek seedhi line: $J$ vs $E$.

Magnitudes use karte hue, $v_d=\frac{e\tau }{m}E$ ko $J=ne{v}_d$ mein daalo:

$J=ne\cdot \frac{e\tau }{m}E=\underset{\text{ise }\sigma \text{ kehte hain}}{\underbrace{\frac{n{e}^2\tau }{m}}}E$

$\boxed{J=\sigma E,\sigma =\frac{n{e}^2\tau }{m}}$

Step 7 — Zoom out: $J=\sigma E$ se $V=IR$ tak

KYA. Local law ko length $L$ aur area $A$ ki poori wire par wrap karo.

YEH STEP KYUN. $J=\sigma E$ har point par ek statement hai. Ise circuit mein use karne ke liye hume whole-wire quantities voltage $V$ aur total current $I$ chahiye.

PICTURE. Wire ek seedhi pipe ki tarah. Iske length $L$ mein voltage $V$ ek uniform field $E=V/L$ banata hai; total current $I=JA$ hai.

$E=\frac{V}{L},I=JA=\sigma EA=\sigma \frac{V}{L}A$

$V$ ke liye rearrange karo:

$V=\underset{R}{\underbrace{\frac{L}{\sigma A}}}I=\frac{\rho L}{A}I\Rightarrow \boxed{V=IR,R=\rho \frac{L}{A}}$

• $L$ : lambi wire → series mein zyada collisions → bada $R$.
• $A$ : moti wire → zyada parallel lanes → chhota $R$.

Yeh geometry factor Resistors in series and parallel mein stack up hota hai.

Step 8 — Edge aur degenerate cases

Hume reader ko kisi aisi scenario mein kabhi nahi chodna chahiye jo humne draw nahi ki.

• $E=0$ (field off). Tab $\vec{a}=0$, toh ${\vec{v}}_d=0$, toh $J=0$. Koi push nahi, koi drift nahi, koi current nahi — sirf random thermal jiggle jo kuch average nahi karti. Figure ek pure random walk dikhata hai zero net displacement ke saath.
• $\tau \to 0$ (non-stop crashes). Drift turant mita di jaati hai: $v_d\to 0$, toh $\sigma =n{e}^2\tau /m\to 0$ aur $\rho \to \infty$. Ek perfect insulator-jaisi limit — kabhi itni free flight nahi milti ki speed build ho sake.
• $\tau \to \infty$ (koi crash nahi). Tab $\sigma \to \infty$, $\rho \to 0$: ek perfect conductor. Lekin saath hi $v_d=a\tau \to \infty$ bhi — Step 2 ka runaway wapas aa jaata hai. Yeh confirm karta hai ki collisions hi hain jo runaway ko tame karti hain.
• Garam karo. Zyada lattice vibration → zyada crashes → $\tau$ girta hai → $\rho =m/(n{e}^2\tau )$ badhta hai. Yahi hai Temperature dependence of resistance. (Semiconductors mein rising $T$ itna $n$ boost karta hai ki $\rho$ iske bajaye gir sakti hai.)
• Non-ohmic materials. Agar $\tau$ ya $n$ khud $E$ ke saath badlein (filaments, diodes, plasmas), toh $\sigma$ constant nahi rehta aur seedhi line mur jaati hai. Ohm's law kabhi sirf constant-$\sigma$ special case hi tha.

Ek-picture summary

Active Recall

Connections

• Ohm's law — microscopic origin, resistivity (index 1.8.16)
• Electric current and current density
• Electric field inside a conductor
• Drude model of conduction
• Drift and diffusion of carriers
• Resistors in series and parallel
• Temperature dependence of resistance
• Semiconductors vs metals
• Joule heating — power dissipation