WHAT we want: explain whyV=IR (the macroscopic law) comes out of what individual electrons do.
WHY it's not obvious: if a constant force acted on a free electron, it would accelerate forever (a=F/m), and the current would grow without limit. But experiments show a steady, constant current for a constant voltage. Something must be stopping the runaway. That "something" is collisions.
Step 1 — Force on one electron.
An electron (charge −e) in field E feels
F=−eE,a=mF=−meE.Why this step? Newton's 2nd law — the only honest starting point for "what does the electron do."
Step 2 — Collisions reset the velocity.
On average an electron travels for a time τ (the relaxation time, the mean time between collisions) before a collision randomizes its velocity. Right after a collision its drift contribution is, on average, zero (collisions scatter it in random directions).
Why this step? The collision is the "brake." After each bump the electron forgets the velocity it had built up — so velocity can't accumulate forever.
Step 3 — Average the velocity gained between collisions.
Starting from zero drift, after time t the electron has v=at. Averaging over the time-since-last-collision (which averages to τ):
vd=aτ=−meτEWhy this step? This is the steady-state balance: gain from the field over one free flight τ, lost at each collision. The result is constant in time — that's why current is steady.
Step 4 — From drift to current density.
Let n = number of free electrons per unit volume. In time dt all electrons within distance vddt of a cross-section A cross it. The charge crossing is dQ=(n)(Avddt)(e), so
I=dtdQ=neAvd,J≡AI=nevd.Why this step? Current = charge per second; count how many carriers sweep through.
Step 5 — Combine. Put vd=meτE (magnitudes) into J=nevd:
J=mne2τE=σE
ρ=m/(ne2τ). Heat the metal → atoms vibrate harder → electrons collide more often → τdrops → ρrises. (In semiconductors the opposite often wins: n grows fast with T, so ρ falls.)
Imagine pushing a shopping cart down a crowded hallway. You push steadily, but you keep bumping into people. You never speed up forever — you just keep moving at a slow, steady walk. The harder you push (bigger voltage), the faster you walk (more current). A narrower hallway with more people in the way is harder to get through — that's higher resistance. Heating things up makes everyone wiggle more and bump you more, so it's even harder to push through.
Dekho, sawaal yeh hai: agar metal ke andar electron par constant electric field se constant force lag raha hai, toh Newton ke hisaab se woh forever accelerate karega aur current infinite ho jaayegi. Par real life mein constant voltage se steady current milti hai. Iska matlab koi "brake" hai — aur woh brake hai collisions. Electron field se thoda speed pakadta hai, phir lattice ke vibrating atoms se takra jaata hai aur uski speed reset ho jaati hai. Average time between collisions ko relaxation timeτ kehte hain.
Is balance se aata hai drift velocity: vd=eEτ/m. Yeh woh chhoti si average speed hai jisse poora electron-sea field ke direction mein slowly khisakta hai. Phir current density J=nevd likh ke combine karo, toh milta hai J=σE jahan σ=ne2τ/m. Yahi microscopic Ohm's law hai — aur resistivity ρ=1/σ=m/(ne2τ). Wire ke liye R=ρL/A.
Important baat: drift speed bahut hi slow hota hai (~0.07 mm/s copper mein), lekin bulb instantly jalta hai kyunki field/signal almost light-speed se travel karta hai aur saare electrons ek saath push hote hain — bilkul nal kholne jaisa, paani turant nikalta hai. Temperature badhao toh atoms zyada hilte hain, collisions badhte hain, τ ghatta hai, isliye metal ka ρ badhta hai. Yaad rakho: ρ material property hai, R mein geometry (L, A) bhi add hoti hai. Ohm's law fundamental nahi hai — diode jaise non-ohmic cheezein bhi hoti hain.
