Derivation of Ohm's Law with Current Drift Velocity Relation

Inside a metal wire, free electrons move randomly at high speed, constantly colliding with fixed metal ions — like people bouncing around in a crowded hallway, going nowhere on average. When a battery is connected, it creates an electric field that gently pushes every electron in one direction between collisions. This small drift, layered on top of the random motion, is what we measure as electric current.

Consider a wire of length l, cross-sectional area A, with potential difference V applied across its ends.

Step 1 - Force and acceleration on an electron.
The field inside the wire is E = V/l. Each electron (charge eee, mass mmm) accelerates as:

Step 2 - Drift velocity from acceleration.
Between collisions, the electron accelerates for an average time τ (relaxation time — the average time between successive collisions):

Step 3 - Drift velocity from current.
With n = free electrons per unit volume:

Step 4 - Combine (2) and (3).

Step 5 - Substitute (1) into (4) and solve for V.

Since m and e are universal constants, and n, l, A, τ stay fixed for a conductor at constant temperature and pressure, equation (5) gives:

This is Ohm's law — the potential difference across a conductor is directly proportional to the current through it, provided physical conditions stay unchanged.

Comparing equation (5) with V = RI:

Comparing (6) with R = ρ\[\frac {l}{A}\]​:

Relaxation time depends on mean free path λ and rms speed v_rms​:

Free electrons behave like gas molecules, so:

Cause-and-effect chain: Temperature rises → v_rms increases (9) → τ decreases (8) → ρ increases (7) → resistance R increases (6). This is why a bulb filament heats up, and its resistance climbs as current flows.

If R₀​ is resistance at 0°C and R_t​ at t°C:

For most metals, α ≈ 1/273 per °C, giving

R_t​ = \[R_0\left(\frac{273+t}{273}\right)=R_0\left(\frac{T}{273}\right)\], so R_t ∝ T — resistance rises roughly linearly with absolute temperature. At very low temperatures, this linearity breaks down, and resistivity approaches a small finite value near absolute zero.

Given: I = 1.0 A, copper wire, length 0.10 m, A = 1.0 × 10⁻⁶ m²; one free electron per atom; density d = 8.9 × 10³ kg m⁻³; atomic weight M = 63.5; N = 6.02 × 10²⁶ kmol⁻¹; e = 1.6 × 10⁻¹⁹ C.

Find: (i) current density j (ii) drift velocity v_d.

Solution:
(i) j = \[\frac {I}{A}\] = \[\frac {1.0}{1.0×10^{−6}}\] = 1.0 × 10⁶ A m⁻²

(ii) n = \[\frac {dN}{M}\] = \[\frac {(8.9×10^3)(6.02×10^{26})}{63.5}\] = 8.4 × 10²⁸ m⁻³

vd = \[\frac {j}{ne}\] = \[\frac {1.0×10^6}{(8.4×10^{28})(1.6×10^{−19})}\] = 7.4 × 10⁻⁵ m s⁻¹

This tiny drift speed (slower than a snail) shows that electrical signals travel almost instantly because the electric field propagates near light speed — not the electrons themselves.

Given: R₁ = 3.15 Ω at 20°C; R₂ = 3.75 Ω at 100°C.

Find: α and R₀​ (resistance at 0°C).

Solution:

• Ohm's law emerges naturally from the microscopic drift-velocity model: V ∝ I, because drift velocity is proportional to the applied field.
• Resistivity ρ = m/(ne²τ) is intrinsic to the material — independent of wire shape or length.
• Metals get more resistive when heated; alloys stay nearly constant; semiconductors and electrolytes get less resistive.
• These behaviours explain real engineering choices: copper for wiring, nichrome for heaters and standard resistors, semiconductors for temperature sensors.