Questions: Microscopic Ohm's Law and Drift Velocity

Question 1

A student reasons: 'When I flip a light switch, the light turns on almost instantly — so electrons must be traveling through the wire at nearly the speed of light.' What is wrong with this reasoning?

• A. The student is correct — electrons in a conductor move at near-light speeds when a current flows
• B. Electrons drift at roughly 0.07 mm/s in a typical household wire; the near-instant response occurs because the electric field configuration propagates through the wire at close to the speed of light, not because electrons race from switch to bulb
• C. The signal travels at the speed of sound in copper, which is faster than in air but far below light speed
• D. Electrons do move quickly, but collisions with the lattice slow them to roughly half the speed of light

Answer: (B)

This is the central misconception about electric current. Drift velocity — the tiny net bias electrons develop in the field direction — is about 0.07 mm/s in a typical copper wire, roughly the speed of a slow ant. But the switch responds instantly because the electromagnetic field configuration (not the electrons themselves) changes at nearly the speed of light throughout the circuit simultaneously. The electrons were already in the wire; they just need to start drifting, and the signal that tells them to do so propagates near-instantly.

Question 2

Using the Drude model (σ = nq²τ/m), which change would INCREASE the conductivity of a metal?

• A. Raising the temperature, which increases lattice vibration and shortens the collision time τ
• B. Adding impurities, which introduces additional scattering sites and shortens τ
• C. Choosing a material with higher carrier density n — more conduction electrons per unit volume
• D. Using a material with larger effective mass m*, so each carrier carries more inertia

Answer: (C)

Conductivity σ = nq²τ/m is larger when n (carrier density) is large, τ (collision time) is long, and m (effective mass) is small. Raising temperature shortens τ by increasing lattice vibrations — this reduces conductivity, which is why most metals become more resistive when heated. Adding impurities also shortens τ through additional scattering. Larger effective mass m* reduces how quickly carriers accelerate in the field, lowering σ. Only increasing n — having more carriers per unit volume — unambiguously increases conductivity.

Question 3

The drift velocity of electrons in a metal carrying a typical household current is comparable in magnitude to their thermal velocity.

• T. True
• F. False

Answer: False

Thermal velocity of conduction electrons in a metal is roughly 10⁶ m/s (about 0.3% of the speed of light). Drift velocity in a copper wire carrying 1 A through a 1 mm² cross-section is approximately 0.07 mm/s — about 10 orders of magnitude slower. The thermal motion is enormous and random, averaging to zero net flow; drift is a tiny directed bias superimposed on top of this frantic random motion. Confusing the two is the most common misconception about microscopic current.

Question 4

When you flip a light switch, the light responds almost instantly not because electrons race through the wire, but because the electric field propagates through the circuit at close to the speed of light.

• T. True
• F. False

Answer: True

This is the key insight linking the microscopic model to everyday experience. Electrons in the wire are always present; the switch closing changes the electric field configuration throughout the circuit almost simultaneously. This field change — not electron transport — is what propagates at near-light speed. Individual electrons drift only millimeters per second, but because the driving signal (the field) is established everywhere at once, current effectively begins flowing at every point in the circuit at nearly the same time.

Question 5

Explain why increasing temperature decreases conductivity in metals, while increasing the number of conduction electrons per unit volume increases it — using the microscopic model.

Model answer: In the Drude model, conductivity σ = nq²τ/m. Temperature increases the amplitude of lattice vibrations, which causes more frequent electron-lattice collisions, shortening the average collision time τ. Shorter τ means electrons accumulate less drift velocity between collisions, so J = σE falls — conductivity decreases. Increasing carrier density n means more charges contribute to the current at any given drift velocity; even if τ and m are unchanged, more carriers means more total charge flow per unit time and area, so conductivity increases.

The two effects act on different parameters in the same formula. Temperature degrades τ (the collision time), which reduces each carrier's contribution to current. Carrier density n multiplies the total number of contributors. This also explains why superconductivity corresponds to τ → ∞ (no scattering whatsoever, so σ diverges), and why insulators have very low n (essentially no free carriers, so σ → 0).