Ohm's law — microscopic origin, resistivity: Question bank

Before we start, three words you must own (every question below leans on them):

True or false — justify

Ohm's law

V = I R

is a fundamental law of nature like Newton's laws.

False. It is an approximation that holds only when

σ

is independent of

E

; diodes, plasmas and hot filaments are non-ohmic and break it.

If you double the voltage across an ohmic wire, the drift velocity doubles.

True.

v_{d} = e E τ / m

and

E = V / L

, so

v_{d} \propto V

— the whole chain from voltage to drift is linear (as long as

τ

stays fixed).

Resistivity

ρ

of a copper sample depends on how long and thick the wire is.

False.

ρ = m / (n e^{2} τ)

depends only on the material (

n

τ

T

). Length and area enter resistance via

R = ρ L / A

, never

ρ

itself.

The random thermal motion of electrons contributes to the current.

False. Thermal velocities point every direction and average to zero net flow; only the small drift

v_{d}

carries current.

In steady state, electrons in a wire move with constant acceleration.

False. They move with constant average velocity

v_{d}

. Between collisions they accelerate, but each collision resets the gain, so the drift is steady, not the acceleration.

Cooling a metal wire lowers its resistance.

True. Colder lattice vibrates less → fewer collisions →

τ

rises →

ρ = m / (n e^{2} τ)

falls, so

R

falls too.

For semiconductors, heating always raises resistance just like metals.

False. In semiconductors heating frees far more carriers, so

n

grows fast and

ρ

falls — the opposite trend (see Semiconductors vs metals).

A lamp lights instantly only because electrons travel from switch to bulb very fast.

False. Drift speed is ~

1 0^{- 4}

m/s (snail-slow). The field/signal that sets every electron drifting propagates near light speed, so all electrons start almost at once.

Conductivity

σ

increases if you apply a stronger electric field.

False.

σ = n e^{2} τ / m

has no

E

in it — that

E

-independence is exactly what makes the material ohmic.

Spot the error

"A constant force acts on the electron, so by

a = F / m

its speed grows without limit, hence current grows forever."

The electron is not free — collisions every

τ

erase the accumulated velocity. The balance between field-gain and collision-loss gives a steady

v_{d}

, so current is constant, not runaway.

"Since

J = n e v_{d}

and

v_{d}

is tiny, the current in a household wire must be tiny too."

The carrier density

n \sim 1 0^{28}

–

1 0^{29} m^{- 3}

is enormous, and it multiplies

v_{d}

. A vast number of slow carriers still delivers amperes.

ρ = m / (n e^{2} τ)

, so resistivity is inversely proportional to the electron's charge."

Charge appears squared (

e^{2}

), not to the first power — so

ρ \propto 1/ e^{2}

. The square comes from

e

entering once in the force and once in the current density.

"Resistance rises with temperature because the electrons themselves get heavier when hot."

Mass

m

is fixed. It's

τ

that shrinks: hotter ions vibrate more, collisions happen more often, so

τ

drops and

ρ

rises.

"Two identical wires in series have half the resistance of one, because charge has two paths."

Series wires share one path, so lengths add — resistance doubles. Two paths (halving) is the parallel arrangement (see Resistors in series and parallel).

"In

v_{d} = - \frac{e τ}{m} E

the minus sign means the current flows opposite to

E

The minus sign means the electrons drift opposite to

E

(they're negative). Conventional current

J = n e v_{d}

(with

e > 0

) still points along

E

— the two minus signs cancel in the current.

Why questions

Why does a longer wire have more resistance, in microscopic terms?

A longer wire means an electron must cross more lattice on its way through, encountering proportionally more collision-causing scattering "in series" —

R = ρ L / A

scales with

L

Why does the drift velocity stay constant instead of climbing during each free flight?

It does climb during a single free flight, but we average over many flights: each collision drops the electron back to zero drift, so the ensemble average is the fixed

v_{d} = e E τ / m

Why is

σ

independent of

E

the crucial fact, rather than just "

J

and

E

both appear"?

σ

depended on

E

, then

J = σ (E) E

would be nonlinear and

V

would not be proportional to

I

. The constancy of

σ

is what makes the relation linear — i.e. Ohm's law.

Why can we say collisions "randomize" velocity rather than "stop" it?

A collision scatters the electron into a random direction, so its drift contribution averages to zero afterward — but its fast thermal speed is unchanged. It forgets the field-gained velocity, not all its energy.

Why does the field inside a current-carrying wire stay non-zero even though it's a conductor?

A static conductor has zero internal field, but a current-carrying one must sustain

E = J / σ

to keep pushing the drifting electrons against collisions (see Electric field inside a conductor).

Edge cases

What happens to

R

as area

A \to \infty

(an infinitely fat wire)?

R = ρ L / A \to 0

: infinitely many parallel paths, so no resistance — the wire behaves like a perfect connection.

What is the drift velocity the instant before you switch on the field (

E = 0

v_{d} = e E τ / m = 0

: with no field there is no net drift, though electrons still zip around thermally with zero average.

If the relaxation time

τ \to 0

(collisions infinitely frequent), what happens to

ρ

ρ = m / (n e^{2} τ) \to \infty

: constant scattering never lets any drift build up, so the material becomes a perfect insulator to current.

If free-carrier density

n \to 0

, does the material conduct?

No:

σ = n e^{2} τ / m \to 0

, so

J = σ E \to 0

. With no carriers there is nothing to drift, regardless of the field.

At a fixed voltage, what limits the current in a real filament as it glows hot?

Rising temperature shortens

τ

, raising

ρ

and hence

R

; so current falls as it heats — this

E

-dependent behaviour is exactly why a lamp filament is non-ohmic.

For a superconductor (idealized

τ \to \infty

, no scattering), what is

ρ

ρ = m / (n e^{2} τ) \to 0

: with no collisions the drift never resets, so current flows with zero resistance — but this is outside the ordinary Drude picture.

Recall One-line survival kit

Constant force but constant velocity (not acceleration) because collisions brake every $τ$ ::: yes $ρ$ = material only; $R = ρ L / A$ adds geometry ::: yes Ohm's law = " $σ$ has no $E$ in it" = linearity ::: yes Metals: heat up → $τ$ down → $ρ$ up. Semiconductors: heat up → $n$ up → $ρ$ down ::: yes

Connections

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