The relaxation time $\tau$ is nearly independent of applied $E$ field whereas it changes significantly with temperature $T$. First fact is (in part) responsible for Ohm's law whereas the second fact leads to variation of $\rho$ with temperature. Elaborate why?

The relaxation time τ is nearly independent of applied E field whereas it changes significantly with temperatu

NCERT Exemplar · Q10

\tau

CBSESubjective· 3mImportance★★★★★

62% · 26/42 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Drift Velocity

Drift Velocity: The Slow March of Electrons

Electrons in a metal are always moving — but randomly. At room temperature they zip around at roughly $10^6$ m/s, colliding with the lattice ions every few trillionths of a second. Without an electric field this motion cancels out: for every electron heading left another heads right, so the net velocity is zero.

Apply a battery and the field gives every electron a tiny, steady push in one direction. Between collisions the electron accelerates only briefly before smashing into an ion and losing its directed motion. What survives is a very small average velocity along the field — the drift velocity.

Note

The random thermal speed is about $10^5$ m/s, but the drift velocity is only about $10^{-4}$ m/s — about a billion times slower. An electron drifts slower than a snail, yet a lamp lights instantly, because the electric field (not the electrons) propagates at nearly the speed of light and starts every electron drifting almost at once.

The precise definition

Drift velocity ( $v_d$ ) is the average velocity acquired by the charge carriers in a conductor under an applied electric field:

v_d = \frac{eE\tau}{m}

where:

$e$ = electron charge ( $1.6 \times 10^{-19}$ C)
$E$ = electric field inside the conductor (V/m)
$\tau$ = average relaxation time — the mean time between collisions (s)
$m$ = electron mass ( $9.1 \times 10^{-31}$ kg)

v_d = \frac{eE\tau}{m}

Linking to current

Drift velocity connects the microscopic motion of electrons to the current an ammeter reads:

I = n e A v_d

where $n$ is the free-electron number density and $A$ the cross-sectional area. A larger $v_d$ means more current, but $v_d$ stays tiny because $\tau$ is tiny (about $10^{-14}$ s in copper).

Tip

For a copper wire carrying $1$ A with area $1$ mm² and $n \approx 8.5 \times 10^{28}$ m $^{-3}$ :

$v_d = \frac{I}{neA} \approx \frac{1}{(8.5\times10^{28})(1.6\times10^{-19})(10^{-6})} \approx 7 \times 10^{-5}\ \text{m/s}$ …

Why this formula?

Drift Velocity: Why the Formula Holds

Let's build this from first principles — understanding why electrons drift the way they do, not just memorizing the formula.

1. The Core Idea: What is Drift Velocity?

In a conductor, free electrons are constantly moving randomly (thermal motion, speeds ~ $10^5$ m/s). Without an electric field, their net displacement is zero — they're like a swarm of bees buzzing in all directions.

When we apply an electric field $\vec{E}$ , it gently nudges each electron in the opposite direction (since electrons are negatively charged). This small, steady net velocity superimposed on the random motion is drift velocity ( $v_d$ ).

Key insight: Drift velocity is not the speed of individual electrons — it's the average velocity of the entire electron cloud.

2. The Derivation: Step by Step

Step 1: Force on a single electron

An electron of charge $-e$ in an electric field $\vec{E}$ experiences:

\vec{F} = -e \vec{E}

The magnitude of acceleration (opposite to $\vec{E}$ ) is:

a = \frac{F}{m} = \frac{eE}{m}

where $m$ is the electron's mass.

Step 2: What happens between collisions?

Electrons don't accelerate forever — they keep colliding with atoms/ions in the metal lattice. Let the average time between collisions be $\tau$ (relaxation time). Just after a collision an electron's velocity is essentially random (zero average in the field direction); it then accelerates for time $\tau$ before the next collision.

Step 3: Drift velocity

Averaging the field-driven velocity over the relaxation time $\tau$ gives the net drift:

\boxed{v_d = \frac{eE\tau}{m}}

Here $\tau$ is the average time since the last collision, so this expression already averages over electrons at every stage between collisions — it is the standard result used in the NCERT treatment.

3. Connecting to Current: The Big Picture

Drift velocity directly gives us current density $J$ :

J = n e v_d

where $n$ = number of free electrons per unit volume. …

Start your 14-day free trial to unlock the full solution →

Drift velocity depends on relaxation time $\tau$ and applied field $E$ ; Ohm’s law ( $J \propto E$ ) follows because $\tau$ is independent of $E$ , while resistivity $\rho$ varies with temperature because $\tau$ changes significantly with $T$ due to increased lattice vibrations.

The key to understanding both Ohm’s law and the temperature dependence of resistivity lies in the microscopic picture of electron motion inside a conductor — specifically, the concept of drift velocity and the relaxation time $\tau$ .

When no electric field is applied, free electrons in a metal move randomly with high speeds (about $10^6$ m/s at room temperature) but with zero net drift. The average time between successive collisions of an electron with the lattice ions (or other electrons) is called the relaxation time $\tau$ . This $\tau$ is determined by how often an electron’s path is interrupted — and that depends on the lattice vibrations (phonons) and impurities, not on the applied field.

Now, when a small electric field $E$ is applied, each electron experiences a constant force $F = -eE$ , giving it a small acceleration $a = -eE/m$ between collisions. But because collisions reset the electron’s velocity randomly, the net effect is a tiny drift velocity superimposed on the random thermal motion.

v_d = -\frac{eE}{m} \tau

This is the average velocity an electron gains in the direction opposite to the field. The negative sign just indicates direction — we often write $v_d = \frac{eE}{m}\tau$ in magnitude.

The current density $J$ is then:

J = n e v_d = \frac{n e^2 \tau}{m} E

where $n$ is the number density of free electrons. This is Ohm’s law in microscopic form: $J = \sigma E$ , with conductivity $\sigma = \frac{n e^2 \tau}{m}$ .

Now let’s address the two facts given in the question.

Why $\tau$ independent of $E$ leads to Ohm’s law

The relaxation time $\tau$ depends on the scattering mechanisms in the metal — primarily collisions with vibrating lattice ions. The applied electric field $E$ (even a strong one, within normal limits) is far too weak to significantly alter the random thermal speeds of electrons. The thermal speed $v_{th} \approx \sqrt{3kT/m}$ is on the order of $10^5$ – $10^6$ m/s, while the drift speed $v_d$ is typically only a few mm/s. So the time between collisions is essentially determined by the thermal motion, not by the tiny drift. Hence $\tau$ is independent of $E$ .

Because $\tau$ does not change with $E$ , the conductivity $\sigma = n e^2 \tau / m$ is a constant for a given temperature. Then $J = \sigma E$ gives a linear relationship — that is Ohm’s law. If $\tau$ depended on $E$ , the $J$ – $E$ graph would curve, and Ohm’s law would fail.

Watch out

A common mistake is to think that $v_d$ itself is proportional to $E$ because of $F = ma$ . That’s true, but the deeper reason Ohm’s law holds is that $\tau$ stays constant — otherwise the proportionality constant $\sigma$ would change with $E$ .
Why $\tau$ changes with $T$ leads to variation of $\rho$ with temperature …

Start your 14-day free trial to unlock the full solution →

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 31 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$Q. Drift velocity Vd varies with the intensity of electric field E as per the relation$
(a) Vd is proportional to E^2
(b) Vd is proportional to 1/E
(c) Vd is proportional to sqrt(E)
(d) Vd is proportional to E
›Reveal solutionSolution

Drift velocity is the (small) average velocity electrons gain between collisions due to the electric field, and it comes out directly proportional to E.

When an electric field E is applied to a conductor, each free electron experiences a force F = eE, giving it an acceleration a = eE/m between collisions with the lattice ions. If tau is the average time between collisions (relaxation time), the average extra velocity gained (the drift velocity) is

vd = a * tau = (eE/m) * tau = (e*tau/m) * E

…

Start your 14-day free trial to see the full solution →
CBSE 2026Set ANNUAL1 mark
$Q. If the current flowing in a copper wire be allowed to flow in another copper wire of same length but of doubled the radius then what will be the effect on the drift velocity of the electron?$
›Reveal solutionSolution

For the same current, $v_d \propto 1/A$ , and doubling the radius quadruples the cross-sectional area.

Current is related to drift velocity by $I = nAev_d$ , so for the same current $I$ (and the same material, hence the same $n$ ), $v_d = \dfrac{I}{nAe} \propto \dfrac{1}{A}$ . If the radius is doubled, the cross-sectional area $A=\pi r^2$ becomes 4 times larger. So the drift velocity becomes

…

Start your 14-day free trial to see the full solution →
CBSE 2026Set ANNUAL1 mark
$Q. State Ohm's law in terms of current density, specific conductance and electric field intensity.$
›Reveal solutionSolution

Microscopic Ohm's law: current density J = σE (σ = conductivity, E = field).

The usual Ohm's law is V = IR. In microscopic (vector) form, it relates the current density J (current per unit cross-sectional area) to the electric field E inside the conductor through the material's specific conductance (conductivity) σ:

J = σ E.

…

Start your 14-day free trial to see the full solution →
CBSE 2026Set SEM31 markMCQ
$Q. Which of the following statement(s) is/are true ? A potential difference of V is applied at the two ends of a conductor of length l and area of cross-section A. Statement I : When potential difference is doubled, current density also gets doubled. Statement II : When potential difference is doubled, drift velocity gets halved. Statement III : When area of cross-section is doubled, current density decreases.$
(a) I and II are true
(b) Only I is true
(c) Only III is true
(d) II and III are true
›Reveal solutionSolution

Doubling V doubles E, so J = σE and v_d = μE both double — Statement I true, Statement II (drift velocity halved) false. J = V/(ρl) is independent of area, so Statement III (J decreases when A doubles) is also false. Only I is true → option (b).

Statement I: J = σE and E = V/l, so doubling V doubles E and hence doubles the current density J. TRUE.

Statement II: drift velocity v_d = (eE/m)τ ∝ E ∝ V. Doubling V doubles v_d, it does not halve it. FALSE.

…

Start your 14-day free trial to see the full solution →
CBSE 2025Set D1 markMCQ
$Q. The relation between drift velocity v of free electrons in conductor in electric conduction and potential difference V between ends of conductor is (A) proportional to V (B) inversely proportional to V (C) proportional to V^2 (D) inversely proportional to V^2$
›Reveal solutionSolution

Drift velocity is directly proportional to the potential difference V.

In a conductor of length L across which a potential difference V is applied, the electric field is E = V/L. Free electrons acquire a drift velocity

$v_d = \frac{eE\tau}{m} = \frac{eV\tau}{mL}$ …

Start your 14-day free trial to see the full solution →
CBSE 2025Set D1 markMCQ
$Q. If the length of a conductor is doubled while keeping the potential difference across it constant, then the drift velocity of electron will (A) remain the same (B) be double (C) be halved (D) increase fourfold$
›Reveal solutionSolution

Doubling the length at constant V halves the drift velocity.

The drift velocity is

$v_d = \frac{eE\tau}{m} = \frac{eV\tau}{mL}$ …

Start your 14-day free trial to see the full solution →
CBSE 2025Set ANNUAL1 markMCQ
$Q. A thick wire is stretched so that its length becomes two times. What is the ratio of change in resistance of the wire to the initial resistance of the wire?$
(i) 2 : 1
(ii) 4 : 1
(iii) 3 : 1
(iv) 1 : 4
›Reveal solutionSolution

New resistance is 4 times the old, so the change is 3 times the original: ratio 3 : 1.

Resistance $R = \rho L/A$ . Stretching keeps the volume $AL$ constant, so if length doubles ( $L \to 2L$ ) the area halves ( $A \to A/2$ ). Then $R' = \rho (2L)/(A/2) = 4\rho L/A = 4R$ . The change in …

Start your 14-day free trial to see the full solution →
CBSE 2025Set ANNUAL1 markMCQ
$Q. Calculate the amount of charge flowing in 2 minutes in a wire of resistance 10 ohm when a potential difference of 20 volts is applied between its ends.$
(i) 120 C
(ii) 240 C
(iii) 20 C
(iv) 4 C
›Reveal solutionSolution

Q = It = (V/R) x t = 2 A x 120 s = 240 C.

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set 55/2/11 markMCQ
$v_d$
›Reveal solutionSolution
Drift speed is directly proportional to the applied potential difference for a given conductor, so halving $V$ halves $v_d$ . The new drift speed is $\dfrac{v_d}{2}$ , which is option (A).

The key here is understanding what drift speed actually depends on. Many students memorise the formula $v_d = \frac{I}{n e A}$ and then try to relate $I$ to $V$ via Ohm's law — that works, but it's easy to lose track of which quantities stay constant. Let's build it from the physics up.

Drift speed is the average velocity electrons acquire due to an electric field inside the conductor. That field is $E = V/L$ , where $L$ is the length of the conductor. The force on each electron is $eE$ , and in the steady state, this force is balanced by collisions with the lattice, giving a constant drift speed proportional to the field. So the fundamental proportionality is:
$v_d \propto E \quad \text{and since} \quad E = \frac{V}{L}, \quad \text{we get} \quad v_d \propto V$
for a fixed conductor (fixed $L$ , fixed material properties like relaxation time $\tau$ , mass $m$ , charge $e$ ).

Now let's walk through it step by step.

Start with the microscopic relation. The drift speed is given by $v_d = \frac{eE\tau}{m}$ , where $\tau$ is the average time between collisions (relaxation time). This comes from $F = eE = m a$ , and then $v_d = a\tau$ . For a given conductor at a fixed temperature, $\tau$ , $m$ , and $e$ are constants.

Express the electric field in terms of the applied voltage. For a conductor of length $L$ , the uniform electric field inside is $E = V/L$ . So:

$v_d = \frac{e (V/L) \tau}{m} = \left(\frac{e\tau}{mL}\right) V$
The quantity in parentheses is constant for a given conductor. So $v_d$ is directly proportional to $V$ .

Apply the change. If $V$ becomes $V/2$ , then:

$v_d' = \left(\frac{e\tau}{mL}\right) \cdot \frac{V}{2} = \frac{1}{2} \left(\frac{e\tau}{mL}\right) V = \frac{v_d}{2}$ …
Start your 14-day free trial to see the full solution →
CBSE 2024Set FS1 markMCQ
$v_d$
(i) $v_d \propto E^2$
(ii) $v_d = \text{Constant}$
(iii) $v_d \propto E$
(iv) $v_d \propto \sqrt{E}$
›Reveal solutionSolution

Drift velocity $v_d=\dfrac{eE\tau}{m}$ is directly proportional to the field $E$ , which is exactly what Ohm's law requires — option (iii).

Concept. Under an electric field $E$ , free electrons acquire a steady drift velocity

$v_d=\frac{eE\tau}{m},$

where $\tau$ is the mean relaxation time.

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set ANNUAL1 mark
$Q. Define the relaxation time of the free electrons drifting in a conductor.$
›Reveal solutionSolution

The average free time an electron travels between collisions with the lattice.

…

Start your 14-day free trial to see the full solution →
CBSE 2023Set 55/1/11 markMCQ
$(A)$
(a)
(b)
(c)
(d)
›Reveal solutionSolution
For a steady current, the product $A v_d$ is constant because $I = n e A v_d$ is fixed. Therefore $v_d \propto 1/A$ , which is a rectangular hyperbola — option (a).

Figure — CBSE 2023 55/1/1 Q4

The key to this question is understanding why drift velocity changes when the wire's cross-section changes — and that comes from the definition of steady current itself.

When we say "a steady current flows", we mean that the same amount of charge passes through every cross-section of the wire per second. The wire is in series with itself: whatever charge flows past the thin end must also flow past the thick end in the same time. If it didn't, charge would pile up somewhere — and that would violate the steady-state condition.

Now, current $I$ is given by:
$I = n e A v_d$
where $n$ is the free electron density (number per unit volume), $e$ is the electron charge, $A$ is the cross-sectional area at that point, and $v_d$ is the drift velocity.

For a metallic wire, $n$ and $e$ are constants (same material throughout). And for a steady current, $I$ is constant along the wire. So:
$n e A v_d = \text{constant}$
which means:
$A v_d = \text{constant}$
Therefore:
$v_d \propto \frac{1}{A}$
This is the relationship we need.

Identify the mathematical form. $v_d \propto 1/A$ is an inverse proportion. Its graph is a rectangular hyperbola — a curve that falls steeply when $A$ is small and flattens out as $A$ grows large. It never touches either axis (asymptotic behaviour).

Check the options against this.

Option (a) shows exactly this: a curve that drops as $A$ increases, shaped like a hyperbola.

Option (b) is a straight line — that would mean $v_d \propto -A + \text{constant}$ , which is wrong.

Option (c) is a horizontal line — that would mean $v_d$ is independent of $A$ , which contradicts $A v_d = \text{constant}$ .

Option (d) is a straight line through the origin — that would mean $v_d \propto A$ , which is the opposite of what we have. …
Start your 14-day free trial to see the full solution →

←Previous questionQ9 — Is the momentum conserved when charge crosses a junction in an electri…→Next questionQ11 — What are the advantages of the null-point method in a Wheatstone bridg…

Q.The relaxation time $\tau$ is nearly independent of applied $E$ field whereas it changes significantly with temperature $T$ . First fact is (in part) responsible for Ohm's law whereas the second fact leads to variation of $\rho$ with temperature. Elaborate why?

Concept understanding — Drift Velocity

Drift Velocity: The Slow March of Electrons

The precise definition

Linking to current

Why this formula?

Drift Velocity: Why the Formula Holds

1. The Core Idea: What is Drift Velocity?

2. The Derivation: Step by Step

Step 1: Force on a single electron

Step 2: What happens between collisions?

Step 3: Drift velocity

3. Connecting to Current: The Big Picture

Method: Connecting a Microscopic Quantity to Two Different Macroscopic Laws

Steps

Q.The relaxation time τ\tauτ is nearly independent of applied EEE field whereas it changes significantly with temperature TTT. First fact is (in part) responsible for Ohm's law whereas the second fact leads to variation of ρ\rhoρ with temperature. Elaborate why?

🎓Unlock everything free for 14 days

Q.The relaxation time $\tau$ is nearly independent of applied $E$ field whereas it changes significantly with temperature $T$ . First fact is (in part) responsible for Ohm's law whereas the second fact leads to variation of $\rho$ with temperature. Elaborate why?

Unlock everything free for 14 days