(a) Establish the relation between drift velocity of electrons ($v_d$) and electric current ($I$) in a conductor. (b) How is $v_d$ affected when the length of the conductor is doubled, keeping the voltage applied across the conductor constant ?

> [!TLDR] > Drift velocity is the average velocity electrons acquire under an electric field, and it relates directly to current via $I = n e A v_d$. When the conductor’s length is doubled at constant voltage, the electric field halves, so drift velocity also halves — $v_d' = v_d / 2$. ## Why drift velocity? The core idea Current in a metal is the flow of free electrons. But electrons don’t zoom straight through — they collide with atoms, bounce around, and only *drift* slowly in the direction of the field. That average net velocity is the **drift velocity** $v_d$. It’s tiny (mm/s), yet it carries the current because there are so many electrons. The key link: current $I$ is the charge passing a cross-section per second. If each electron carries charge $e$, and $n$ electrons per unit volume drift with speed $v_d$ through area $A$, then in one second the charge crossing is $n e A v_d$. That’s the current. --- ## (a) Deriving $I = n e A v_d$ 1. **Set up the picture** Consider a conductor of cross-sectional area $A$. Let $n$ be the number of free electrons per unit volume. Each electron has charge $e$ (magnitude $1.6 \times 10^{-19} \,\text{C}$). Under an electric field $E$, electrons drift with average velocity $v_d$ along the field direction. 2. **Charge crossing in time $\Delta t$** In a small time $\Delta t$, an electron moves a distance $v_d \Delta t$. So all electrons within a cylinder of length $v_d \Delta t$ and area $A$ will cross a given cross-section in that time. The volume of this cylinder is $A v_d \Delta t$. 3. **Number of electrons and total charge** Number of electrons in that volume = $n \cdot (A v_d \Delta t)$. Total charge crossing = $(n A v_d \Delta t) \cdot e$. 4. **Current = charge per unit time** $$I = \frac{\text{charge crossing}}{\Delta t} = \frac{n e A v_d \Delta t}{\Delta t} = n e A v_d.$$ > [!FORMULA] > $$I = n e A v_d$$ That’s the fundamental relation. Rearranging gives $v_d = \frac{I}{n e A}$. > [!TIP]

(a) Establish the relation between drift velocity of electrons (vd) and electric current (I) in a conductor. (

Question

v_d

(b) How is

v_d

affected when the length of the conductor is doubled, keeping the voltage applied across the conductor constant ?

CBSECBSE Class XII Board 2026Subjective· 3mImportance★★★★★

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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. …

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Drift velocity is the average velocity electrons acquire under an electric field, and it relates directly to current via $I = n e A v_d$ . When the conductor’s length is doubled at constant voltage, the electric field halves, so drift velocity also halves — $v_d' = v_d / 2$ .

Why drift velocity? The core idea

Current in a metal is the flow of free electrons. But electrons don’t zoom straight through — they collide with atoms, bounce around, and only drift slowly in the direction of the field. That average net velocity is the drift velocity $v_d$ . It’s tiny (mm/s), yet it carries the current because there are so many electrons.

The key link: current $I$ is the charge passing a cross-section per second. If each electron carries charge $e$ , and $n$ electrons per unit volume drift with speed $v_d$ through area $A$ , then in one second the charge crossing is $n e A v_d$ . That’s the current.

(a) Deriving $I = n e A v_d$

Set up the picture

Consider a conductor of cross-sectional area $A$ . Let $n$ be the number of free electrons per unit volume. Each electron has charge $e$ (magnitude $1.6 \times 10^{-19} \,\text{C}$ ). Under an electric field $E$ , electrons drift with average velocity $v_d$ along the field direction.
Charge crossing in time $\Delta t$

In a small time $\Delta t$ , an electron moves a distance $v_d \Delta t$ . So all electrons within a cylinder of length $v_d \Delta t$ and area $A$ will cross a given cross-section in that time. The volume of this cylinder is $A v_d \Delta t$ .
Number of electrons and total charge

Number of electrons in that volume = $n \cdot (A v_d \Delta t)$ .

Total charge crossing = $(n A v_d \Delta t) \cdot e$ .
Current = charge per unit time

I = \frac{\text{charge crossing}}{\Delta t} = \frac{n e A v_d \Delta t}{\Delta t} = n e A v_d.

I = n e A v_d

That’s the fundamental relation. Rearranging gives $v_d = \frac{I}{n e A}$ .

Tip

Notice $v_d$ is inversely proportional to $A$ — thinner wires have faster drift for the same current. That’s why thin wires heat up more: faster electrons collide more often.

(b) Effect of doubling length at constant voltage

Now the conductor’s length $L$ is doubled, but the applied voltage $V$ stays the same. What happens to $v_d$ ?

Electric field depends on length For a uniform conductor, the electric field inside is $E = V / L$ . If $L$ doubles, $E$ halves:

E' = \frac{V}{2L} = \frac{E}{2}.

Drift velocity is proportional to field …

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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

…

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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

…

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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.

…

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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.

…

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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

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

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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

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

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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 …

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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.

…

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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:

$v_d' = \left(\frac{e\tau}{mL}\right) \cdot \frac{V}{2} = \frac{1}{2} \left(\frac{e\tau}{mL}\right) V = \frac{v_d}{2}$ …
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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.

…

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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.

…

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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. …
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Q.(a) Establish the relation between drift velocity of electrons (vdv_dvd​) and electric current (III) in a conductor.

Why drift velocity? The core idea

(a) Deriving I=neAvdI = n e A v_dI=neAvd​

(b) Effect of doubling length at constant voltage

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Q.(a) Establish the relation between drift velocity of electrons ( $v_d$ ) and electric current ( $I$ ) in a conductor.

(a) Deriving $I = n e A v_d$

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