Which of the following characteristics of electrons determines the current in a conductor? (a) Drift velocity

NCERT Exemplar · Q4

Q. Which of the following characteristics of electrons determines the current in a conductor?

(a) Drift velocity alone.

(b) Thermal velocity alone.

(d) Neither drift nor thermal velocity.

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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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The current in a conductor is determined by the drift velocity of electrons — the net average velocity they acquire under an electric field. The correct answer is drift velocity.

Why drift velocity, not random motion?

Inside a conductor, electrons are constantly moving in random directions due to thermal energy — their average velocity is zero. If you put an electric field across the conductor, it gives a tiny net push to each electron in one direction. This net push doesn't stop the random jiggling; it just shifts the whole random motion slightly forward. The result is a small, steady average velocity called the drift velocity ( $v_d$ ).

Current is the net flow of charge. Random motion cancels out — it contributes nothing to current. Only the drift velocity gives a net transport of charge. So the characteristic that matters is drift velocity.

Watch out

A common mistake is to think the speed of individual electrons (their thermal speed, ~ $10^6$ m/s) determines current. That speed is huge, but it's random — it produces zero net current. Drift velocity is tiny (mm/s), but it's the net velocity, and that's what matters.

Step-by-step reasoning

Current is defined as the rate of flow of charge. For a conductor of cross-sectional area $A$ , with $n$ free electrons per unit volume, each carrying charge $e$ , the current is:

I = n e A v_d

where $v_d$ is the drift velocity. This formula comes directly from counting how many electrons cross a section per second.

Drift velocity itself depends on the electric field and the material. Under an electric field $E$ , an electron accelerates between collisions. The average time between collisions is the relaxation time $\tau$ . The drift velocity is:

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

So $v_d$ is proportional to the field and the relaxation time.

What about the other options?
- Thermal velocity: Random thermal motion gives zero net current. …

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Q.Which of the following characteristics of electrons determines the current in a conductor?

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

Why drift velocity, not random motion?

Step-by-step reasoning

Method: Deciding what "determines the current" in a conductor

Steps

Q.Which of the following characteristics of electrons determines the current in a conductor?

Why drift velocity, not random motion?

Step-by-step reasoning

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