Q.(a) Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area 1.0×10−7 m2 carrying a current of 1.5 A. Assume that each copper atom contributes roughly one conduction electron. The density of copper is 9.0×103 kg/m3, and its atomic mass is 63.5 u.
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 106 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.
The random thermal speed is about 105 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 (vd) is the average velocity acquired by the charge carriers in a conductor under an applied electric field:
vd=meEτ
where:
- e = electron charge (1.6×10−19 C)
- E = electric field inside the conductor (V/m)
- τ = average relaxation time — the mean time between collisions (s)
- m = electron mass (9.1×10−31 kg)
vd=meEτ
Linking to current
Drift velocity connects the microscopic motion of electrons to the current an ammeter reads:
I=neAvd
where n is the free-electron number density and A the cross-sectional area. A larger vd means more current, but vd stays tiny because τ is tiny (about 10−14 s in copper).
For a copper wire carrying 1 A with area 1 mm² and n≈8.5×1028 m−3:
vd=neAI≈(8.5×1028)(1.6×10−19)(10−6)1≈7×10−5 m/s
That is about 0.07 mm per second — slower than a garden snail.
Common misconception
Electrons do not race through wires near light speed. The field propagates almost instantly, so all electrons begin drifting together, but each one only crawls. It is like a hose already full of water: open the tap and water leaves the far end at once, though the individual molecules have barely moved. Drift velocity is that slow, directed crawl superimposed on the electrons' frantic random jitter.
Drift velocity of electrons and its link to current via I = neAv_d is a defining topic of the NCERT Class 12 Physics chapter on current electricity, tested through both conceptual and numerical questions in CBSE boards, JEE Main and NEET. Anyone searching "drift velocity formula and derivation class 12 physics" will find this relaxation-time explanation is the standard NCERT-aligned answer.
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 ~105 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 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 (vd).
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 E experiences:
F=−eE
The magnitude of acceleration (opposite to E) is:
a=mF=meE
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 τ (relaxation time). Just after a collision an electron's velocity is essentially random (zero average in the field direction); it then accelerates for time τ before the next collision.
Step 3: Drift velocity
Averaging the field-driven velocity over the relaxation time τ gives the net drift:
vd=meEτ
Here τ 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=nevd
where n = number of free electrons per unit volume.
Substituting vd=meEτ:
J=mne2τE
Comparing with Ohm's law J=σE, we get:
σ=mne2τ
Why this matters: conductivity depends on:
- n — more free electrons → better conductor
- τ — fewer collisions → higher conductivity
- m — lighter electrons → faster drift
4. Key Takeaways for Exams
| Concept | Formula | Why |
|---|---|---|
| Drift velocity | vd=meEτ | Acceleration × average time between collisions |
| Current density | J=nevd | Charge × number density × drift speed |
| Conductivity | σ=mne2τ | From combining above two |
Remember: Drift velocity is tiny — typically 10−4 m/s for copper wires — yet current flows almost instantly because the electric field propagates at near light speed, pushing all electrons simultaneously.
Concept: Drift Velocity — the small net velocity electrons acquire under an applied field, linked to current by I=neAvd, so vd=neAI.
(a) Free-electron number density n. Each Cu atom gives one conduction electron, so
n=MρNA=63.5×10−3(9.0×103)(6.022×1023)≈8.5×1028 m−3.
Drift speed. With I=1.5 A, A=1.0×10−7 m2, e=1.6×10−19 C:
vd=(8.5×1028)(1.6×10−19)(1.0×10−7)1.5≈1.1×10−3 m/s.
(b) (i) Thermal (rms) speed of Cu atoms at ∼300 K is ≈3.4×102 m/s, about 3×105 times larger than vd. (ii) The field propagates at ≈3×108 m/s, about 3×1011 times larger than vd.
vd≈1.1×10−3 m/s — far smaller than the atoms' thermal speed (∼102 m/s) and the field-propagation speed (∼3×108 m/s).
Using I=neAvd, the drift speed of electrons in the copper wire is vd≈1.1×10−3 m/s — negligible next to the atoms' thermal speed (∼102 m/s, i.e. ≈343 m/s) and the field-propagation speed (∼3×108 m/s).
Principle
The current in a metal is carried by free electrons that drift with a tiny average velocity vd superimposed on their fast random thermal motion. Current and drift speed are related by
I=neAvd⇒vd=neAI
where n is the free-electron number density, e the electron charge, and A the cross-sectional area.
(a) Drift speed
Step 1 — number density n. Each copper atom donates one conduction electron, so n equals the atomic number density:
n=MρNA=63.5×10−3 kg/mol(9.0×103 kg/m3)(6.022×1023 mol−1).
Computing: 63.5×10−39.0×103=1.417×105 mol/m3, and
n=(1.417×105)(6.022×1023)≈8.5×1028 m−3.
Convert the atomic mass to kg/mol: 63.5 u→63.5×10−3 kg/mol. Skipping this factor of 103 is the usual error.
Step 2 — substitute. With I=1.5 A, A=1.0×10−7 m2, e=1.6×10−19 C:
neA=(8.5×1028)(1.6×10−19)(1.0×10−7)≈1.37×103 C/(m⋅s)⋅(units of A/vd).
vd=1.37×1031.5≈1.1×10−3 m/s.
(b) Comparisons
- Thermal speed of copper atoms. From kinetic theory 21mvrms2=23kBT, with atomic mass m=6.022×102363.5×10−3≈1.05×10−25 kg and T=300 K:
Thus vrms/vd≈343/(1.1×10−3)≈3×105: the thermal speed exceeds the drift speed by about five orders of magnitude.
vrms=m3kBT=1.05×10−253(1.38×10−23)(300)≈3.4×102 m/s.
- Field-propagation speed. The electric field that drives the drift travels along the conductor at nearly the speed of light, c≈3×108 m/s, so
The field reaches every electron almost instantly, which is why the bulb lights immediately even though each electron only crawls.
vdc≈1.1×10−33×108≈3×1011.
✓Final answervd≈1.1×10−3 m/s; the thermal speed of Cu atoms is ≈3.4×102 m/s (about 3×105 times larger) and the field propagates at ≈3×108 m/s (about 3×1011 times larger).
Method: Drift Velocity Formula from Current–Charge Relation
This method uses the fundamental relation between current, charge carrier density, and drift velocity.
Steps
Step 1: Write the drift velocity formula
The current I in a conductor is given by:
I=neAvd
where:
- n = number density of conduction electrons (m−3)
- e = charge of an electron = 1.6×10−19 C
- A = cross-sectional area (m2)
- vd = drift velocity (m/s)
Rearranging for vd:
vd=neAI
Step 2: Find n, the number density of conduction electrons
Given: each copper atom contributes one conduction electron.
So n = number of copper atoms per cubic metre.
First, find number of atoms per mole: Avogadro’s number NA=6.02×1023 mol−1.
Mass of one mole of copper = atomic mass = 63.5 g=63.5×10−3 kg.
Volume of one mole of copper:
Volume=densitymass=9.0×10363.5×10−3=7.06×10−6 m3
Number of atoms per cubic metre:
n=Volume of one moleNA=7.06×10−66.02×1023=8.53×1028 m−3
Step 3: Substitute into drift velocity formula
Given:
- I=1.5 A
- A=1.0×10−7 m2
- e=1.6×10−19 C
- n=8.53×1028 m−3
vd=(8.53×1028)(1.6×10−19)(1.0×10−7)1.5
First compute denominator:
neA=(8.53×1028)×(1.6×10−19)×(1.0×10−7)=1.365×103
Thus:
vd=1.365×1031.5=1.1×10−3 m/s
Answer (a): 1.1×10−3 m/s
(b) Comparisons
- Thermal speed of copper atoms at ordinary temperatures
At room temperature (T≈300 K), the root-mean-square speed of copper atoms is:
where k=1.38×10−23 J/K and mass of one copper atom m=6.02×102363.5×10−3=1.05×10−25 kg.
vth=m3kT
Comparison: Drift speed (∼10−3 m/s) is about 105 times smaller than thermal speed (∼102 m/s).vth=1.05×10−253×1.38×10−23×300≈1.18×105≈3.4×102 m/s
- Speed of propagation of electric field The electric field propagates at nearly the speed of light: c≈3×108 m/s. Comparison: Drift speed is about 1011 times smaller than the field propagation speed.
Key Insight
The drift velocity is extremely slow — electrons move at millimetres per second — yet the electric signal travels near light speed. This is like a long pipe full of marbles: push one end, and the pulse reaches the other end almost instantly, even though each marble moves only a tiny distance.
Common Mistakes & How to Avoid Them — Drift Velocity
Mistake 1: Forgetting to convert atomic mass unit (u) to kg
The error: Students use 63.5 u directly in calculations without converting to kg. Since 1 u=1.66×10−27 kg, the mass of one copper atom is:
m=63.5×1.66×10−27 kg
How to avoid: Always check units — density is in kg/m3, so atomic mass must be in kg for consistency. Write the conversion step explicitly.
Mistake 2: Confusing number density (n) with mass density (ρ)
The error: Using ρ (density of copper) directly as n (number of conduction electrons per unit volume).
Correct approach: Number density n is found by:
n=atoms per electronNumber of atoms per unit volume
Since each atom contributes 1 electron:
n=Mρ×NA
where:
- ρ=9.0×103 kg/m3
- NA=6.02×1023 mol−1
- M=63.5×10−3 kg/mol (molar mass in kg)
How to avoid: Remember: n is number per volume, not mass per volume. Use Avogadro's number to bridge mass → number.
Mistake 3: Using wrong formula for drift velocity
The error: Writing vd=nAI instead of the correct:
vd=neAI
where e=1.6×10−19 C is the electron charge.
How to avoid: Drift velocity comes from I=neAvd. Always check dimensions — current is charge per time, so charge e must appear.
Mistake 4: Arithmetic errors in powers of 10
The error: Mismanaging exponents when calculating n or vd, especially with 1023 and 10−19.
How to avoid: Write all numbers in scientific notation before multiplying/dividing. Group powers of 10 separately:
n=63.5×10−3(9.0×103)(6.02×1023)=63.59.0×6.02×103+23+3
Mistake 5: Not comparing magnitudes correctly in part (b)
The error: Giving numerical values without meaningful comparison.
Correct comparison:
- Drift speed vd≈10−4 m/s (very slow — like a snail)
- Thermal speed of copper atoms at 300 K: vth≈m3kT≈102 m/s — 106 times larger
- Electric field propagation speed ≈ speed of light 3×108 m/s — 1012 times larger
How to avoid: Always express comparisons as ratios (e.g., "thermal speed is 106 times drift speed"). This shows conceptual understanding.
Mistake 6: Thinking drift speed is the same as signal speed
The error: Assuming electrons move at near light speed because the bulb lights instantly.
The truth: Individual electrons drift at mm/s, but the electric field signal propagates at nearly c. It's like a hose already full of water — turning on the tap sends a pressure wave instantly, but the water itself moves slowly.
How to avoid: Distinguish clearly between:
- Drift velocity — actual motion of electrons
- Drift velocity — actual motion of electrons
- Signal velocity — speed of energy/information transfer
Quick Summary Table
| Mistake | Fix |
|---|---|
| Using u instead of kg | Convert: 1 u=1.66×10−27 kg |
| Confusing n with ρ | Use n=MρNA |
| Omitting e in formula | vd=neAI |
| Exponent errors | Group powers of 10 separately |
| No ratio comparison | Express as "X times larger/smaller" |
| Confusing drift vs signal | Signal speed ≈c, drift ≈10−4 m/s |
Showing the 12 most recent of 13 on this concept.
- KEAM 2026Set eng-2026-04174 marksMCQQ.A uniform wire of area of cross section 1×10−7m2 carries a current of 1.6 A. If the number density of electrons is 5×1028m−3, the drift velocity of electrons (in mm s−1) is (A) 1 (B) 3 (C) 2 (D) 4 (E) 1.5
›Reveal solutionSolution
Drift velocity vd=I/(neA)=2×10−3 m/s=2 mm s−1.
The drift velocity is related to current I, number density n, electron charge e, and cross-sectional area A by I=neAvd, so:
vd=neAI
Substituting I=1.6 A, n=5×1028 m−3, e=1.6×10−19 C, A=1×10−7 m2:
neA=5×1028×1.6×10−19×1×10−7=800
vd=8001.6=2×10−3 ms−1=2 mm s−1
✓Final answerThe correct option is (C).
- KEAM 2026Set eng-2026-04194 marksMCQQ.If the drift velocity of electrons in a copper wire of cross-sectional area 2mm2 carrying current I is v1 and that in another copper wire of cross-sectional area 1.5mm2 carrying current 2I is v2, then the ratio v1:v2 is (A) 3:8 (B) 2:4 (C) 8:3 (D) 4:2 (E) 1:3
›Reveal solutionSolution
Drift velocity v=I/(nAe)∝I/A; compute the ratio for the two wires.
Drift velocity relation: I=nAevd⇒vd=nAeI, so for the same material vd∝AI.
Wire 1: current I, area 2 mm2⇒v1∝2I.
Wire 2: current 2I, area 1.5 mm2⇒v2∝1.52I=34I.
Ratio:
v2v1=4I/3I/2=21⋅43=83⇒3:8.
✓Final answerThe correct option is (A).
- KEAM 2026Set eng-2026-04204 marksMCQQ.When the electric field applied across a conductor is doubled without changing its temperature, the drift velocity (A) doubles (B) triples (C) quadruples (D) becomes half (E) remains unchanged
›Reveal solutionSolution
vd∝E, so doubling the field doubles the drift velocity.
Reasoning. The drift velocity is vd=meEτ, where the relaxation time τ depends on temperature (held constant). Since vd is directly proportional to E, doubling E doubles vd.
✓Final answerThe correct option is (A).
- KEAM 2026Set pha-2026-0419F4 marksMCQQ.The drift velocity of electrons in a linear conductor of given length under a potential can be doubled by (A) decreasing the potential 4 times (B) increasing the potential 2 times (C) decreasing the potential 8 times (D) increasing the potential 4 times (E) decreasing the potential 2 times
›Reveal solutionSolution
vd∝V, so doubling the applied potential doubles the drift velocity.
The drift velocity of electrons in a conductor of length L under potential difference V is
vd=meEτ=mLeVτ.
For a given conductor (L, τ, m fixed), vd is directly proportional to V. To double vd, the potential must be increased 2 times.
✓Final answerThe correct option is (B).
- KEAM 2025Set eng-2025-04234 marksMCQQ.The dependence of drift velocity vd on the electric field E, for which Ohm's law is obeyed is (A) vd∝E2 (B) vd∝E (C) vd∝E (D) vd∝E1 (E) vd∝E21
›Reveal solutionSolution
For Ohm's law to hold, the drift velocity is directly proportional to the electric field, vd∝E.
Concept and Intuition
Drift velocity is vd=meEτ, linear in E. This linearity is exactly what makes current proportional to voltage, i.e. Ohm's law.
Step-by-Step Solution
- vd=meEτ.
- With τ constant, vd∝E.
- This gives I=neAvd∝E∝V, i.e. Ohm's law.
Common Mistakes
- Confusing drift velocity's linear dependence on E with a square-root or inverse dependence.
✓Final answerThe correct option is (B) — vd∝E.
ANSWER: B
- KEAM 2025Set eng-2025-04254 marksMCQQ.If τ is the average time between any two successive collisions for the electrons in a metal wire under the application of an electric field E, then the mobility μ of the electrons is (A) mEeτ (B) mEeτ2 (C) meτ2 (D) emτ2 (E) meτ
›Reveal solutionSolution
Mobility is drift speed per unit field; since vd=eEτ/m, dividing by E gives μ=eτ/m.
Drift velocity of electrons under field E:
vd=meEτ.
Mobility is defined as drift speed per unit electric field:
μ=Evd=E1⋅meEτ=meτ.
✓Final answerThe correct option is (E).
- KEAM 2025Set eng-2025-04284 marksMCQQ.Mobility is the drift velocity per unit (A) charge (B) volume (C) electric field (D) current (E) time
›Reveal solutionSolution
Mobility is defined as drift velocity per unit electric field: μ=vd/E.
The mobility of a charge carrier is the magnitude of drift velocity acquired per unit applied electric field:
μ=E∣vd∣,
with units m2V−1s−1.
✓Final answerThe correct option is (C).
- KEAM 2025Set eng-2025-04284 marksMCQQ.The resistivity of a metallic wire is directly proportional to (T – temperature; τ average time of collisions of free electrons; n – number of free electrons per unit volume; A – area of cross-section) (A) n (B) τ (C) A (D) n1 (E) T1
›Reveal solutionSolution
Resistivity ρ=ne2τm, so it is directly proportional to n1.
From the free-electron model,
ρ=ne2τm.
Here ρ is inversely proportional to n and to τ, and independent of area A. Among the choices, resistivity is directly proportional to n1.
✓Final answerThe correct option is (D).
- KEAM 2024Set eng-2024-06064 marksMCQQ.n number of electrons flowing in a copper wire for 1 minute constitute a current of 0.5 A. Twice the number of electrons flowing through the same wire for 20 s will constitute a current of (A) 0.25 A (B) 3 A (C) 1 A (D) 1.25 A (E) 2.25 A
›Reveal solutionSolution
Find the charge of n electrons from the first case, double it, then divide by the new time.
In the first case n electrons flow in 1 min (60 s) giving 0.5 A:
Q1=ne=It=0.5×60=30 C.
In the second case twice the electrons (2n) flow, so the charge is Q2=2ne=60 C, over 20 s:
I=tQ2=2060=3 A.
✓Final answerThe correct option is (B).
- KEAM 2024Set eng-2024-06074 marksMCQQ.Mobility μ of an electron is related to average collision time τ as (e=electronic charge, m=mass of the electron) (A) τ1=mμ (B) μ=emτ (C) μ1=τ (D) μ=meτ (E) μτ=em
›Reveal solutionSolution
μ=meτ.
Drift velocity vd=meEτ. Mobility is defined as μ=Evd=meτ, where e is the electronic charge, m the electron mass and τ the average collision (relaxation) time.
✓Final answerThe correct option is (D).
- KEAM 2024Set eng-2024-06084 marksMCQQ.A steady current of 2A is flowing through a conducting wire. The number of electrons flowing per second in it is (A) 1.25×107 (B) 1.25×1019 (C) 2.50×1010 (D) 0.125×1025 (E) 2.5×1017
›Reveal solutionSolution
Number per second =I/e=2/(1.6×10−19)=1.25×1019.
Current is charge per unit time: I=ne, where n is the number of electrons per second and e=1.6×10−19 C.
n=eI=1.6×10−192=1.25×1019 electrons/s.
✓Final answerThe correct option is (B).
- KEAM 2024Set eng-2024-06094 marksMCQQ.Magnitude of drift velocity per unit electric field is known as (A) displacement current (B) mobility (C) electric resistance (D) electrical conductivity (E) relaxation time
›Reveal solutionSolution
Mobility is defined as the drift speed acquired per unit applied electric field, μ=vd/E.
For a conductor the drift velocity is vd=meEτ, so the magnitude of drift velocity per unit field is
μ=E∣vd∣=meτ.
This quantity is called the mobility of the charge carrier (SI unit m2V−1s−1). Conductivity (σ=neμ) and resistance are related but are not defined as vd/E; relaxation time τ and displacement current are unrelated.
✓Final answerThe correct option is (B).
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