JEE Mains Physics · Current Electricity

Current, Drift Velocity and Current Density

Current is the rate at which charge crosses a section, I = dq/dt; inside a wire it is carried by free electrons drifting slowly against the field, so I = neAv, where v is the drift speed.

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Why this matters

Eighteen PYQs, eleven of them multiple choice, and one from 2026. Seven are about charge: four integrate a current that changes with time, and the rest count electrons per second, find the least value of a current, or turn a battery's mAh rating into energy. Eleven are about the drift of electrons: drift speed from I = neAv, mobility, how drift speed depends on length, area and temperature, the current through part of a wire, and the field inside a wire from E = ρJ.

Concept 1 of 2: Charge, current and the area under an I–t graph

Current tells you how many coulombs pass a section each second. If the current changes with time, add up the small amounts

I\,dt

: that is an integral, or the area under the I–t graph. Going the other way, if the charge is given as a function of time, the current is its slope.

Definition

$I = \dfrac{dq}{dt}$ . For a steady current, $q = It$ .
For a changing current, $q = \displaystyle\int_{t_1}^{t_2} I\,dt$ , the area under the I–t graph between the two times asked.
If $q(t)$ is given, $I = dq/dt$ . The current is least where $dI/dt = 0$ and $d^{2}I/dt^{2} > 0$ .
Electrons per second: $n = I/e$ , with $e = 1.6 \times 10^{-19}$ C. For a device rated P at V, first find $I = P/V$ .
Battery capacity: 1 mAh = $10^{-3} \times 3600 = 3.6$ C. Energy stored = charge × voltage.

Charge and current

I = \frac{dq}{dt}, \qquad q = \int_{t_1}^{t_2} I\,dt, \qquad n = \frac{I}{e}

Worked example

The current in a wire is

I = (2 + 6t)

A, with t in seconds. Find the charge that passes a section between

t = 2

s and

t = 5

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 1 February 2024 · Q29Moderate

Example 1 · Current Electricity · Current, Drift Velocity and Current Density

The current in a conductor is expressed as

I= 3t^{2}+ 4t^{3}

, where

I

is in Ampere and

t

is in second. The amount of electric charge that flows through a section of the conductor during

t= 1

s to

t = 2\text{ }s

is ____ C.

Use the limits the question gives

"From t = 1 s to t = 2 s" means the lower limit is 1, not 0. Integrating from zero adds the charge of the first second and lands on a wrong option.

mAh is not coulombs

Multiply mAh by 3.6 to get coulombs. Then energy = charge × voltage. Forgetting the 3600 s in an hour gives an answer smaller by a factor of 3600.

Least current comes from dI/dt, not dq/dt

When the charge is given as q(t), differentiate once to get I and once more to find where I is least. Setting dq/dt = 0 finds where the current is zero instead.

Drill 6 more on charge, current and the area under an i–t graph

Concept 2 of 2: Drift velocity, mobility and current density

Free electrons in a metal move fast and in random directions, colliding all the time. A field adds a small push between collisions, so on average they creep along the wire: that slow average is the drift velocity. Electrons are negative, so they drift against the field, from lower to higher potential, while conventional current flows the other way.

Definition

$I = neAv_d$ , where n is the number of free electrons per m³. Current density $J = I/A = nev_d$ .
$v_d = \dfrac{eE\tau}{m}$ , where $\tau$ is the mean time between collisions. Mobility $\mu = v_d/E = e\tau/m$ .
Inside a wire of length l across a voltage V: $E = V/l$ . Also $J = \sigma E$ , so $E = \rho J$ , and the force on one electron is $eE$ .
At a fixed voltage, $v_d = \dfrac{e\tau}{m}\cdot\dfrac{V}{l}$ : it does not depend on the area, and it falls as the length rises. At a fixed current, $v_d \propto 1/A$ .
A hotter metal has more collisions (smaller $\tau$ ), so its drift speed at a given field falls.
For part of a cross-section, current = J × that part's area (for uniform J), or $\int J\,dA$ in general.

Drift and current density

I = neAv_d, \qquad v_d = \frac{eE\tau}{m} = \mu E, \qquad J = nev_d = \sigma E

Worked example

A wire of cross-section 1 mm² carries 1.92 A. It has

6 \times 10^{28}

free electrons per m³. Find the drift speed.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 8 Apr 2023 · Q114Moderate

Example 2 · Current Electricity · Current, Drift Velocity and Current Density

The number density of free electrons in copper is nearly

8 \times10^{28}{\text{ }m}^{- 3}

. A copper wire has its area of cross section

= 2 \times10^{- 6}{\text{ }m}^{2}

and is carrying a current of 3.2 A. The drift speed of the electrons is___

\times10^{- 6}{\text{ }ms}^{- 1}

mm² is 10⁻⁶ m²

Areas are usually given in mm². Leaving them in mm² makes the drift speed a million times too small.

Electrons drift towards higher potential

Conventional current runs from high to low potential. Electrons are negative, so they drift the other way, against the field. A statement that electrons drift from higher to lower potential describes conventional current, not the electrons.

At fixed voltage the area does not matter

Doubling the area at the same voltage doubles the current but leaves the drift speed, (eτ/m)(V/l), unchanged. Only at a fixed current does a larger area mean a smaller drift speed.

Drill 10 more on drift velocity, mobility and current density

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Charge, current and the area under an I–t graph
Charge and current
$I = \frac{dq}{dt}, \qquad q = \int_{t_1}^{t_2} I\,dt, \qquad n = \frac{I}{e}$
Drift velocity, mobility and current density
Drift and current density
$I = neAv_d, \qquad v_d = \frac{eE\tau}{m} = \mu E, \qquad J = nev_d = \sigma E$

Watch out for (6)

Use the limits the question gives
→ Charge, current and the area under an I–t graph
mAh is not coulombs
→ Charge, current and the area under an I–t graph
Least current comes from dI/dt, not dq/dt
→ Charge, current and the area under an I–t graph
mm² is 10⁻⁶ m²
→ Drift velocity, mobility and current density
Electrons drift towards higher potential
→ Drift velocity, mobility and current density
At fixed voltage the area does not matter
→ Drift velocity, mobility and current density

Test yourself on Current Electricity

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Current Electricity chapter test Drill all 18 questions