NCERT text

Currents are not always steady, and hence, more generally, we define the current as follows. Let ΔQ be the net charge flowing across a cross-section of a conductor during the time interval Δt [i.e., between times t and (t + ΔQ t)]. Then, the current at time t across the cross-section of the conductor is defined as the value of the ratio of ΔQ to Δt in the limit of Δt tending to zero,

Explanation :

Earlier we used: $I=\frac{Q}{t}$

This works only when:

current is steady
same charge flows every second

Example:

2 C in 1st second
2 C in 2nd second
2 C in 3rd second

Then current is constant.

But Real Current Often Changes

Suppose:

Time	Charge Flow
First second	2 C
Second second	5 C
Third second	1 C

Now, the current is changing every moment.

So the ordinary formula becomes incomplete.

Scientists needed:

“current at one exact instant.”

That is why Equation (3.2) was created.

Meaning of Symbols

Symbol 1:

$\Delta Q$

Means:

Small amount of charge

Example:

tiny charge crossing wire

Symbol 2:

$\Delta t$

Means:

Small time interval

Example:

0.001 second
0.000001 second

Symbol 3:

$\frac{\Delta Q}{\Delta t}$

Means:

Charge flow rate

How much charge flows during tiny time?

Symbol 4:

$\lim_{\Delta t \to 0}$

This is the MOST IMPORTANT part.

It means:

Make the time interval extremely tiny

Almost zero.

Why?

Because we want:

current at one exact instant.

Not average over large time.

Final Meaning of Equation (3.2)
$I(t)=\lim_{\Delta t \to0}\frac{\Delta Q}{\Delta t}$ Current at a particular instant equals the charge crossing per unit time during an extremely tiny time interval.

Relation with Calculus

This equation becomes: $I=\frac{dQ}{dt}$

because: $\frac{dQ}{dt}$

is calculus notation for:

infinitely small change in charge
divided by an infinitely small change in time.

Final Meaning of Equation (3.2)

$I(t)=\lim_{\Delta t \to0}\frac{\Delta Q}{\Delta t}$
Current at a particular instant equals the charge crossing per unit time during an extremely tiny time interval.