General Definition of Instantaneous Current as a Limit of Charge Flow

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:

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.

Numerical Example

Suppose:$$Q = 4t^2$$

Find current at:$$t=2s$$

Using:$$I=\frac{dQ}{dt}$$

Differentiate:$$I=\frac{d}{dt}(4t^2)$$$$I=8t$$

At $t=2s$$$I=8(2)$$$$I=16A$$