Question

A current flowing through a wire depends on time as$I=3t^2+2t+5$. The charge flowing through the cross-section of the wire in time from $t=0$ to $t=2s$ is:

• A. $22C$
• B. $20C$
• C. $18C$
• D. $5C$

Answer 1

The correct option is A

$22C$

Step 1: Given data

Current-time variation, $I=3t^2+2t+5$

Initial time, $t=0$

Final time, $t=2s$

Step 2: Formula used

$I=\frac{dQ}{dt}$ …………………….(a)

Where ($dQ$) is the small charge flowing in a small time duration ($dt$)

Step 3: Calculation of charge flowing through the wire

Substituting the given values in equation (a), we get

$3t^2+2t+5=\frac{dQ}{dt}$

$dQ=(3t^2+2t+5)dt$ …………………(b)

The total charge flowing through the wire can be calculated by integrating the equation (b) over limits $t=0$ to $t=2s$

Integration equation (b), we get

$$\begin{gathered} \int dQ={\int }_0^2(3t^2+2t+5)dt \\ Q={\int }_0^2\frac{3t^3}{3}+\frac{2t^2}{2}+5t \\ Q={\left[t^3\right]}_0^2+{\left[t^2\right]}_0^2+5{\left[t\right]}_0^2 \\ Q={\left(2\right)}^3+{\left(2\right)}^2+5\left(2\right) \\ Q=8+4+10 \\ Q=22C \end{gathered}$$

Hence, option (a) is correct.