Question

A current through a wire depends on time as $i={\alpha }_{0}t+\beta t^2$ where ${\alpha }_0=20A/s$ and $\beta =8A{s}^{-2}$. Find the charge crossed through a section of the wire in $15s$.

• A. $2100C$
• B. $260C$
• C. $2250C$
• D. $11250C$

Answer 1

The correct option is D

$11250C$

Step 1: Given Data:

Equation of current is $i={\alpha }_{0}t+\beta t^2$

${\alpha }_0=20A/s$

$\beta =8A{s}^{-2}$

Time $t=15s$

Step 2: Calculating the charge flown:

Current $i$ is given as,

$i=\frac{dQ}{dt}$

$dQ=idt$

${\int }_0^{Q}dQ={\int }_0^{t}idt$

$Q={\int }_0^{15}\left({\alpha }_{0}t+\beta t^2\right)dt$

$={\left[{\alpha }_0\frac{t^2}{2}+\beta \frac{t^3}{3}\right]}_{t=15}-{\left[{\alpha }_0\frac{t^2}{2}+\beta \frac{t^3}{3}\right]}_{t=0}$

$=\left[20\times \frac{15\times 15}{2}+8\times \frac{15\times 15\times 15}{3}\right]-0$

$=2250+9000$

$=11250C$

Hence, option D is the correct answer.