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  4. A current through a wire depends on time as i =α0 t +β t 2 where α0=20 A / s and β=8 As -2 . Find the charge crossed through a section of the wire in 15 s.

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

JEE MainJEE Main 2021Current Electricity

Solution:

$i =20 t +8 t ^{2}$
$i =\frac{ dq }{ dt } $
$\Rightarrow \int d q =\int idt$
$\Rightarrow q =\int\limits_{0}^{15}\left(20 t +8 t ^{2}\right) dt$
$q =\left(\frac{20 t ^{2}}{2}+\frac{8 t ^{3}}{3}\right)_{0}^{15}$
$q =10 \times(15)^{2}+\frac{8(15)^{3}}{3}$
$q =2250+9000$
$q =11250 C$