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 $15 \,s$ (in $C$)
JEE MAIN 2021, Medium
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$i =20 t +8 t ^{2}$
$i =\frac{ dq }{ dt } \Rightarrow \int d q =\int idt$
$\Rightarrow q =\int_{0}^{15}\left(20 t +8 t ^{2}\right) dt$
$q =\left(\frac{20 t ^{2}}{2}+\frac{8 t ^{3}}{3}\right)_{0}^{15}$
$q =10 \times(15)^{2}+\frac{8(15)^{3}}{3}$
$q =2250+9000$
$q =11250 C$
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