Question

In an ac circuit, the current is given by $i=4\sin (100\pi t+{30}^{\circ })$ ampere. The first time when the current becomes maximum (after $t=0$) at $t$ is equal to -

Answer 1

Given,
$i=4\sin (100\pi t+{30}^o)$ ........(1)

From the above equation, the maximum value of current is given by, $i_0=4$

Substituting $i=4$ in (1), we get

$\sin (100\pi t+{30}^o)=1$

$\Rightarrow 100\pi t+\frac{\pi }{6}=\frac{(2n+1)\pi }{2}$

For first time, $n=0$

$\Rightarrow 100\pi t=\frac{\pi }{3}$

$\Rightarrow t=\frac{1}{300}\text{sec}$

Alter:

$\frac{di}{dt}=400\pi \cos (100\pi t+\frac{\pi }{6})$

For maximum, $\frac{di}{dt}=0$

$\cos (100\pi t+\frac{\pi }{6})=0$

$\Rightarrow 100\pi t+\frac{\pi }{6}=\frac{(2n+1)\pi }{2}$

For first time, $n=0$

$\Rightarrow 100\pi t=\frac{\pi }{3}$

$\Rightarrow t=\frac{1}{300}\text{sec}$