Question

The instantaneous value of current in an \(\mathrm{AC}\). circuit is \(\mathrm{I}=2 \sin [100 \pi \mathrm{t}+(\pi / 3)] \mathrm{A}\). The current will be maximum for the first time at, (a) \(\mathrm{t}=(1 / 100) \mathrm{sec}\) (b) \(\mathrm{t}=(1 / 200) \mathrm{sec}\) (c) \(t=(1 / 400) \mathrm{sec}\) (d) \(t=(1 / 600) \mathrm{sec}\)

Short answer

The current will be maximum for the first time when \(t=\frac{1}{600}\) sec. So, the correct answer is (d) \(t=(1 / 600) sec\).

Step-by-step solution

Set up the equation for maximum current

We know that the current is maximum when the sine function is maximum. So, we will set up the equation as follows: \(100\pi t + (\pi/3) = \pi/2 + 2n\pi\)

Solve for t

To solve for t, we will isolate t in the equation as follows: \(100\pi t = \pi/2 + 2n\pi - \pi/3\) Now, divide both sides by \(100\pi\): \(t = \frac{\pi/2 + 2n\pi - \pi/3}{100\pi}\)

Find the smallest positive value of t

To get the smallest positive value of t, we can set n = 0 then solve for t: \(t = \frac{\pi/2 - \pi/3}{100\pi} = \frac{\pi/6}{100\pi} = \frac{1}{600} sec\) So, the current will be maximum for the first time at t = 1/600 sec, and the correct answer is (d) \(t=(1 / 600) sec\).