An alternating current is represented by the equation, $i=100 \sqrt{2} \sin (100 \pi t)$ ampere. The RMS value of current and the frequency of the given alternating current are

<p>To solve the problem, we need to determine both the RMS value of the current and the frequency from the given equation:</p> <p>$i(t)=100\sqrt{2}\sin(100\pi t).$</p> <p>Here's how we do it step by step:</p> <p><strong>RMS Current Calculation:</strong></p> <p><p>For a sinusoidal current of the form $i(t)=I_0 \sin(\omega t),$ the RMS (root-mean-square) current is given by:</p> <p>$I_{\text{RMS}} = \frac{I_0}{\sqrt{2}}.$</p></p> <p><p>In our case, the amplitude $I_0 = 100\sqrt{2}\, \text{A}.$ Plugging into the formula:</p> <p>$I_{\text{RMS}} = \frac{100\sqrt{2}}{\sqrt{2}} = 100\, \text{A}.$</p></p> <p><strong>Frequency Calculation:</strong></p> <p><p>The argument of the sine function is $100\pi t.$ In the general form $\sin(\omega t),$ $\omega$ is the angular frequency which relates to the frequency $f$ by the equation:</p> <p>$\omega = 2\pi f.$</p></p> <p><p>Given $\omega = 100\pi,$ we can solve for $f$:</p> <p>$f = \frac{\omega}{2\pi} = \frac{100\pi}{2\pi} = 50\, \text{Hz}.$</p></p> <p>With these calculations, we find that the RMS current is $100\, \text{A}$ and the frequency is $50\, \text{Hz}.$ </p> <p>Thus, the correct option is:</p> <p>Option D: $100\,\text{A}, \, 50\,\text{Hz}.$</p>