Question

In an ac circuit consist of resistance(R) in series with inductor(L) or capacitor(C) the plot of current and source emf with time is shown in figure. If angular frequency of source is $100\pi$ rad $s^{-1}$ and value of R is $\frac{10\sqrt{3}}{\pi }\Omega$ then C or L is equal to.

• A. $\frac{100}{{\pi }^2}$mH
• B. $100\sqrt{3}\mu F$
• C. $1000\mu F$
• D. $1000$mH

Answer 1

The correct option is C $1000\mu F$

In the given graph,

$I=I_0\sin (\omega t+\varphi )$

At $t=0I=\frac{I_0}{2}$

$\frac{I_0}{2}=I_0\sin (\omega x_0+\varphi )$

$\sin \varphi =\frac{1}{2}$

$\varphi =\frac{\pi }{6}={\tan }^{-1}\frac{X}{R}$

$\frac{X}{R}=\frac{1}{\sqrt{3}}$

$X=\frac{R}{\sqrt{3}}=\frac{100\sqrt{3}}{\pi \sqrt{3}}=\frac{10}{\pi }$

$X=\frac{1}{\omega C}=\omega L$ but in this case current is leading therefore $X=\frac{1}{\omega C}$

$C=\frac{\pi }{100\omega }=\frac{\pi }{10\times 100\pi }=1000\mu F$