At any instant t the values of voltage applied and the current through a series LCR circuit are given as:
\( V=\displaystyle\frac{1}{\sqrt{2}}\sin(100\pi t)\,\mathrm{V} \)
\( I=\displaystyle\frac{1}{\sqrt{2}}\sin\left(100\pi t+\frac{\pi}{3}\right)\mathrm{A} \)
The average power consumed in the circuit is given by:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{1}{8}\,\mathrm{W} \)
The average power in an a.c. circuit is \( P = V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi \), where \( \phi \) is the phase difference between voltage and current. The given values are the peak values, \( V_0 = I_0 = \frac{1}{\sqrt{2}} \), so convert them to rms values first.
\( \displaystyle V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}} = \frac{1}{\sqrt{2}\times\sqrt{2}} = \frac{1}{2}\,\mathrm{V},\qquad I_{\mathrm{rms}} = \frac{1}{2}\,\mathrm{A} \)
The current leads the voltage by \( \phi = \frac{\pi}{3} \), so \( \cos\phi = \cos60^{\circ} = \frac{1}{2} \).
\( \displaystyle P = \frac{1}{2}\times\frac{1}{2}\times\frac{1}{2} = \frac{1}{8}\,\mathrm{W} \)