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The instantaneous values of alternating current and voltages in a circuit are given as $i = \frac{1}{\sqrt2}\sin(100\pi t)$ ampere and $e = \frac{1}{\sqrt2}\sin\left(100\pi t + \frac{\pi}{3}\right)$ volt. The average power in Watts consumed in the circuit is
A
$\frac{1}{8}$
B
$\frac{1}{4}$
C
$\frac{\sqrt3}{4}$
D
$\frac{1}{2}$
Detailed Solution
$P = V_{rms}I_{rms}\cos\phi = \frac{1}{2}V_0I_0\cos\phi$
$V_0 = \frac{1}{\sqrt2}$ V, $I_0 = \frac{1}{\sqrt2}$ A, $\phi = \frac{\pi}{3}$
$P = \frac{1}{2}\times\frac{1}{\sqrt2}\times\frac{1}{\sqrt2}\times\cos\frac{\pi}{3} = \frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}$
$P = \frac{1}{8}$ W
$V_0 = \frac{1}{\sqrt2}$ V, $I_0 = \frac{1}{\sqrt2}$ A, $\phi = \frac{\pi}{3}$
$P = \frac{1}{2}\times\frac{1}{\sqrt2}\times\frac{1}{\sqrt2}\times\cos\frac{\pi}{3} = \frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}$
$P = \frac{1}{8}$ W