The average power ($P$) consumed in an AC circuit is determined by the RMS values of voltage and current and the cosine of the phase difference (power factor). The formula is $P = V_{rms} I_{rms} \cos \phi$ .

1. Identify Peak Values: From the given equations $i = I_m \sin(\omega t)$ and $e = V_m \sin(\omega t + \phi)$, we have peak current $I_m = 1/\sqrt{2}$ A and peak voltage $V_m = 1/\sqrt{2}$ V.
2. Calculate RMS Values: $I_{rms} = \frac{I_m}{\sqrt{2}} = \frac{1/\sqrt{2}}{\sqrt{2}} = \frac{1}{2}$ A. $V_{rms} = \frac{V_m}{\sqrt{2}} = \frac{1/\sqrt{2}}{\sqrt{2}} = \frac{1}{2}$ V.
3. Identify Phase Difference: The phase difference $\phi$ is given as $\pi/3$ radians (or $60^\circ$).
4. Calculate Power: $P = (\frac{1}{2}) (\frac{1}{2}) \cos(\frac{\pi}{3})$ $P = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$ W.