Physics - EMI/AC Question with Solution | TestHub

For current to lag voltage by $\pi /4$, the phase angle $\varphi =\pi /4$, so $\tan \varphi =1$, which means $X_L-X_C=R$. For power factor $\cos \varphi =0.5$, $\varphi =\pm \pi /3$, so $|\tan \varphi |=\sqrt{3}$, implying $|X_L-X_C|=R\sqrt{3}$. Voltage across inductor equals voltage across capacitor implies $V_L=V_C$, so $X_L=X_C$, which is the resonance condition $\omega =\frac{1}{\sqrt{LC}}$. Maximum average power is dissipated at resonance, where $Z=R$, so $P_{max}=\frac{V_{rms}^2}{R}=\frac{(V_0/\sqrt{2}{)}^2}{R}=\frac{V_0^2}{2R}$.