Ta có:
$Z_{L}Z_{C}=\frac{\omega L}{\omega C}=\frac{L}{C}=R^{2}$
Theo bài ra:
$U_{RL}=\sqrt{3}U_{RC}$
$\Rightarrow Z_{RL}=\sqrt{3}Z_{RC}$
$\Rightarrow R^{2}+Z_{L}^{2}=3(R^{2}+Z_{C}^{2})$
$\Rightarrow 3Z_{C}^{2}+2R^{2}-Z_{L}^{2}=0$
$\Rightarrow 3Z_{C}^{2}+2Z_{L}Z_{C}-Z_{L}^{2}=0$
$\Rightarrow 3Z_{C}(Z_{C}+Z_{L})-Z_{L}(Z_{C}+Z_{L})=0$
$\Rightarrow (Z_{L}+Z_{C})(3Z_{C}-Z_{L})=0$
Suy ra: $Z_{L}=3Z_{C}$
Lại có: $Z_{L}Z_{C}=R^{2}$ suy ra: $3Z_{C}^{2}=R^{2}\Rightarrow Z_{C}^{2}=\frac{R^{2}}{3}$
Hệ số công suất:
$cos\varphi=\frac{R}{\sqrt{R^{2}+(Z_{L}-Z_{C})^{2}}}$
$=\frac{R}{\sqrt{R^{2}+4Z_{C}^{2}}}$
$= \frac{R}{\sqrt{R^{2}+\frac{4R^{2}}{3}}}=\sqrt{\frac{3}{7}}$