Given below are two electric circuits A and B.
Calculate the ratio of power factor of the circuit B to the power factor of circuit A.
CBSE DELHI - SET 1 2007
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$\text{cos}\phi = \frac{\text{R}}{\text{Z}}$Where $\text{Z} = \sqrt{\text{R}^{2}+ \big(\text{X}_{L} - {\text{X}_{C}}\big)^{2}}$
$\frac{\text{cos}\phi_B}{\text{cos}\phi_A}$ $ = \frac{\text{Z}_A}{\text{Z}_{B}}$ $=\frac{\sqrt{\text{R}^{2}_{A} + \text{X}}_{L_{A}}^{2}}{\sqrt{\text{R}_{B}^{2}+\big(\text{X}_{L_{B}}-\text{X}_{C_{B}}}\big)^{2}}$
$=\frac{\text{R}\sqrt{10}}{\text{R}\sqrt{5}} = \sqrt{2}$
$\frac{\text{cos}\phi_B}{\text{cos}\phi_A}$ $ = \frac{\text{Z}_A}{\text{Z}_{B}}$ $=\frac{\sqrt{\text{R}^{2}_{A} + \text{X}}_{L_{A}}^{2}}{\sqrt{\text{R}_{B}^{2}+\big(\text{X}_{L_{B}}-\text{X}_{C_{B}}}\big)^{2}}$
$=\frac{\text{R}\sqrt{10}}{\text{R}\sqrt{5}} = \sqrt{2}$
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