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(A) The power factor of an $LR$ circuit is given by $\cos \phi_{LR} = \frac{R}{Z_{LR}} = \frac{R}{\sqrt{R^2 + X_L^2}}$.Given $X_L = R$,we have $\cos \

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$1: 1$

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(A) The power factor of an $LR$ circuit is given by $\cos \phi_{LR} = \frac{R}{Z_{LR}} = \frac{R}{\sqrt{R^2 + X_L^2}}$.Given $X_L = R$,we have $\cos \phi_{LR} = \frac{R}{\sqrt{R^2 + R^2}} = \frac{R}{R\sqrt{2}} = \frac{1}{\sqrt{2}}$.The power factor of an $LCR$ circuit is given by $\cos \phi_{LCR} = \frac{R}{Z_{LCR}} = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}}$.Given $X_L = R$ and $X_C = 2R$,we have $\cos \phi_{LCR} = \frac{R}{\sqrt{R^2 + (R - 2R)^2}} = \frac{R}{\sqrt{R^2 + (-R)^2}} = \frac{R}{R\sqrt{2}} = \frac{1}{\sqrt{2}}$.The ratio of the power factor of the $LR$ circuit to the $LCR$ circuit is $\frac{1/\sqrt{2}}{1/\sqrt{2}} = 1:1$.

$A$ resistor of resistance $R$ and an inductor of inductive reactance $R$ are connected in series to an $AC$ source. $A$ capacitor of capacitive reactance $2R$ is then connected in series with $L$ and $R$. The ratio of the power factors of the $LR$ and $LCR$ circuits is:

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