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(B) For an $LR$ series circuit,the time constant is given by $	au = \frac{L}{R}$.When the battery is replaced by an $AC$ source,the impedance $Z$ of

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$\frac{1}{\sqrt{1 + (\omega 	au)^2}}$

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(B) For an $LR$ series circuit,the time constant is given by $	au = \frac{L}{R}$.When the battery is replaced by an $AC$ source,the impedance $Z$ of the $LR$ circuit is $Z = \sqrt{R^2 + X_L^2} = \sqrt{R^2 + (\omega L)^2}$.The power factor $\cos \phi$ is defined as $\cos \phi = \frac{R}{Z}$.Substituting the expression for $Z$,we get $\cos \phi = \frac{R}{\sqrt{R^2 + \omega^2 L^2}}$.Dividing the numerator and denominator by $R$,we get $\cos \phi = \frac{1}{\sqrt{1 + \omega^2 (\frac{L}{R})^2}}$.Since $	au = \frac{L}{R}$,substituting this into the equation gives $\cos \phi = \frac{1}{\sqrt{1 + \omega^2 	au^2}}$.

The time constant of the given $LR$ circuit is $\tau$. If the battery is replaced by an $AC$ source having voltage $V = V_0 \cos \omega t$,the power factor of the circuit will be

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