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(C) For a series $L, R$ circuit,the power factor is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X_L^2}}$,where $X_L = \omega L$.Given the

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$\frac{1}{\sqrt{5}}$

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(C) For a series $L, R$ circuit,the power factor is given by $\cos \phi = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X_L^2}}$,where $X_L = \omega L$.Given the initial angular frequency $\omega_1 = 1000 \ rad/s$ and power factor $\cos \phi_1 = \frac{1}{\sqrt{2}}$.Since $\cos \phi_1 = \frac{1}{\sqrt{2}}$,the phase angle $\phi_1 = 45^{\circ}$.Thus,$	an \phi_1 = \frac{X_{L1}}{R} = \frac{\omega_1 L}{R} = 	an 45^{\circ} = 1$.This implies $R = \omega_1 L = 1000 L$.When the source is changed to $E = (20 \sin 2000 t) \ V$,the new angular frequency is $\omega_2 = 2000 \ rad/s$.The new inductive reactance is $X_{L2} = \omega_2 L = 2000 L = 2(\omega_1 L) = 2R$.The new power factor is $\cos \phi_2 = \frac{R}{\sqrt{R^2 + X_{L2}^2}} = \frac{R}{\sqrt{R^2 + (2R)^2}} = \frac{R}{\sqrt{R^2 + 4R^2}} = \frac{R}{\sqrt{5R^2}} = \frac{1}{\sqrt{5}}$.

$A$ series $L, R$ circuit connected with an ac source $E = (25 \sin 1000 t) \ V$ has a power factor of $\frac{1}{\sqrt{2}}$. If the source of emf is changed to $E = (20 \sin 2000 t) \ V$,the new power factor of the circuit will be:

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