Question

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

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\frac{1}{\sqrt{5}}$
• D. $\frac{1}{\sqrt{7}}$

Answer 1

Answer: (C)

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

Answer 2

Step 1: Determine Initial Reactance $X_L$: Since the initial power factor $\cos \theta = \frac{1}{\sqrt{2}}$, we have $\tan \theta = 1$, meaning $X_L = R$. With the initial angular frequency $\omega_1 = 1000 \, \text{rad/s}$, we can write $X_L = \omega_1 L = R$.

Step 2: Calculate Reactance at New Frequency: For the new frequency, $\omega_2 = 2000 \, \text{rad/s}$, the new inductive reactance becomes:

$X'_L = \omega_2 L = 2 \omega_1 L = 2R$

Step 3: Determine New Power Factor: With the new reactance $X'_L = 2R$, we find $\tan \theta' = \frac{X'_L}{R} = 2$, which gives:

$\cos \theta' = \frac{1}{\sqrt{1 + (2)^2}} = \frac{1}{\sqrt{5}}$

Conclusion: The new power factor is therefore:

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