Question

A sinusoidal voltage V(t) = 210 sin 3000 t volt is applied to a series LCR circuit in which L = 10 mH, C = 25 μF and R = 100 Ω. The phase difference (Φ) between the applied voltage and resultant current will be:

• A. tan^–1(0.17)
• B. tan^–1(9.46)
• C. tan^–1(0.30)
• D. tan^–1(13.33)

Answer 1

The Correct Option is A

1. Given the sinusoidal voltage V(t) = 210 \sin 3000t volts, we need to determine the phase difference \Phi between the applied voltage and the resultant current in the series LCR circuit.
2. Key parameters of the circuit are:
   • Inductance L = 10 \text{ mH} = 0.01 \text{ H}
   • Capacitance C = 25 \mu\text{F} = 25 \times 10^{-6} \text{ F}
   • Resistance R = 100 \Omega
3. The angular frequency \omega of the voltage is 3000 \text{ rad/s}.
4. Calculate the inductive reactance X_L: X_L = \omega L = 3000 \times 0.01 = 30 \, \Omega.
5. Calculate the capacitive reactance X_C: X_C = \frac{1}{\omega C} = \frac{1}{3000 \times 25 \times 10^{-6}} = \frac{1}{0.075} \approx 13.33 \, \Omega.
6. The net reactance X\_net is given by: X\_net = X_L - X_C = 30 - 13.33 = 16.67 \, \Omega.
7. The phase difference \Phi is determined by the formula: \tan\Phi = \frac{X\_net}{R} = \frac{16.67}{100} = 0.1667.
8. Thus, the phase difference \Phi is: \tan^{-1}(0.17), which matches the correct answer.

Conclusion: The phase difference between the applied voltage and the resultant current is \tan^{-1}(0.17).