Question

For the given figures, choose the correct options:

• A. At resonance, current in (b) is less than that in (a)
• B. The rms current in circuit (b) can never be larger than that in (a)
• C. The rms current in figure(a) is always equal to that in figure (b)
• D. The rms current in circuit (b) can be larger than that in (a)

Answer 1

The Correct Option is B

To solve this problem, we need to analyze the two circuits provided in the images:

1. Circuit (a): This circuit consists of a simple resistor of 40 \, \Omega connected to an AC source of 220 \, \text{V}, 50 \, \text{Hz}.
2. Circuit (b): This circuit consists of a resistor, inductor, and capacitor in series. The resistor is 40 \, \Omega, the inductor is 50 \, \text{mH}, and the capacitor is 0.5 \, \mu\text{F} connected to the same AC source of 220 \, \text{V}, 50 \, \text{Hz}.

Now, let's calculate and compare the RMS currents:

1. RMS Current in Circuit (a): I_a = \frac{V}{R} = \frac{220 \, \text{V}}{40 \, \Omega} = 5.5 \, \text{A}
2. Impedance of Circuit (b): The impedance Z_b in a series circuit with resistance R, inductance L, and capacitance C is calculated as: Z_b = \sqrt{R^2 + (X_L - X_C)^2}, where X_L = 2\pi f L and X_C = \frac{1}{2 \pi f C}.
3. Calculations for Circuit (b):
   • X_L = 2 \pi \times 50 \times 0.05 = 15.7 \, \Omega
   • X_C = \frac{1}{2 \pi \times 50 \times 0.5 \times 10^{-6}} \approx 636.6 \, \Omega
   • Z_b = \sqrt{40^2 + (15.7 - 636.6)^2} \approx 636.18 \, \Omega
4. RMS Current in Circuit (b): I_b = \frac{V}{Z_b} = \frac{220 \, \text{V}}{636.18 \, \Omega} \approx 0.35 \, \text{A}

Comparison and Conclusion:

• The RMS current in circuit (b) is much smaller compared to circuit (a) due to the high impedance caused by the resonance effects.
• Therefore, the option "The RMS current in circuit (b) can never be larger than that in (a)" is the correct choice.