Alternating Current Set-1

Q1. The instantaneous current $i(t)$ and the capacitor voltage $v_C(t)$ are plotted for an AC circuit. At $t=0$ the current crosses zero going positive ($i(0)=0,\;di/dt>0$) while the capacitor voltage is at its positive maximum ($v_C(0)=V_{\max}$). Based on these waveforms, which statement is correct?

Q2. A series $RLC$ circuit has $R=10\ \Omega,\;L=0.20\ \text{H},\;C=50\ \mu\text{F}$ and is driven by $v(t)=100\sin(\omega t)\ \text{V}$. Find the resonant frequency $f_0$ (in Hz) and the amplitude (peak) of the steady‑state current at resonance.

Q3. Assertion (A): When a pure capacitor is connected to a sinusoidal source, it absorbs a positive average power over one complete cycle.
Reason (R): For a pure capacitor the current leads the voltage by $90^\circ$, so instantaneous power $p(t)=v(t)i(t)$ is a sinusoid (proportional to $\sin2\omega t$) that integrates to zero over a period.

Q4. A series $RL$ circuit is connected to an AC source of RMS voltage $130\ \text{V}$. The line RMS current is $2\ \text{A}$. Measured RMS voltages across $R$ and $L$ are $50\ \text{V}$ and $120\ \text{V}$ respectively. Which statement is correct?

Q5. The magnitude of impedance $|Z|$ of a series $RLC$ circuit versus frequency has a minimum $|Z|_{\min}=5\ \Omega$ at $f_0=5.00\ \text{kHz}$. The frequencies at which the current amplitude falls to $1/\sqrt2$ of its resonant value are $f_1=4.90\ \text{kHz}$ and $f_2=5.10\ \text{kHz}$. From this data, which of the following is correct (take $\omega_0=2\pi f_0$)?

Q6. A resistor $R=8\ \Omega$ in series with a capacitor $C=100\ \mu\text{F}$ is connected to an AC source of frequency $f=50\ \text{Hz}$. The line current is $i(t)=5\sqrt2\sin(\omega t)\ \text{A}$ (so $I_{\text{rms}}=5\ \text{A}$). The RMS voltage across the capacitor is

Q7. Assertion (A): At resonance in a series $RLC$ circuit, the power factor becomes zero.
Reason (R): At resonance the inductive and capacitive reactances cancel each other and the circuit behaves as purely resistive; hence current is in phase with the supply voltage and the power factor is unity.

Q8. An AC source supplies $V_{\text{rms}}=100\ \text{V}$ to a load drawing $I_{\text{rms}}=5\ \text{A}$. A wattmeter reads the average (real) power $P=300\ \text{W}$. Which of the following is correct?

Q9. A series $RLC$ circuit with $R=2.0\ \Omega,\ L=0.400\ \text{H}$ and $C=25.33\ \mu\text{F}$ is driven at resonance $f_0=50\ \text{Hz}$ by $v(t)=20\sqrt2\sin(\omega t)\ \text{V}$. The amplitude (peak) of the voltage across the inductor at resonance is approximately:

Q10. Assertion (A): In a parallel resonant (anti‑resonant) LC circuit connected to a sinusoidal source, the input impedance reaches a maximum at resonance and the current drawn from the source is minimum.
Reason (R): At resonance the currents through the inductive and capacitive branches are equal in magnitude and opposite in phase, so they largely cancel and only a small net current (determined by parallel resistance) is drawn from the source.