Electrical oscillations and AC circuits

An ideal oscillating circuit consists of an inductor of inductance $50\mathrm{mH}$ and a capacitor of capacitance $20\mathrm{nF}$. Compute the frequency of the oscillations.

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A capacitor of capacitance $5.0\mu \mathrm{F}$, charged to $20\mathrm{V}$, is connected to an inductor of inductance $0.20\mathrm{H}$. Neglecting the resistance of the circuit, compute the maximum value of the current.

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Compute the capacitance to be connected to an inductor of inductance $10\mathrm{mH}$ so that the $LC$ circuit oscillates at the frequency of $1.0\mathrm{kHz}$.

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In a circuit the current is $i(t)=2.5\sin (100\pi t)\mathrm{A}$, with $t$ in seconds. Compute the rms value and the frequency of the current.

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A capacitor of capacitance $47\mu \mathrm{F}$ is connected to the grid, with rms voltage $230\mathrm{V}$ and frequency $50\mathrm{Hz}$. Compute the rms current.

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An ideal inductor of inductance $0.15\mathrm{H}$ is connected to a source with rms voltage $120\mathrm{V}$ and frequency $60\mathrm{Hz}$. Compute the inductive reactance and the rms current.

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An $80\Omega$ resistor and an ideal $0.30\mathrm{H}$ inductor are connected in series to the grid, at $230\mathrm{V}$ rms and $50\mathrm{Hz}$. Compute the impedance, the rms current and the phase shift between voltage and current.

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A series $RLC$ circuit has, at the frequency of the source, $R=50\Omega$, $X_L=150\Omega$ and $X_C=30\Omega$, and is supplied with rms voltage $260\mathrm{V}$. Compute the impedance, the rms current and the phase shift.

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A series $RLC$ circuit has $R=20\Omega$, $L=0.40\mathrm{H}$ and $C=25\mu \mathrm{F}$, and is supplied by a source of rms voltage $10\mathrm{V}$ and adjustable frequency. Compute the resonant frequency and the rms current at resonance.

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A load connected to the grid at $230\mathrm{V}$ rms draws the rms current $5.0\mathrm{A}$ with power factor $0.80$. Compute the average power absorbed.

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A series $RLC$ circuit has $R=10\Omega$, $L=1.0\mathrm{mH}$ and $C=10\mathrm{nF}$. Compute the resonant angular frequency, the quality factor and the width of the bandwidth in angular frequency.

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The primary of an ideal transformer has $500$ turns and is connected to an rms voltage of $230\mathrm{V}$; the secondary has $2500$ turns and supplies a resistive load that draws the rms current $0.20\mathrm{A}$. Compute the rms voltage at the secondary and the rms current in the primary.

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A free $RLC$ circuit has $L=50\mathrm{mH}$, $C=2.0\mu \mathrm{F}$ and $R=100\Omega$. Compute the angular frequency of the damped oscillations and the time after which the energy of the circuit has decreased by a factor $e$, in the approximation $U(t)\approx U_0{e}^{-\frac{R}{L}t}$.

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