Category: Part 4: Oscillations & Waves

Problem Statement Analyze the circuit: An RLC circuit has initial charge $Q_0$. Find the condition on $R$, $L$, $C$ for oscillations to occur, and the minimum $R$ for purely exponential decay. The characteristic equation: $Lp^2 + Rp + 1/C = 0$, roots $p = -R/(2L) \pm \sqrt{R^2/(4L^2) – 1/(LC)}$. Underdamped (oscillatory): $\beta^2 $$\box Given Information…

Problem Statement Analyze the circuit: Find the impedance of a parallel LC circuit and the frequency at which it becomes infinite (anti-resonance). For ideal parallel LC (no resistance), the admittances add: $$Y = Y_L + Y_C = \frac{1}{i\omega L} + i\omega C = i\left(\omega C – \frac{1}{\omega L}\right)$$ Impedance: $$Z = \frac{1}{Y} = \f Given…

Problem Statement Solve the oscillation/wave problem: In a series RLC circuit at resonance, compare energies stored in $L$ and $C$ as functions of time. Show total stored energy is constant. At resonance, $Q(t) = Q_0\cos(\omega_0 t)$ and $I(t) = I_0\sin(\omega_0 t)$ where $I_0 = \omega_0 Q_0$. Electric energy: $U_C = Q^2/(2C) = \frac{Q_0^2}{2C}\cos^2(\ Given Information…

Problem Statement Analyze the circuit: In an AC circuit with impedance $Z$ and phase angle $\psi$, find the average power consumed and define the power factor. Instantaneous power: $p(t) = V(t)I(t) = V_0\cos(\omega t)\cdot I_0\cos(\omega t – \psi)$ $$p(t) = \frac{V_0 I_0}{2}[\cos\psi + \cos(2\omega t – \psi)]$$ Time-averaging (the oscill Given Information See problem statement…

Problem Statement Analyze the circuit: An RLC series circuit is driven by EMF $\mathcal{E} = \mathcal{E}_0\cos(\omega t)$. Find the steady-state current amplitude and phase. The circuit equation: $$L\ddot Q + R\dot Q + \frac{Q}{C} = \mathcal{E}_0\cos(\omega t)$$ For current $I = I_0\cos(\omega t – \psi)$, using impedance $Z = R + i(X_L – Given Information…

Problem Statement Analyze the circuit: At resonance in a series RLC circuit, find the voltage across $L$ and $C$ relative to the source voltage. At resonance $\omega = \omega_0 = 1/\sqrt{LC}$, current $I_0 = \mathcal{E}_0/R$. Voltage across $L$: $V_L = I_0 \cdot \omega_0 L = \mathcal{E}_0 \omega_0 L/R = Q\mathcal{E}_0$ Voltage across $C$ Given Information…

Problem Statement Analyze the circuit: An RLC series circuit has $R$, $L$, $C$. Initially $Q(0) = Q_0$, $I(0) = 0$. Find $Q(t)$ for underdamped case. Kirchhoff’s law: $$L\ddot Q + R\dot Q + \frac{Q}{C} = 0$$ Defining $\beta = R/(2L)$ and $\omega_0 = 1/\sqrt{LC}$, for $\beta $$\omega_1 = \sqrt{\omega_0^2 – \beta^2} = \sqrt{\frac{1}{LC} – Given…

Problem Statement Analyze the circuit: For a damped RLC circuit, find the logarithmic decrement $\lambda$ and quality factor $Q$ in terms of $R$, $L$, $C$. With $\beta = R/(2L)$ and $\omega_1 = \sqrt{1/(LC) – R^2/(4L^2)}$: $$\lambda = \beta T_1 = \frac{R}{2L} \cdot \frac{2\pi}{\omega_1} = \frac{\pi R}{L\omega_1}$$ For small damping ($R\s Given Information See problem statement…

Problem Statement Analyze the circuit: Write the complete mechanical–electrical analogy for an LC circuit and a spring-mass system. Comparing $m\ddot{x} + kx = 0$ and $L\ddot{Q} + Q/C = 0$: Mechanical Electrical Mass $m$ Inductance $L$ Spring constant $k$ $1/C$ (elastance) Displacement $x$ Charge $Q$ Velocity $\dot x$ Current $I = \dot Q Given Information…

Problem Statement Analyze the circuit: An inductor $L$ and capacitor $C$ form a closed circuit. Initially $C$ is charged to $Q_0$ and no current flows. Describe the subsequent oscillation. Kirchhoff’s voltage law around the loop: $$L\frac{dI}{dt} + \frac{Q}{C} = 0, \quad I = -\dot Q$$ $$L\ddot Q + \frac{Q}{C} = 0 \implies \ddot Q +…