Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A series RLC circuit is driven by $\mathcal{E} = \mathcal{E}_0\cos(\omega t)$. Find the charge amplitude on $C$ as a function of $\omega$ and identify the resonance frequency. The charge amplitude: $$Q_0(\omega) = \frac{\mathcal{E}_0/L}{\sqrt{(\omega_0^2-\omega^2) Given Information See problem statement for all given quantities. Physical Concepts &…

Problem Statement Define the quality factor $Q$ for a damped harmonic oscillator. Express it in terms of (a) the natural frequency $\omega_0$ and damping coefficient $\beta$, and (b) energy stored vs. energy lost per cycle. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: An oscillating circuit resonates at $f_0 = 1.0$ MHz with bandwidth $\Delta f = 2.0$ kHz. Find $Q$, $L$ if $C = 100$ pF, and $R$. $$Q = \frac{f_0}{\Delta f} = \frac{1.0\times10^6}{2.0\times10^3} = 500$$ From $\omega_0 = 1/\sqrt{LC}$: $$L = \frac{1}{\omega_0^2 C} = Given Information…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: An RLC oscillating circuit has $R = 0.40$ $\Omega$, $L = 1.0$ mH, $C = 4.0$ $\mu$F. Find the number of oscillations before the charge amplitude falls by half. $$\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{10^{-3}\times4\times10^{-6}}} = \frac{10^3}{2} = 500\sq Given Information See problem statement for all…

Problem Statement Solve the capacitor/capacitance problem: Solve the capacitor/capacitance problem: A capacitor ($C = 2.0$ $\mu$F) is charged to $V_0 = 200$ V and connected at $t=0$ to an inductor $L = 5.0$ mH. Find $\omega_0$, the current at $t = T/4$, and the energy stored in $L$ at $t = T/4$. $$\omega_0 = \frac{1}{\sqrt{LC}}…

Problem Statement Solve the nuclear physics problem: Solve the oscillation/wave problem: A damped RLC circuit ($R$, $L$, $C$) starts with charge $Q_0$. Find the charge after exactly one half-period $T_1/2$. The charge: $Q(t) = Q_0 e^{-\beta t}\cos(\omega_1 t + \phi_0)$ where $\phi_0$ accounts for the initial condition $\dot{Q}(0) = 0$. With $Q(0) = Q_ Given…

Problem Statement Analyze the circuit: In an LC circuit, the charge on the capacitor is $Q = Q_0\cos(\omega_0 t)$. Write expressions for current, voltage across $L$, and voltage across $C$, and describe their phases. Given $Q(t) = Q_0\cos(\omega_0 t)$: $$I(t) = -\dot{Q} = Q_0\omega_0\sin(\omega_0 t) = I_0\sin(\omega_0 t)$$ Voltage across Given Information See problem statement…

Problem Statement Solve the oscillation/wave problem: A series RLC circuit is driven at resonance by $\mathcal{E}_0 = 10$ V, $R = 1.0$ $\Omega$, $L = 1.0$ mH, $C = 1.0$ $\mu$F. Find $I_0$, $V_L$, $V_C$, and power consumed. $$\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{10^{-3}\times10^{-6}}} = \frac{10^3}{1} = 10^3\sqrt{10} \approx 31{,}623 \text{ r Given Information See problem…

Problem Statement Solve the oscillation/wave problem: An RLC circuit ($R = 5.0$ $\Omega$, $L = 2.0$ mH, $C = 10$ $\mu$F) is given initial charge. Find $\omega_0$, $\omega_1$, $\beta$, $Q$, and $\lambda$. $$\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{2\times10^{-3}\times10\times10^{-6}}} = \frac{1}{\sqrt{2\times10^{-8}}} = \frac{10^4}{\sqrt{2}} \app Given Information See problem statement for all given quantities. Physical Concepts & Formulas…

Problem Statement Solve the oscillation/wave problem: A charged capacitor ($C = 1.0$ $\mu$F, $V_0 = 100$ V) is connected to an uncharged inductance ($L = 1.0$ mH). Find the maximum current and the period. This is pure LC oscillation (no resistance). Energy conservation: $$\frac{1}{2}CV_0^2 = \frac{1}{2}LI_{\max}^2$$ $$I_{\max} = V_0\sqrt{\frac{C}{L}} = Given Information See problem statement…