# a. Write down a Kirchoff loop equation for the circuit when the switch is flipped to on. **Step 1:** Apply Kirchhoff’s voltage law (sum of voltage drops = ) for the loop: - Voltage across resistor: \( V_R = IR \) - Voltage across inductor: \( V_L = L \frac{dI}{dt} \) - Voltage across capacitor: \( V_C = \frac{q}{C} \) The sum is: \[ IR + L\frac{dI}{dt} + \frac{q}{C} = \] **:** This equation states that the total voltage drop across all three elements (R, L, and C) in the series circuit is zero, which matches Kirchhoff's loop. --- # b. Take the equation from part a and express it solely in terms of \( R, L, C, q, \) and \( t \). **Step 2:** Recall that current \( I = \frac{dq}{dt} \): \[ R\frac{dq}{dt} + L\frac{d^2q}{dt^2} + \frac{q}{C} = \] **Explanation:** By substituting \( I = \frac{dq}{dt} \) and \( \frac{dI}{dt} = \frac{d^2q}{dt^2} \), the equation is rewritten entirely in terms of charge \( q \), time \( t \), and the circuit parameters. --- # c. Write down a conservation of energy equation for the circuit when the switch is flipped to on. \[ \frac{q^2}{2C} + \frac{1}{2}LI^2 \] Energy lost by resistor = energy dissipated as heat: \[ \frac{q^2}{2C} + \frac{1}{2}LI^2 + \int_^t I^2R\,dt = \text{constant} \] **Explanation:** Energy is stored in the capacitor and inductor, and is dissipated as heat in the resistor. The sum remains constant. --- # d. Take the time derivative of the equation in part c. Differentiate: \[ \frac{d}{dt} \left( \frac{q^2}{2C} + \frac{1}{2}L I^2 + \int_^t I^2R\,dt \right) = \] \[ \frac{q}{C} \frac{dq}{dt} + L I \frac{dI}{dt} + I^2 R = \] **Explanation:** Use the chain rule for differentiation. \( \frac{dq}{dt} = I \). --- # e. Take the equation from part d and express it solely in terms of \( R, L, C, q, t \). Substitute \( I = \frac{dq}{dt} \), \( \frac{dI}{dt} = \frac{d^2q}{dt^2} \): \[ \frac{q}{C} \frac{dq}{dt} + L \frac{dq}{dt} \frac{d^2q}{dt^2} + R \left(\frac{dq}{dt}\right)^2 = \] **Explanation:** Every term is now in terms of \( q \), its derivatives, and the circuit parameters. --- # f. Draw a graph of what you expect the charge on the capacitor to be with respect to time. - **With \( R > \):** The charge oscillates and exponentially decays to zero. - **Graph:** Damped oscillation (decaying sine wave). --- # g. Draw the same graph but with what you would expect if \( R = \). - **With \( R = \):** The charge oscillates forever (no decay). - **Graph:** Pure sinusoidal oscillation (constant amplitude sine wave). --- ## **Summary Table** | Condition | Graph Description | |-------------|---------------------------------------| | \( R > \) | Damped oscillations (decay to zero) | | \( R = \) | Undamped oscillations (never decay) | --- **Final Answer:** For a series RLC circuit, the charge \( q(t) \) follows a second-order differential equation. If resistance is present, the solution is a decaying oscillation. If resistance is zero, the solution is a perpetual oscillation.

# a. Write down a Kirchhoff loop equation for the circuit when the switch is flipped to on. **Step 1:** Using Kirchhoff’s voltage law (sum of voltage drops = 0) for the loop, we have: - Voltage across the resistor: \( V_R = IR \) - Voltage across the inductor: \( V_L = L \frac{dI}{dt} \) - Voltage across the capacitor: \( V_C = \frac{q}{C} \) The equation is: \[ V_R + V_L + V_C = 0 \implies IR + L\frac{dI}{dt} + \frac{q}{C} = 0 \] **Explanation:** This equation states that the total voltage drop across the resistor, inductor, and capacitor in the series circuit equals zero, which aligns with Kirchhoff's loop law. --- # b. Take the equation from part a and express it solely in terms of \( R, L, C, q, \) and \( t \). **Step 2:** Recall that current \( I = \frac{dq}{dt} \): \[ R\frac{dq}{dt} + L\frac{d^2q}{dt^2} + \frac{q}{C} = 0 \] **Explanation:** By substituting \( I = \frac{dq}{dt} \) and \( \frac{dI}{dt} = \frac{d^2q}{dt^2} \), we rewrite the equation using only charge \( q \) and its derivatives. --- # c. Write down a conservation of energy equation for the circuit when the switch is flipped to on. \[ \frac{q^2}{2C} + \frac{1}{2}LI^2 = \text{constant} \] Energy lost by the resistor (dissipated as heat): \[ \frac{q^2}{2C} + \frac{1}{2}LI^2 + \int_0^t I^2R\,dt = \text{constant} \] **Explanation:** The energy stored in the capacitor and inductor is equal to the total energy at any point, accounting for energy dissipated in the resistor. --- # d. Take the time derivative of the equation in part c. Differentiate the conservation of energy equation: \[ \frac{d}{dt} \left( \frac{q^2}{2C} + \frac{1}{2}L I^2 + \int_0^t I^2R\,dt \right) = 0 \] This results in: \[ \frac{q}{C} \frac{dq}{dt} + L I \frac{dI}{dt} + I^2 R = 0 \] **Explanation:** Using the chain rule, we differentiate each term with respect to time, recognizing that \( \frac{dq}{dt} = I \). --- # e. Take the equation from part d and express it solely in terms of \( R, L, C, q, t \). Substituting \( I = \frac{dq}{dt} \) and \( \frac{dI}{dt} = \frac{d^2q}{dt^2} \): \[ \frac{q}{C} \frac{dq}{dt} + L \frac{dq}{dt} \frac{d^2q}{dt^2} + R \left( \frac{dq}{dt} \right)^2 = 0 \] **Explanation:** All terms are expressed in terms of charge \( q \), its derivatives, and the circuit parameters. --- # f. Draw a graph of what you expect the charge on the capacitor to be with respect to time. - **With \( R > 0 \):** The charge oscillates and exhibits exponential decay. - **Graph:** Damped oscillation (decaying sine wave). --- # g. Draw the same graph but with what you would expect if \( R = 0 \). - **With \( R = 0 \):** The charge oscillates indefinitely without decay. - **Graph:** Pure sinusoidal oscillation (constant amplitude sine wave). --- ## **Summary Table** | Condition | Graph Description | |-------------|---------------------------------------| | \( R > 0 \) | Damped oscillations (decay to zero) | | \( R = 0 \) | Undamped oscillations (never decay) | --- **Final Answer:** The charge \( q(t) \) in a series RLC circuit behaves according to a second-order differential equation. If resistance is present, the solution represents a decaying oscillation; without resistance, it results in perpetual oscillation.