Oscillations in LC and RLC Circuits · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Charge on a spring: LC ringing and RLC damping

Science · Electricity & Magnetism · ages 20-21
Name ______________________   Date ____________

  1. Compare L q'' + q/C = 0 with m x'' + k x = 0. What corresponds to inductance L?

    • The spring stiffness k
    • The inertial mass m
    • The damping constant b

  2. What is the natural angular frequency omega0 of an ideal source-free LC circuit?

    • 1 / sqrt(L C)
    • 1 / sqrt(L / C)
    • 1 / sqrt(C / L)

  3. Adding positive resistance to a source-free LC loop allows its nonzero oscillations to continue forever at constant amplitude.

    Circle one:   True   False

  4. Compare source-free series RLC circuits with the same L, C, nonzero initial capacitor charge and zero initial current. Which regime gives the fastest nonoscillatory return toward zero charge?

    • Critical damping, at R = 2 sqrt(L/C)
    • Light damping, at R < 2 sqrt(L/C)
    • Heavy damping, at R > 2 sqrt(L/C)

  5. A nonzero ideal LC oscillation reaches an instant with capacitor charge q = 0. Where is its stored energy?

    • Stored in the capacitor, with zero current
    • Dissipated in a resistor, with zero current
    • Stored in the inductor, with peak current magnitude

  6. Choose the LC reference direction so I = dq/dt. What mechanical quantity corresponds to current in the spring analogy?

    • Displacement, the distance from equilibrium
    • Velocity, the rate of change of position
    • Acceleration, the rate of change of velocity

  7. In an ideal source-free LC loop, vL + vC = 0, vL = L dI/dt, vC = q/C and I = dq/dt. Which normalized charge equation follows?

    • q'' + q/(L C) = 0
    • q'' + (R/L) q' + q/(L C) = 0
    • q' + q/(L C) = 0

  8. A student says resistance wastes no energy and always leaves free RLC oscillations near omega0. Which correction is accurate?

    • Resistance stores energy; any resistance raises oscillation frequency
    • Resistance supplies energy; enough resistance sustains oscillation
    • Resistance dissipates energy as heat; enough resistance prevents oscillation
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Charge on a spring: LC ringing and RLC damping W1-mt_QJsqX7xzkZ-s1

  1. The inertial mass m · L multiplies the second derivative of charge, just as m multiplies the second derivative of displacement.
  2. 1 / sqrt(L C) · Comparing q'' + q/(LC) = 0 with q'' + omega0 squared q = 0 gives omega0 = 1/sqrt(LC).
  3. False · Resistance dissipates energy at rate R I squared. The free response loses stored energy instead of maintaining constant amplitude.
  4. Critical damping, at R = 2 sqrt(L/C) · For this common initial state, critical damping is the boundary giving the fastest nonoscillatory return. Greater resistance slows the return.
  5. Stored in the inductor, with peak current magnitude · The capacitor energy q squared/(2C) is zero. Conserved energy is entirely L I squared/2, so current magnitude is greatest.
  6. Velocity, the rate of change of position · Charge corresponds to displacement. Their time derivatives, current and velocity, therefore correspond.
  7. q'' + q/(L C) = 0 · Substitution gives L q'' + q/C = 0. Dividing by L gives q'' + q/(LC) = 0, with no resistor term.
  8. Resistance dissipates energy as heat; enough resistance prevents oscillation · Positive resistance dissipates energy at R I squared. For 0 < R < 2 sqrt(L/C), omega_d is below omega0 and close only for weak damping. At or above critical resistance, the response is nonoscillatory.
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