§ 26. Electromagnetic Oscillations in an LC / RLC Circuit (Grade 11)
Free electromagnetic oscillations between capacitor C and inductor coil L, Thomson period. In an oscillating tank circuit, electric field energy in the capacitor periodically transforms into magnetic field energy in the inductor coil.
Theory Summary & Scientific Mechanism
A series RLC circuit comprising capacitance C, inductance L, and resistance R is the exact electrodynamic dual of a damped mechanical oscillator. Capacitor charge q(t) corresponds to displacement x, current I(t) = q to velocity x, inductance L to mass m (via self-induced EMF ℰ_L = -LI), and inverse capacitance 1/C to spring stiffness k.
Kirchhoff’s voltage loop rule U_L + U_R + U_C = 0 yields the second-order differential equation Lq + Rq + (1)/(C)q = 0. In an ideal lossless LC tank (R = 0), charge oscillates harmonically at natural angular frequency ω₀ = (1)/(√(LC)) with Thomson’s period T₀ = 2π√(LC).
Four times per period, energy sloshes between the capacitor’s electric field W_E = (q²)/(2C) = (1)/(2)CU² and the inductor’s magnetic field W_B = (1)/(2)LI². Because I = q, current leads charge in phase by π/2. With nonzero resistance (R > 0), amplitude decays exponentially at rate β = (R)/(2L) as electromagnetic energy dissipates into Joule heat P = I² R.
Everyday Analogy
An RLC circuit in a radio tuner or smartphone is the electrical twin of a swinging pendulum! Charge stored on the capacitor acts like height (potential energy E_C), while current in the inductor acts like motion (magnetic energy E_L).
Interactive Question & Derivation
An ideal LC circuit consists of a capacitor C and inductor L. Initially the capacitor is fully charged (U = U₀, current I = 0). What happens at the exact instant when the capacitor becomes completely discharged (q = 0)?
- A: The current in the inductor also drops to zero and oscillations stop
- B: Due to self-induction, the current reaches its peak Iₘax = U₀√(C/L) and all circuit energy is stored in the inductor's magnetic field W_B = (1)/(2)LIₘax²
- C: The oscillation period depends only on the initial voltage U₀
Study Hint: According to Thomson’s formula T = 2π√(LC) and Joule heating I² R: what happens to oscillation amplitude and total electromagnetic energy when resistance R > 0?
Proof & Explanation: When the charge on the capacitor plates reaches zero (q = 0, W_E = 0), conservation of energy requires that all initial electric energy (1)/(2)CU₀² has converted into magnetic energy (1)/(2)LIₘax² inside the inductor. Current is at its maximum and continues flowing by self-induction, recharging the capacitor with opposite polarity with Thomson's period T = 2π√(LC).
Key Equations & 3D Simulation Parameters
T = 2π√(LC), W_E,max = (C U₀²)/(2) = (L Iₘax²)/(2), Iₘax = U₀√((C)/(L))
- Coil inductance L: 0.8 … 4 H (default: 2 H)
- Capacitor capacitance C: 0.8 … 4 mF (default: 2 mF)
- Circuit resistance R (damping): 0 … 0.5 Ω (default: 0.05 Ω)
- Initial capacitor voltage U₀: 2 … 8 V (default: 5 V)