A source-free parallel circuit consists of R = 8 Ω, L = 2 H, and C = 1/32 F

A source-free parallel circuit consists of R = 8 Ω, L = 2 H, and C = 1/32 F. The continuous initial state is v(0+) = 0 and i_L(0+) = 4 A, with i_L directed from the positive node to ground through L. Determine α, ω0, the characteristic roots, and the damping classification. Determine v(t) and i_L(t) for t ≥ 0. Compute the initial stored energy and the stored energy remaining as t → ∞. The resistor voltage is v(t). Determine the energy dissipated in the resistor, E_R = ∫_0^∞ [v(t)^2 / R] dt, by evaluating the integral using the waveform of v, and confirm that E_R equals the loss of stored energy. Also evaluate E_R by the shortcut that, in a source-free RLC circuit with R > 0, all initially stored energy is eventually dissipated in R.

Form the initial stored energy from the given state. For \(t\to\infty\) in a source-free network with \(R>0\), stored energy vanishes.