Lc Oscillations

An ideal LC loop continually trades energy between the capacitor's electric field and the inductor's magnetic field, producing undamped simple-harmonic oscillation of charge at the natural angular frequency (\omega = 1/\sqrt{LC}). Computing the product, (LC = (2\times10^{-3})(8\times10^{-6}) = 1.6\times10^{-8}) s^2, so (\sqrt{LC} = 4\times10^{-4}) s and (\omega = 1/(4\times10^{-4}) = 2.5\times10^{3}) rad/s. The option 5.0 (\times) 10^3 rad/s omits the square root of the product. The option 1.25 (\times) 10^3 rad/s halves the correct value through an arithmetic slip. The option 1.0 (\times) 10^4 rad/s misplaces the exponent. This parallels the NCERT analogy between LC oscillations and a frictionless mass-spring system, with charge playing the role of displacement and current the role of velocity, while the inductance behaves like inertial mass and the reciprocal of capacitance like the spring constant. Energy is conserved throughout: when the capacitor is fully charged the current is zero and all energy is electrical, and a quarter period later the capacitor is empty while the current and magnetic energy are maximal. A plausibility check confirms (1/\sqrt{H\cdot F}) reduces to rad/s, and increasing either L or C would lower the frequency, just as a heavier mass or a softer spring slows a mechanical oscillator.