Let's apply the principle of conservation of energy to a mass oscillating horizontally back and forth on the end of a spring. The total energy E equals the energy stored in the spring, one-half k A-squared, and the kinetic energy, one-half m-v-squared. The energy of motion, one-half m-v-squared reaches a maximum in the middle of the oscillation, just at the point the energy in the spring, one-half k A-squared, reaches zero. As the oscillation continues, the energy stored in the spring reaches a maximum at the point the energy of motion reaches zero. If an inductor and a capacitor are hooked in a circuit, it behaves in a manner similar to a mass-spring system. The energy of the inductor's magnetic field, one-half L-I-squared, reaches a maximum as the electric field of the capacitor, Q-squared divided by 2C, reaches zero. Then, the energy stored in the capacitor reaches a maximum as the energy of the conductor decreases to zero. The mass-spring system exhibits simple harmonic motion. The force acting on the spring according to Hooke's Law is equal to minus the spring constant, k, times the distance of the mass from the equilibrium point. From Newton's second law stating force equals mass times acceleration, we know that mass times acceleration plus the spring constant times the displacement x equals zero. Rewriting acceleration as the second derivative of distance with respect to time yields the differential equation for the mass-spring system. Another characteristic of simple harmonic motion is that the distance from the equilibrium point equals the amplitude times the cosine of the angle, theta, ...where theta equals the angular speed, omega, times time. The energy in the capacitor equals the charge squared divided by two times the capacitance. This capacitor energy is analogous to the energy stored in the stretched (or compressed) spring, one-half k-x-squared. As the capacitor discharges, the energy goes into the magnetic energy of the inductor. This energy is one-half times the inductance times the square of the current. This inductance energy is analogous to the kinetic energy of the oscillating block. In both the electrical and the mechanical systems, energy is lost as heat. The total energy of the circuit system, U, equals the energy stored in the capacitor plus the kinetic energy in the inductor. The total energy of the system, U, is constant. If we take the derivative of this constant with respect to time, we get zero equals the derivative of the inductor and capacitor energies. d-Q d-t is the current, I. Thus, the second derivative of the charge is the same as the first derivative of the current. Substituting this result in our equation and canceling the current gives us a second order differential equation. Next we solve for Q. Note the similarity of form between the mass-spring system differential equation and this one. The mass-spring system is a harmonic oscillator, so the LC circuit must also be a harmonic oscillator. For a simple harmonic oscillator the second derivative of x with respect to time equals negative omega squared times x, ...where omega is the angular frequency and equals the square root of the spring constant divided by the mass. The general solution of this differential equation can be written as x equals the amplitude times the cosine of the angular frequency times the time plus the phase constant theta. Using our comparison chart we can write that the charge of the capacitor Q equals the maximum charge times the cosine of the angular frequency times the time... ...plus the phase constant theta, where in this case the angular frequency equals the square root of one divided by L times C. To obtain an expression for the current as a function of time we take the derivative of the charge of the capacitor with respect to time to find the current equals negative angular... ...frequency times maximum charge capacitor times the sine of the angular frequency times the time plus the phase constant theta. Thus we find that the LC circuit is a simple harmonic oscillator that follows the same conservation of energy rules that apply to the analogous mechanical systems. For an LC circuit with L= 3 Milli-Henrys, C = 10 Pico-Farads and an e-m-f of 12Volts, determine the charge and current as functions of time.

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