31.10: LC Circuits
An LC circuit consists of an inductor and a capacitor, either in series or parallel. Consider a charged capacitor connected with an inductor in series. Before the switch is closed, all the energy of the circuit is stored in the electric field of the capacitor. When the switch is closed, the capacitor begins to discharge, producing a current in the circuit. The current, in turn, creates a magnetic field in the inductor. Because of the induced emf in the inductor, the current cannot change instantaneously; it starts at zero and eventually builds up to a maximum value. During this buildup, the capacitor is discharging. At each instant, the capacitor potential equals the induced emf, so as the capacitor discharges, the rate of change of current decreases. When the capacitor potential becomes zero, the induced emf is also zero, and the current has leveled off at its maximum value.
The net effect of this process is a transfer of energy from the capacitor with its diminishing electric field, to the inductor with its increasing magnetic field. Since there is no resistance in the circuit, no energy is lost through Joule heating; thus, the maximum energy stored in the capacitor is equal to the maximum energy stored later in the inductor. At any arbitrary time, the total energy of an LC circuit is the sum of electrical and magnetic energy. Therefore, energy remains conserved in an LC circuit. This is analogous to the mechanical oscillations of a mass attached at the end of a spring. In this case, energy is transferred back and forth between the mass, which has kinetic energy, and the spring, which has potential energy. In the LC circuit, the capacitor charge q plays the role of the displacement x, and the current is analogous to the particle's velocity. The inductance is analogous to the mass m, and the reciprocal of the capacitance is analogous to the force constant k.
