Simple Nature - Light and Matter

gap. For a given amount of charge on the plates, how does this affectthe amount of energy stored in the electric field? How does this affect thecapacitance?Now redo the analysis in terms of the mechanical work needed in orderto charge up the plates.10.5.2 OscillationsFigure j shows the simplest possible oscillating circuit. For anyuseful application it would actually need to include more components.For example, if it was a radio tuner, it would need to beconnected to an antenna and an amplifier. Nevertheless, all theessential physics is there.We can analyze it without any sweat or tears whatsoever, simplyby constructing an analogy with a mechanical system. In amechanical oscillator, k, we have two forms of stored energy,U spring = 1 2 kx2 (1)j / A series LRC circuit.K = 1 2 mv2 . (2)In the case of a mechanical oscillator, we have usually assumeda friction force of the form that turns out to give the nicest mathematicalresults, F = −bv. In the circuit, the dissipation of energyinto heat occurs via the resistor, with no mechanical force involved,so in order to make the analogy, we need to restate the role of thefriction force in terms of energy. The power dissipated by frictionequals the mechanical work it does in a time interval dt, divided bydt, P = W/ dt = F dx/ dt = F v = −bv 2 , sok / A mechanical analogy forthe LRC circuit.rate of heat dissipation = −bv 2 . (3)self-check FEquation (1) has x squared, and equations (2) and (3) have v squared.Because they’re squared, the results don’t depend on whether thesevariables are positive or negative. Does this make physical sense? ⊲Answer, p. 926In the circuit, the stored forms of energy areU C = 12C q2 (1 ′ )U L = 1 2 LI2 , (2 ′ )and the rate of heat dissipation in the resistor israte of heat dissipation = −RI 2 . (3 ′ )Comparing the two sets of equations, we first form analogies betweenquantities that represent the state of the system at some momentSection 10.5 LRC Circuits 593