Assignment:
Consider a basic electric circuit with a resistor, capacitor, and inductor and input voltage V(t). It follows Kirchoff's Laws that the charge on the capacitor Q = Q(t) solves the differential equation:
LQ" + RQ' + 1/C(Q) = V(t),
where L (inductance), R (resistance), and C (capacitance) are positive constants (depending on material). The current in the circuit is the rate of change of the charge: I(t) = Q'(t).
(a) Compare this to the mass-spring equation. What plays the role of the mass, spring constant, damping coefficient, displacement, velocity, external forcing?
(b) Suppose a given circuit has no input voltage (V(t) Ξ 0) and a positive initial charge Q(0) = Qo > 0. Find a condition on R > 0 so that the equation has oscillatory solutions.
(c) If you wanted to build a circuit that would oscillate forever without any input voltage (V(t) Ξ 0) what could you do?
Provide complete and step by step solution for the question and show calculations and use formulas.