Question

RLC circuit in series A resistor R is connected in series to an inductor L and a capacitor C, without any external emf sources.

(a) Using the fact that the energy stored in both the capacitor and the inductor is being dissipated in the resistor, show that the charge on the capacitor q(t) satisfies the differential equation

d^2 q/ dt^2 + Rdq/Ldt + q/LC = 0.

This is the equation of a damped oscillator and it has a solution of the form (feel free to check it on your own)

q(t) = Ae^[−(R/2L)t ]*cos[root(1/LC − R^2/4L^2) t + φ]

when R^2 < 4L/C, the underdamped limit.

(b) Why do we need this restriction in the value of R?

(c) Make a plot of q(t) for both R = 0 and R 6= 0. What is the role of the resistance in this circuit? Does the name ”damped oscillator” make sense? Explain.

(d) What are the values of the initial charge q0 and the initial current i0 as a function of A and φ.

Answer 1

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