A singleloop circuit consists of a 740 resistor a 160 H indu

A single-loop circuit consists of a 7.40 resistor, a 16.0 H inductor, and a 3.10 F capacitor. Initially the capacitor has a charge of 6.20 C and the current is zero. Calculate the charge on the capacitor N complete cycles later for the following values of N.

Solution

We have an undriven RLC circuit with the initial condition of a charged capacitor. Note that this is analogous to an unforced spring-mass-damper system with an initially stretched spring, and the effect of damping can be treated similarly. We know natural frequency 0 = 1/sqrt(LC), and the oscillation period T = 2/0. After an integer number of cycles N the system is in the same phase of the oscillation as it was at t=0. So the charge Q at such a time is just the initial charge Q0 multiplied by the term representing reduced amplitude (decay) due to damping.
Amplitude of a system subject to exponential decay is
A = A0 * EXP(-0t), where is the damping coefficient.
Since we are interested in the decay over N cycles, we can rewrite this as
A = A0 * EXP(-N) where is the per-cycle decay (\"logarithmic decrement\") = 2, and N = t/T.
Damping in an RLC circuit is defined to be at the critical value when 0 = R/2L or R = 20L; thus we can define (actual/critical damping) = R/20L.

So to solve the problem, we find , , and the values of Q0*EXP(-N) for N = 5, 10 and 100. I get:

0 = 141.99 rad/s
= 0.00163
= 0.010233
Q(5) = 5.89 C
Q(10) = 5.597 C
Q(100) = 2.228 C