Power dissipated over 1 cycle of a AC source-free RLC series circuit?

[testobesto]

Homework Statement

382* A 0:01 µF capacitor, a 0.1 H inductor whose resistance is 1000 Ohms, and a switch are connected in a series circuit. The capacitor is initially charged to a potential difference of 400 V.
The switch is then closed.
(b)/ (c) How much energy is converted to heat in the first complete cycle? How much energy is converted to heat in the complete train of oscillations?

Relevant Equations

a = R/2L, w_0 = 1/sqrt(LC), w_d = sqrt(a^2 - w_0^2), i(t) = e^(-at)[A_1 cos(w_d t) + A_2sin(w_d t)]

I tried to start by assuming that we need to integrate something over 1 period (2pi). Therefore, we need i(t)^2 R integrated over something. From there, I recognized that this is an underdamped model since R/2L < 1/sqrt(LC). I believe that i(t) can be represented by i(t) = e^(-at)[A_1 cos(w_d t) + A_2sin(w_d t)]. However, I am stuck on finding A1 and A2, and even if I found it, I don't know if this is the correct approach. I also tried letting 1 period = (1/2pi*(w_0))^-1 and working with that, but I can't see if it works.

I feel like there is supposed to be a much simpler approach, but I am not sure what it could be.


(Answer is given to be 6.9e-4 J, but I don't get process)

 

[BvU]

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I feel like there is supposed to be a much simpler approach

I can think of a simpler approach for part (b)/ (c) ...
(How much energy is present in the circuit at $t=0$ )

But for part (a) some work is unavoidable. Fortunately you already have an expression for the solution. Notice that in the plot in the link all responses start with $i(0)=0$. With $A_1 = 0$, what do you get for the fraction $\int_0^T/\int_0^\infty$ (where $T =$1 period) ?

$\$

 

[nasu]

Did you learn about RMS values of voltage and current? (this is for the OP)

 

[testobesto]

Did you learn about RMS values of voltage and current? (this is for the OP)

Yes.

 

[testobesto]

​

I can think of a simpler approach for part (b)/ (c) ...
(How much energy is present in the circuit at $t=0$ )

But for part (a) some work is unavoidable. Fortunately you already have an expression for the solution. Notice that in the plot in the link all responses start with $i(0)=0$. With $A_1 = 0$, what do you get for the fraction $\int_0^T/\int_0^\infty$ (where $T =$1 period) ?

$\$

At t=0, the capacitor holds all charge, 8e-4 C. I take it that since A_1 = 0, we just evaluate for A_2? Or use phase shifts for a new identity? I also immediately recognized that for (c), the answer is the same at 8e-4 C, but only because I know that all charge can only leave through the resistor as t->inf

 

[DaveE]

As you observed this is a highly resonant circuit (Q=10K !). As such, a good approximation for the peak inductor current can be found using conservation of energy at the 1st quarter cycle. Likewise, the resonant frequency can be approximated by assuming R=0. Although I doubt that you need to know the period since you are integrating over 1 period, ω will drop out of the result.

PS: The approximate solution to simple resonance like this has a nice simple form that is worth memorizing. You're likely to see it many more times.

 

[testobesto]

Would there be some function that gives the charge in the inductor and capacitor as a function of omega or time, then we evaluate that function at t=0 and t = 1 period, and take it so that the charge on the capacitor at 0 minus the charge on both the inductor and capacitor at 1 period would equal the charge that was dissipated on the resistor, due to the charges on the RHS and LHS on each side needing to agree? I have that as R = 0, the resonant frequency = 1/√LC = 31622.8 Hz