Maximum Charge on a Capacitor in an LC Ciruit

[forestmine]

Homework Statement

A charged capacitor is connected to an ideal inductor to form an LC circuit with a frequency of oscillation f = 1.6 Hz. At time t = 0 the capacitor is fully charged. At a given instant later the charge on the capacitor is measured to be 3.0 μC and the current in the circuit is equal to 75 μA. What is the maximum charge of the capacitor?

Homework Equations

q = Qcos(ωt)
i = -ωQsin(ωt)

The Attempt at a Solution

Alright, this problem has been driving me nuts. I'm honestly not sure how to go about this one, and I think there are some errors in the steps I took, but here's what I've done so far.

Since we're given the frequency, I solved for ω. ω=2pi*f = 10.05 rad.

Then, I used dq/dt to find the time. Now, I feel weird about this step, since dq/dt is the rate of change of the charge, but essentially, I thought of it as i = q/t (I believe I made a huge error in making this assumption, but I wasn't sure how else to go about this...)

Substituting, I found t = q/i = .04s.

And then I used both of the above equations to attempt to find a value for Q. Both give me two different numbers, both of which are incorrect.

Conceptually, I understand what's going on here. Initially, the capacitor is fully charged. When there's a current running through the circuit and a charge on the capacitor, the capacitor is in the process of discharging (or charging, considering we don't know the specifics).

I just don't quite see how to go about arriving at the correct answer. I'm pretty sure my method of solving for t is absolutely wrong, but I'm stumped!

Any help would be greatly appreciated!

 

[gneill]

Yeah, bit of a problem with your determination of the time. You'd need to set up a differential equation involving the current and charge to see how they are interrelated with respect to time.

Have you considered a conservation of energy approach?

 

[forestmine]

I have considered a conservation of energy approach, but even that hasn't led me far...

I know that when the capacitor is fully charged, all the energy is electrical, and in the form Q^2/2C.

At the instant where there is a charge and a current flowing, the total energy is givey by 1/2LI^2 + q^2/2C = Q^2/2C, where there is both energy stored in the magnetic field and the electric field, but as I'm not given values for L or C, I'm not sure how I could use this.

 

[gneill]

I have considered a conservation of energy approach, but even that hasn't led me far...

I know that when the capacitor is fully charged, all the energy is electrical, and in the form Q^2/2C.

At the instant where there is a charge and a current flowing, the total energy is givey by 1/2LI^2 + q^2/2C = Q^2/2C, where there is both energy stored in the magnetic field and the electric field, but as I'm not given values for L or C, I'm not sure how I could use this.

Ah. But you ARE given the natural frequency of the LC oscillations...

 

[forestmine]

Ah, you're absolutely right. So, you're referring to w = 1/(LC)^1/2, correct? And since I'm given f, I can easily solve for w.

 

[forestmine]

Excellent, I got it! Thank you so much. I need to remember to just make whatever substitutions as needed...thanks!

 

[gneill]

Glad to be of help